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passage: A "Hello, World!" program is usually a simple computer program that emits (or displays) to the screen (often the console) a message similar to "Hello, World!". A small piece of code in most general-purpose programming languages, this program is used to illustrate a language's basic syntax. Such a program is often the first written by a student of a new programming language, but it can also be used as a sanity check to ensure that the computer software intended to compile or run source code is correctly installed, and that its operator understands how to use it.
## History
While several small test programs have existed since the development of programmable computers, the tradition of using the phrase "Hello, World!" as a test message was influenced by an example program in the 1978 book The C Programming Language, with likely earlier use in BCPL. The example program from the book prints , and was inherited from a 1974 Bell Laboratories internal memorandum by Brian Kernighan, Programming in C: A Tutorial:
```c
main( ) {
printf("hello, world");
}
```
In the above example, the function defines where the program should start executing. The function body consists of a single statement, a call to the function, which stands for "print formatted"; it outputs to the console whatever is passed to it as the parameter, in this case the string .
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passage: The example program from the book prints , and was inherited from a 1974 Bell Laboratories internal memorandum by Brian Kernighan, Programming in C: A Tutorial:
```c
main( ) {
printf("hello, world");
}
```
In the above example, the function defines where the program should start executing. The function body consists of a single statement, a call to the function, which stands for "print formatted"; it outputs to the console whatever is passed to it as the parameter, in this case the string .
The C-language version was preceded by Kernighan's own 1972 A Tutorial Introduction to the Language B, where the first known version of the program is found in an example used to illustrate external variables:
```text
main( ) {
extrn a, b, c;
putchar(a); putchar(b); putchar(c); putchar('!*n');
}
a 'hell';
b 'o, w';
c 'orld';
```
The program above prints on the terminal, including a newline character. The phrase is divided into multiple variables because in B a character constant is limited to four ASCII characters. The previous example in the tutorial printed on the terminal, and the phrase was introduced as a slightly longer greeting that required several character constants for its expression.
The Jargon File reports that "hello, world" instead originated in 1967 with the language BCPL. Outside computing, use of the exact phrase began over a decade prior; it was the catchphrase of New York radio disc jockey William B. Williams beginning in the 1950s.
## Variations
"Hello, World!"
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passage: Outside computing, use of the exact phrase began over a decade prior; it was the catchphrase of New York radio disc jockey William B. Williams beginning in the 1950s.
## Variations
"Hello, World!" programs vary in complexity between different languages. In some languages, particularly scripting languages, the "Hello, World!" program can be written as one statement, while in others (more so many low-level languages) many more statements can be required. For example, in Python, to print the string followed by a newline, one only needs to write
```python
print("Hello, World!")
```
. In contrast, the equivalent code in C++ requires the import of the C++ standard library, the declaration of an entry point (main function), and a call to print a line of text to the standard output stream.
The phrase "Hello, World!" has seen various deviations in casing and punctuation, such as "hello world" which lacks the capitalization of the leading H and W, and the presence of the comma or exclamation mark. Some devices limit the format to specific variations, such as all-capitalized versions on systems that support only capital letters, while some esoteric programming languages may have to print a slightly modified string. Other human languages have been used as the output; for example, a tutorial for the Go language emitted both English and Chinese or Japanese characters, demonstrating the language's built-in Unicode support. Another notable example is the Rust language, whose management system automatically inserts a "Hello, World" program when creating new projects.
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passage: Other human languages have been used as the output; for example, a tutorial for the Go language emitted both English and Chinese or Japanese characters, demonstrating the language's built-in Unicode support. Another notable example is the Rust language, whose management system automatically inserts a "Hello, World" program when creating new projects.
Some languages change the function of the "Hello, World!" program while maintaining the spirit of demonstrating a simple example. Functional programming languages, such as Lisp, ML, and Haskell, tend to substitute a factorial program for "Hello, World!", as functional programming emphasizes recursive techniques, whereas the original examples emphasize I/O, which violates the spirit of pure functional programming by producing side effects. Languages otherwise able to print "Hello, World!" (assembly language, C, VHDL) may also be used in embedded systems, where text output is either difficult (requiring added components or communication with another computer) or nonexistent. For devices such as microcontrollers, field-programmable gate arrays, and complex programmable logic devices (CPLDs), "Hello, World!" may thus be substituted with a blinking light-emitting diode (LED), which demonstrates timing and interaction between components.
The Debian and Ubuntu Linux distributions provide the "Hello, World!" program through their software package manager systems, which can be invoked with the command . It serves as a sanity check and a simple example of installing a software package.
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passage: The Debian and Ubuntu Linux distributions provide the "Hello, World!" program through their software package manager systems, which can be invoked with the command . It serves as a sanity check and a simple example of installing a software package. For developers, it provides an example of creating a .deb package, either traditionally or using debhelper, and the version of used, GNU Hello, serves as an example of writing a GNU program.
Variations of the "Hello, World!" program that produce a graphical output (as opposed to text output) have also been shown. Sun demonstrated a "Hello, World!" program in Java based on scalable vector graphics, and the XL programming language features a spinning Earth "Hello, World!" using 3D computer graphics. Mark Guzdial and Elliot Soloway have suggested that the "hello, world" test message may be outdated now that graphics and sound can be manipulated as easily as text.
In computer graphics, rendering a trianglecalled "Hello Triangle"is sometimes used as an introductory example for graphics libraries.
## Time to Hello World
"Time to hello world" (TTHW) is the time it takes to author a "Hello, World!" program in a given programming language. This is one measure of a programming language's ease of use. Since the program is meant as an introduction for people unfamiliar with the language, a more complex "Hello, World!" program may indicate that the programming language is less approachable.
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passage: This is one measure of a programming language's ease of use. Since the program is meant as an introduction for people unfamiliar with the language, a more complex "Hello, World!" program may indicate that the programming language is less approachable. For instance, the first publicly known "Hello, World!" program in Malbolge (which actually output "HEllO WORld") took two years to be announced, and it was produced not by a human but by a code generator written in Common Lisp .
The concept has been extended beyond programming languages to APIs, as a measure of how simple it is for a new developer to get a basic example working; a shorter time indicates an easier API for developers to adopt.
## Wikipedia articles containing "Hello, World!" programs
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passage: In mathematics, and more specifically in abstract algebra, a
### *-algebra
(or involutive algebra; read as "star-algebra") is a mathematical structure consisting of two involutive rings and , where is commutative and has the structure of an associative algebra over . Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.
However, it may happen that an algebra admits no involution.
## Definitions
-ring
In mathematics, a
### *-ring
is a ring with a map that is an antiautomorphism and an involution.
More precisely, is required to satisfy the following properties:
-
-
-
-
for all in .
This is also called an involutive ring, involutory ring, and ring with involution. The third axiom is implied by the second and fourth axioms, making it redundant.
Elements such that are called self-adjoint.
Archetypical examples of a *-ring are fields of complex numbers and algebraic numbers with complex conjugation as the involution. One can define a sesquilinear form over any *-ring.
Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: and so on.
*-rings are unrelated to star semirings in the theory of computation.
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passage: Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: and so on.
*-rings are unrelated to star semirings in the theory of computation.
-algebra
A *-algebra is a *-ring, with involution * that is an associative algebra over a commutative *-ring with involution , such that .
The base *-ring is often the complex numbers (with acting as complex conjugation).
It follows from the axioms that * on is conjugate-linear in , meaning
for .
A *-homomorphism is an algebra homomorphism that is compatible with the involutions of and , i.e.,
- for all in .
### Philosophy of the *-operation
The *-operation on a *-ring is analogous to complex conjugation on the complex numbers. The *-operation on a *-algebra is analogous to taking adjoints in complex matrix algebras.
### Notation
The * involution is a unary operation written with a postfixed star glyph centered above or near the mean line:
, or
(TeX: `x^*`),
but not as ""; see the asterisk article for details.
## Examples
- Any commutative ring becomes a *-ring with the trivial (identical) involution.
- The most familiar example of a *-ring and a *-algebra over reals is the field of complex numbers where * is just complex conjugation.
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passage: ## Examples
- Any commutative ring becomes a *-ring with the trivial (identical) involution.
- The most familiar example of a *-ring and a *-algebra over reals is the field of complex numbers where * is just complex conjugation.
- More generally, a field extension made by adjunction of a square root (such as the imaginary unit ) is a *-algebra over the original field, considered as a trivially-*-ring. The * flips the sign of that square root.
- A quadratic integer ring (for some ) is a commutative *-ring with the * defined in the similar way; quadratic fields are *-algebras over appropriate quadratic integer rings.
- Quaternions, split-complex numbers, dual numbers, and possibly other hypercomplex number systems form *-rings (with their built-in conjugation operation) and *-algebras over reals (where * is trivial). None of the three is a complex algebra.
- Hurwitz quaternions form a non-commutative *-ring with the quaternion conjugation.
- The matrix algebra of matrices over R with * given by the transposition.
- The matrix algebra of matrices over C with * given by the conjugate transpose.
- Its generalization, the Hermitian adjoint in the algebra of bounded linear operators on a Hilbert space also defines a *-algebra.
- The polynomial ring over a commutative trivially-*-ring is a *-algebra over with .
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passage: - Its generalization, the Hermitian adjoint in the algebra of bounded linear operators on a Hilbert space also defines a *-algebra.
- The polynomial ring over a commutative trivially-*-ring is a *-algebra over with .
- If is simultaneously a *-ring, an algebra over a ring (commutative), and , then is a *-algebra over (where * is trivial).
- As a partial case, any *-ring is a *-algebra over integers.
- Any commutative *-ring is a *-algebra over itself and, more generally, over any its -subring.
- For a commutative *-ring , its quotient by any its -ideal is a *-algebra over .
- For example, any commutative trivially-*-ring is a *-algebra over its dual numbers ring, a *-ring with non-trivial *, because the quotient by makes the original ring.
- The same about a commutative ring and its polynomial ring : the quotient by restores .
- In Hecke algebra, an involution is important to the Kazhdan–Lusztig polynomial.
- The endomorphism ring of an elliptic curve becomes a *-algebra over the integers, where the involution is given by taking the dual isogeny. A similar construction works for abelian varieties with a polarization, in which case it is called the Rosati involution (see Milne's lecture notes on abelian varieties).
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passage: - The endomorphism ring of an elliptic curve becomes a *-algebra over the integers, where the involution is given by taking the dual isogeny. A similar construction works for abelian varieties with a polarization, in which case it is called the Rosati involution (see Milne's lecture notes on abelian varieties).
Involutive Hopf algebras are important examples of *-algebras (with the additional structure of a compatible comultiplication); the most familiar example being:
- The group Hopf algebra: a group ring, with involution given by .
## Non-Example
Not every algebra admits an involution:
Regard the 2×2 matrices over the complex numbers. Consider the following subalgebra:
$$
\mathcal{A} := \left\{\begin{pmatrix}a&b\\0&0\end{pmatrix} : a,b\in\Complex\right\}
$$
Any nontrivial antiautomorphism necessarily has the form:
$$
\varphi_z\left[\begin{pmatrix}1&0\\0&0\end{pmatrix}\right] = \begin{pmatrix}1&z\\0&0\end{pmatrix} \quad \varphi_z\left[\begin{pmatrix}0&1\\0&0\end{pmatrix}\right] = \begin{pmatrix}0&0\\0&0\end{pmatrix}
$$
for any complex number
$$
z\in\Complex
$$
.
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passage: Regard the 2×2 matrices over the complex numbers. Consider the following subalgebra:
$$
\mathcal{A} := \left\{\begin{pmatrix}a&b\\0&0\end{pmatrix} : a,b\in\Complex\right\}
$$
Any nontrivial antiautomorphism necessarily has the form:
$$
\varphi_z\left[\begin{pmatrix}1&0\\0&0\end{pmatrix}\right] = \begin{pmatrix}1&z\\0&0\end{pmatrix} \quad \varphi_z\left[\begin{pmatrix}0&1\\0&0\end{pmatrix}\right] = \begin{pmatrix}0&0\\0&0\end{pmatrix}
$$
for any complex number
$$
z\in\Complex
$$
.
It follows that any nontrivial antiautomorphism fails to be involutive:
$$
\varphi_z^2\left[\begin{pmatrix}0&1\\0&0\end{pmatrix}\right] = \begin{pmatrix}0&0\\0&0\end{pmatrix}\neq\begin{pmatrix}0&1\\0&0\end{pmatrix}
$$
Concluding that the subalgebra admits no involution.
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passage: Consider the following subalgebra:
$$
\mathcal{A} := \left\{\begin{pmatrix}a&b\\0&0\end{pmatrix} : a,b\in\Complex\right\}
$$
Any nontrivial antiautomorphism necessarily has the form:
$$
\varphi_z\left[\begin{pmatrix}1&0\\0&0\end{pmatrix}\right] = \begin{pmatrix}1&z\\0&0\end{pmatrix} \quad \varphi_z\left[\begin{pmatrix}0&1\\0&0\end{pmatrix}\right] = \begin{pmatrix}0&0\\0&0\end{pmatrix}
$$
for any complex number
$$
z\in\Complex
$$
.
It follows that any nontrivial antiautomorphism fails to be involutive:
$$
\varphi_z^2\left[\begin{pmatrix}0&1\\0&0\end{pmatrix}\right] = \begin{pmatrix}0&0\\0&0\end{pmatrix}\neq\begin{pmatrix}0&1\\0&0\end{pmatrix}
$$
Concluding that the subalgebra admits no involution.
## Additional structures
Many properties of the transpose hold for general *-algebras:
- The Hermitian elements form a Jordan algebra;
- The skew Hermitian elements form a Lie algebra;
- If 2 is invertible in the *-ring, then the operators and are orthogonal idempotents, called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of modules (vector spaces if the *-ring is a field) of symmetric and anti-symmetric (Hermitian and skew Hermitian) elements.
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passage: These spaces do not, generally, form associative algebras, because the idempotents are operators, not elements of the algebra.
### Skew structures
Given a *-ring, there is also the map .
It does not define a *-ring structure (unless the characteristic is 2, in which case −* is identical to the original *), as , neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where .
Elements fixed by this map (i.e., such that ) are called skew Hermitian.
For the complex numbers with complex conjugation, the real numbers are the Hermitian elements, and the imaginary numbers are the skew Hermitian.
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passage: The (EGA; from French: "Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné) is a rigorous treatise on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the . In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes, which he defined. The work is now considered the foundation and basic reference of modern algebraic geometry.
## Editions
Initially thirteen chapters were planned, but only the first four (making a total of approximately 1500 pages) were published. Much of the material which would have been found in the following chapters can be found, in a less polished form, in the Séminaire de géométrie algébrique (known as SGA). Indeed, as explained by Grothendieck in the preface of the published version of SGA, by 1970 it had become clear that incorporating all of the planned material in EGA would require significant changes in the earlier chapters already published, and that therefore the prospects of completing EGA in the near term were limited. An obvious example is provided by derived categories, which became an indispensable tool in the later SGA volumes, but was not yet used in EGA III as the theory was not yet developed at the time. Considerable effort was therefore spent to bring the published SGA volumes to a high degree of completeness and rigour.
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passage: An obvious example is provided by derived categories, which became an indispensable tool in the later SGA volumes, but was not yet used in EGA III as the theory was not yet developed at the time. Considerable effort was therefore spent to bring the published SGA volumes to a high degree of completeness and rigour. Before work on the treatise was abandoned, there were plans in 1966–67 to expand the group of authors to include Grothendieck's students Pierre Deligne and Michel Raynaud, as evidenced by published correspondence between Grothendieck and David Mumford. Grothendieck's letter of 4 November 1966 to Mumford also indicates that the second-edition revised structure was in place by that time, with Chapter VIII already intended to cover the Picard scheme. In that letter he estimated that at the pace of writing up to that point, the following four chapters (V to VIII) would have taken eight years to complete, indicating an intended length comparable to the first four chapters, which had been in preparation for about eight years at the time.
Grothendieck nevertheless wrote a revised version of EGA I which was published by Springer-Verlag. It updates the terminology, replacing "prescheme" by "scheme" and "scheme" by "separated scheme", and heavily emphasizes the use of representable functors. The new preface of the second edition also includes a slightly revised plan of the complete treatise, now divided into twelve chapters.
Grothendieck's EGA V which deals with Bertini type theorems is to some extent available from the Grothendieck Circle website.
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passage: The new preface of the second edition also includes a slightly revised plan of the complete treatise, now divided into twelve chapters.
Grothendieck's EGA V which deals with Bertini type theorems is to some extent available from the Grothendieck Circle website. Monografie Matematyczne in Poland has accepted this volume for publication, but the editing process is quite slow (as of 2010).
James Milne has preserved some of the original Grothendieck notes and a translation of them into English. They may be available from his websites connected with the University of Michigan in Ann Arbor.
## Chapters
The following table lays out the original and revised plan of the treatise and indicates where (in SGA or elsewhere) the topics intended for the later, unpublished chapters were treated by Grothendieck and his collaborators.
# First edition Second edition Comments I Le langage des schémas Le langage des schémas Second edition brings in certain schemes representing functors such as Grassmannians, presumably from intended Chapter V of the first edition. In addition, the contents of Section 1 of Chapter IV of first edition was moved to Chapter I in the second edition. II Étude globale élémentaire de quelques classes de morphismes Étude globale élémentaire de quelques classes de morphismes First edition complete, second edition did not appear. III Étude cohomologique des faisceaux cohérents Cohomologie des Faisceaux algébriques cohérents. Applications.
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passage: III Étude cohomologique des faisceaux cohérents Cohomologie des Faisceaux algébriques cohérents. Applications. First edition complete except for last four sections, intended for publication after Chapter IV: elementary projective duality, local cohomology and its relation to projective cohomology, and Picard groups (all but projective duality treated in SGA II). IV Étude locale des schémas et des morphismes de schémas Étude locale des schémas et des morphismes de schémas First edition essentially complete; some changes made in last sections; the section on hyperplane sections made into the new Chapter V of second edition (draft exists) V Procédés élémentaires de construction de schémas Complements sur les morphismes projectifs Did not appear. Some elementary constructions of schemes apparently intended for first edition appear in Chapter I of second edition. The existing draft of Chapter V corresponds to the second edition plan. It includes also expanded treatment of some material from SGA VII. VI Technique de descente. Méthode générale de construction des schémas Techniques de construction de schémas Did not appear. Descent theory and related construction techniques summarised by Grothendieck in FGA. By 1968 the plan had evolved to treat algebraic spaces and algebraic stacks. VII Schémas de groupes, espaces fibrés principaux Schémas en groupes, espaces fibrés principaux Did not appear. Treated in detail in SGA III. VIII Étude différentielle des espaces fibrés Le schéma de Picard Did not appear.
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passage: Treated in detail in SGA III. VIII Étude différentielle des espaces fibrés Le schéma de Picard Did not appear. Material apparently intended for first edition can be found in SGA III, construction and results on Picard scheme are summarised in FGA. IX Le groupe fondamental Le groupe fondamental Did not appear. Treated in detail in SGA I. X Résidus et dualité Résidus et dualité Did not appear. Treated in detail in Hartshorne's edition of Grothendieck's notes "Residues and duality" XI Théorie d'intersection, classes de Chern, théorème de Riemann-Roch Théorie d'intersection, classes de Chern, théorème de Riemann-Roch Did not appear. Treated in detail in SGA VI. XII Schémas abéliens et schémas de Picard Cohomologie étale des schémas Did not appear. Étale cohomology treated in detail in SGA IV, SGA V. XIII Cohomologie de Weil none Intended to cover étale cohomology in the first edition.
In addition to the actual chapters, an extensive "Chapter 0" on various preliminaries was divided between the volumes in which the treatise appeared. Topics treated range from category theory, sheaf theory and general topology to commutative algebra and homological algebra. The longest part of Chapter 0, attached to Chapter IV, is more than 200 pages.
Grothendieck never gave permission for the 2nd edition of EGA I to be republished, so copies are rare but found in many libraries.
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passage: The longest part of Chapter 0, attached to Chapter IV, is more than 200 pages.
Grothendieck never gave permission for the 2nd edition of EGA I to be republished, so copies are rare but found in many libraries. The work on EGA was finally disrupted by Grothendieck's departure first from IHÉS in 1970 and soon afterwards from the mathematical establishment altogether. Grothendieck's incomplete notes on EGA V can be found at Grothendieck Circle.
In historical terms, the development of the EGA approach set the seal on the application of sheaf theory to algebraic geometry, set in motion by Serre's basic paper FAC. It also contained the first complete exposition of the algebraic approach to differential calculus, via principal parts. The foundational unification it proposed (see for example unifying theories in mathematics) has stood the test of time.
EGA has been scanned by NUMDAM and is available at their website under "Publications mathématiques de l'IHÉS", volumes 4 (EGAI), 8 (EGAII), 11 (EGAIII.1re), 17 (EGAIII.2e), 20 (EGAIV.1re), 24 (EGAIV.2e), 28 (EGAIV.3e) and 32 (EGAIV.4e).
## Bibliographic information
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passage: Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of
### Lie groups
, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He also made significant contributions to general relativity and indirectly to quantum mechanics. He is widely regarded as one of the greatest mathematicians of the twentieth century.
His son Henri Cartan was an influential mathematician working in algebraic topology.
## Life
Élie Cartan was born 9 April 1869 in the village of Dolomieu, Isère to Joseph Cartan (1837–1917) and Anne Cottaz (1841–1927). Joseph Cartan was the village blacksmith; Élie Cartan recalled that his childhood had passed under "blows of the anvil, which started every morning from dawn", and that "his mother, during those rare minutes when she was free from taking care of the children and the
house, was working with a spinning-wheel". Élie had an elder sister Jeanne-Marie (1867–1931) who became a dressmaker; a younger brother Léon (1872–1956) who became a blacksmith working in his father's smithy; and a younger sister Anna Cartan (1878–1923), who, partly under Élie's influence, entered École Normale Supérieure (as Élie had before) and chose a career as a mathematics teacher at a lycée (secondary school).
Élie Cartan entered an elementary school in Dolomieu and was the best student in the school.
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passage: Élie had an elder sister Jeanne-Marie (1867–1931) who became a dressmaker; a younger brother Léon (1872–1956) who became a blacksmith working in his father's smithy; and a younger sister Anna Cartan (1878–1923), who, partly under Élie's influence, entered École Normale Supérieure (as Élie had before) and chose a career as a mathematics teacher at a lycée (secondary school).
Élie Cartan entered an elementary school in Dolomieu and was the best student in the school. One of his teachers, M. Dupuis, recalled "Élie Cartan was a shy student, but an unusual light of great intellect was shining in his eyes, and this was combined with an excellent memory". Antonin Dubost, then the representative of Isère, visited the school and was impressed by Cartan's unusual abilities. He recommended Cartan to participate in a contest for a scholarship in a lycée. Cartan prepared for the contest under the supervision of M. Dupuis and passed at the age of ten. He spent five years (1879–1884) at the College of Vienne and then two years (1884–1886) at the Lycée of Grenoble. In 1886 he moved to the Lycée Janson de Sailly in Paris to study sciences for two years; there he met and befriended his classmate Jean-Baptiste Perrin (1870–1942) who later became a famous physicist in France.
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passage: He spent five years (1879–1884) at the College of Vienne and then two years (1884–1886) at the Lycée of Grenoble. In 1886 he moved to the Lycée Janson de Sailly in Paris to study sciences for two years; there he met and befriended his classmate Jean-Baptiste Perrin (1870–1942) who later became a famous physicist in France.
Cartan enrolled in the École Normale Supérieure in 1888, where he attended lectures by Charles Hermite (1822–1901), Jules Tannery (1848–1910), Gaston Darboux (1842–1917), Paul Appell (1855–1930), Émile Picard (1856–1941), Édouard Goursat (1858–1936), and Henri Poincaré (1854–1912) whose lectures were what Cartan thought most highly of.
After graduation from the École Normale Superieure in 1891, Cartan was drafted into the French army, where he served one year and attained the rank of sergeant. For the next two years (1892–1894) Cartan returned to ENS and, following the advice of his classmate Arthur Tresse (1868–1958) who studied under Sophus Lie in the years 1888–1889, worked on the subject of classification of simple Lie groups, which was started by Wilhelm Killing. In 1892 Lie came to Paris, at the invitation of Darboux and Tannery, and met Cartan for the first time.
Cartan defended his dissertation, The structure of finite continuous groups of transformations in 1894 in the Faculty of Sciences in the Sorbonne.
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passage: In 1892 Lie came to Paris, at the invitation of Darboux and Tannery, and met Cartan for the first time.
Cartan defended his dissertation, The structure of finite continuous groups of transformations in 1894 in the Faculty of Sciences in the Sorbonne. Between 1894 and 1896 Cartan was a lecturer at the University of Montpellier; during the years 1896 through 1903, he was a lecturer in the Faculty of Sciences at the University of Lyon.
In 1903, while in Lyon, Cartan married Marie-Louise Bianconi (1880–1950); in the same year, Cartan became a professor in the Faculty of Sciences at the University of Nancy. In 1904, Cartan's first son, Henri Cartan, who later became an influential mathematician, was born; in 1906, another son, Jean Cartan, who became a composer, was born. In 1909 Cartan moved his family to Paris and worked as a lecturer in the Faculty of Sciences in the Sorbonne. In 1912 Cartan became Professor there, based on the reference he received from Poincaré. He remained in Sorbonne until his retirement in 1940 and spent the last years of his life teaching mathematics at the École Normale Supérieure for girls.
As a student of Cartan, the geometer Shiing-Shen Chern wrote:Usually the day after [meeting with Cartan] I would get a letter from him. He would say, “After you left, I thought more about your questions...”—he had some results, and some more questions, and so on. He knew all these papers on simple Lie groups, Lie algebras, all by heart.
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passage: He would say, “After you left, I thought more about your questions...”—he had some results, and some more questions, and so on. He knew all these papers on simple Lie groups, Lie algebras, all by heart. When you saw him on the street, when a certain issue would come up, he would pull out some old envelope and write something and give you the answer. And sometimes it took me hours or even days to get the same answer... I had to work very hard. In 1921 he became a foreign member of the Polish Academy of Learning and in 1937 a foreign member of the Royal Netherlands Academy of Arts and Sciences. In 1938 he participated in the International Committee composed to organise the International Congresses for the Unity of Science.
He died in 1951 in Paris after a long illness.
In 1976, a lunar crater was named after him. Before, it was designated Apollonius D.
## Work
In the Travaux, Cartan breaks down his work into 15 areas. Using modern terminology, they are:
1. Lie theory
1. Representations of Lie groups
1. Hypercomplex numbers, division algebras
1. Systems of PDEs, Cartan–Kähler theorem
1. Theory of equivalence
1. Integrable systems, theory of prolongation and systems in involution
1. Infinite-dimensional groups and pseudogroups
1.
### Differential geometry
and moving frames
1. Generalised spaces with structure groups and connections, Cartan connection, holonomy, Weyl tensor
1. Geometry and topology of Lie groups
1. Riemannian geometry
1. Symmetric spaces
1.
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passage: Riemannian geometry
1. Symmetric spaces
1. Topology of compact groups and their homogeneous spaces
1. Integral invariants and classical mechanics
1. Relativity, spinors
Cartan's mathematical work can be described as the development of analysis on differentiable manifolds, which many now consider the central and most vital part of modern mathematics and which he was foremost in shaping and advancing. This field centers on Lie groups, partial differential systems, and differential geometry; these, chiefly through Cartan's contributions, are now closely interwoven and constitute a unified and powerful tool.
Lie groups
Cartan was practically alone in the field of Lie groups for the thirty years after his dissertation. Lie had considered these groups chiefly as systems of analytic transformations of an analytic manifold, depending analytically on a finite number of parameters. A very fruitful approach to the study of these groups was opened in 1888 when Wilhelm Killing systematically started to study the group in itself, independent of its possible actions on other manifolds. At that time (and until 1920) only local properties were considered, so the main object of study for Killing was the Lie algebra of the group, which exactly reflects the local properties in purely algebraic terms.
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passage: A very fruitful approach to the study of these groups was opened in 1888 when Wilhelm Killing systematically started to study the group in itself, independent of its possible actions on other manifolds. At that time (and until 1920) only local properties were considered, so the main object of study for Killing was the Lie algebra of the group, which exactly reflects the local properties in purely algebraic terms. Killing's great achievement was the determination of all simple complex Lie algebras; his proofs, however, were often defective, and Cartan's thesis was devoted mainly to giving a rigorous foundation to the local theory and to proving the existence of the exceptional Lie algebras belonging to each of the types of simple complex Lie algebras that Killing had shown to be possible. Later Cartan completed the local theory by explicitly solving two fundamental problems, for which he had to develop entirely new methods: the classification of simple real Lie algebras and the determination of all irreducible linear representations of simple Lie algebras, by means of the notion of weight of a representation, which he introduced for that purpose. It was in the process of determining the linear representations of the orthogonal groups that Cartan discovered in 1913 the spinors, which later played such an important role in quantum mechanics.
After 1925 Cartan grew more and more interested in topological questions.
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passage: It was in the process of determining the linear representations of the orthogonal groups that Cartan discovered in 1913 the spinors, which later played such an important role in quantum mechanics.
After 1925 Cartan grew more and more interested in topological questions. Spurred by Weyl's brilliant results on compact groups, he developed new methods for the study of global properties of Lie groups; in particular he showed that topologically a connected Lie group is a product of a Euclidean space and a compact group, and for compact Lie groups he discovered that the possible fundamental groups of the underlying manifold can be read from the structure of the Lie algebra of the group. Finally, he outlined a method of determining the Betti numbers of compact Lie groups, again reducing the problem to an algebraic question on their Lie algebras, which has since been completely solved.
### Lie pseudogroups
After solving the problem of the structure of Lie groups which Cartan (following Lie) called "finite continuous groups" (or "finite transformation groups"), Cartan posed the similar problem for "infinite continuous groups", which are now called Lie pseudogroups, an infinite-dimensional analogue of Lie groups (there are other infinite generalizations of Lie groups). The Lie pseudogroup considered by Cartan is a set of transformations between subsets of a space that contains the identical transformation and possesses the property that the result of composition of two transformations in this set (whenever this is possible) belongs to the same set. Since the composition of two transformations is not always possible, the set of transformations is not a group (but a groupoid in modern terminology), thus the name pseudogroup.
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passage: The Lie pseudogroup considered by Cartan is a set of transformations between subsets of a space that contains the identical transformation and possesses the property that the result of composition of two transformations in this set (whenever this is possible) belongs to the same set. Since the composition of two transformations is not always possible, the set of transformations is not a group (but a groupoid in modern terminology), thus the name pseudogroup. Cartan considered only those transformations of manifolds for which there is no subdivision of manifolds into the classes transposed by the transformations under consideration. Such pseudogroups of transformations are called primitive. Cartan showed that every infinite-dimensional primitive pseudogroup of complex analytic transformations belongs to one of the six classes: 1) the pseudogroup of all analytic transformations of n complex variables; 2) the pseudogroup of all analytic transformations of n complex variables with a constant Jacobian (i.e., transformations that multiply all volumes by the same complex number); 3) the pseudogroup of all analytic transformations of n complex variables whose Jacobian is equal to one (i.e., transformations that preserve volumes); 4) the pseudogroup of all analytic transformations of 2n > 4 complex variables that preserve a certain double integral (the symplectic pseudogroup); 5) the pseudogroup of all analytic transformations of 2n > 4 complex variables that multiply the above-mentioned double integral by a complex function; 6) the pseudogroup of all analytic transformations of 2n + 1 complex variables that multiply a certain form by a complex function (the contact pseudogroup).
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passage: Such pseudogroups of transformations are called primitive. Cartan showed that every infinite-dimensional primitive pseudogroup of complex analytic transformations belongs to one of the six classes: 1) the pseudogroup of all analytic transformations of n complex variables; 2) the pseudogroup of all analytic transformations of n complex variables with a constant Jacobian (i.e., transformations that multiply all volumes by the same complex number); 3) the pseudogroup of all analytic transformations of n complex variables whose Jacobian is equal to one (i.e., transformations that preserve volumes); 4) the pseudogroup of all analytic transformations of 2n > 4 complex variables that preserve a certain double integral (the symplectic pseudogroup); 5) the pseudogroup of all analytic transformations of 2n > 4 complex variables that multiply the above-mentioned double integral by a complex function; 6) the pseudogroup of all analytic transformations of 2n + 1 complex variables that multiply a certain form by a complex function (the contact pseudogroup). There are similar classes of pseudogroups for primitive pseudogroups of real transformations defined by analytic functions of real variables.
### Differential systems
Cartan's methods in the theory of differential systems are perhaps his most profound achievement. Breaking with tradition, he sought from the start to formulate and solve the problems in a completely invariant fashion, independent of any particular choice of variables and unknown functions. He thus was able for the first time to give a precise definition of what is a "general" solution of an arbitrary differential system.
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passage: Breaking with tradition, he sought from the start to formulate and solve the problems in a completely invariant fashion, independent of any particular choice of variables and unknown functions. He thus was able for the first time to give a precise definition of what is a "general" solution of an arbitrary differential system. His next step was to try to determine all "singular" solutions as well, by a method of "prolongation" that consists in adjoining new unknowns and new equations to the given system in such a way that any singular solution of the original system becomes a general solution of the new system. Although Cartan showed that in every example which he treated his method led to the complete determination of all singular solutions, he did not succeed in proving in general that this would always be the case for an arbitrary system; such a proof was obtained in 1955 by Masatake Kuranishi.
Cartan's chief tool was the calculus of exterior differential forms, which he helped to create and develop in the ten years following his thesis and then proceeded to apply to a variety of problems in differential geometry, Lie groups, analytical dynamics, and general relativity. He discussed a large number of examples, treating them in an extremely elliptic style that was made possible only by his uncanny algebraic and geometric insight.
Differential geometry
Cartan's contributions to differential geometry are no less impressive, and it may be said that he revitalized the whole subject, for the initial work of Riemann and Darboux was being lost in dreary computations and minor results, much as had happened to elementary geometry and invariant theory a generation earlier.
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passage: He discussed a large number of examples, treating them in an extremely elliptic style that was made possible only by his uncanny algebraic and geometric insight.
Differential geometry
Cartan's contributions to differential geometry are no less impressive, and it may be said that he revitalized the whole subject, for the initial work of Riemann and Darboux was being lost in dreary computations and minor results, much as had happened to elementary geometry and invariant theory a generation earlier. His guiding principle was a considerable extension of the method of "moving frames" of Darboux and Ribaucour, to which he gave a tremendous flexibility and power, far beyond anything that had been done in classical differential geometry. In modern terms, the method consists in associating to a fiber bundle E the principal fiber bundle having the same base and having at each point of the base a fiber equal to the group that acts on the fiber of E at the same point. If E is the tangent bundle over the base (which since Lie was essentially known as the manifold of "contact elements"), the corresponding group is the general linear group (or the orthogonal group in classical Euclidean or Riemannian geometry). Cartan's ability to handle many other types of fibers and groups allows one to credit him with the first general idea of a fiber bundle, although he never defined it explicitly. This concept has become one of the most important in all fields of modern mathematics, chiefly in global differential geometry and in algebraic and differential topology.
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passage: Cartan's ability to handle many other types of fibers and groups allows one to credit him with the first general idea of a fiber bundle, although he never defined it explicitly. This concept has become one of the most important in all fields of modern mathematics, chiefly in global differential geometry and in algebraic and differential topology. Cartan used it to formulate his definition of a connection, which is now used universally and has superseded previous attempts by several geometers, made after 1917, to find a type of "geometry" more general than the Riemannian model and perhaps better adapted to a description of the universe along the lines of general relativity.
Cartan showed how to use his concept of connection to obtain a much more elegant and simple presentation of Riemannian geometry. His chief contribution to the latter, however, was the discovery and study of the symmetric Riemann spaces, one of the few instances in which the initiator of a mathematical theory was also the one who brought it to its completion. Symmetric Riemann spaces may be defined in various ways, the simplest of which postulates the existence around each point of the space of a "symmetry" that is involutive, leaves the point fixed, and preserves distances. The unexpected fact discovered by Cartan is that it is possible to give a complete description of these spaces by means of the classification of the simple Lie groups; it should therefore not be surprising that in various areas of mathematics, such as automorphic functions and analytic number theory (apparently far removed from differential geometry), these spaces are playing a part that is becoming increasingly important.
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passage: Symmetric Riemann spaces may be defined in various ways, the simplest of which postulates the existence around each point of the space of a "symmetry" that is involutive, leaves the point fixed, and preserves distances. The unexpected fact discovered by Cartan is that it is possible to give a complete description of these spaces by means of the classification of the simple Lie groups; it should therefore not be surprising that in various areas of mathematics, such as automorphic functions and analytic number theory (apparently far removed from differential geometry), these spaces are playing a part that is becoming increasingly important.
## Alternative theory to general relativity
Cartan created a competitor theory of gravity also Einstein–Cartan theory.
## Publications
Cartan's papers have been collected in his Oeuvres complètes, 6 vols. (Paris, 1952–1955).
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passage: ## Publications
Cartan's papers have been collected in his Oeuvres complètes, 6 vols. (Paris, 1952–1955). Two excellent obituary notices are S. S. Chern and C. Chevalley, in Bulletin of the American Mathematical Society, 58 (1952); and J. H. C. Whitehead, in Obituary Notices of the Royal Society (1952).
-
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- Leçons sur les invariants intégraux, Hermann, Paris, 1922
-
-
-
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- La parallelisme absolu et la théorie unitaire du champ, Hermann, 1932
- Les Espaces Métriques Fondés sur la Notion d'Arie, Hermann, 1933
- La méthode de repère mobile, la théorie des groupes continus, et les espaces généralisés, 1935
- Leçons sur la théorie des espaces à connexion projective, Gauthiers-Villars, 1937
- La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Gauthiers-Villars, 1937
-
- Les systèmes différentiels extérieurs et leurs applications géométriques, Hermann, 1945
- Oeuvres complètes, 3 parts in 6 vols., Paris 1952 to 1955, reprinted by CNRS 1984:
- Part 1: Groupes de Lie (in 2 vols.), 1952
- Part 2, Vol. 1: Algèbre, formes différentielles, systèmes différentiels, 1953
- Part 2, Vol. 2: Groupes finis, Systèmes différentiels, théories d'équivalence, 1953
- Part 3, Vol. 1: Divers, géométrie différentielle, 1955
- Part 3, Vol. 2: Géométrie différentielle, 1955
- Élie Cartan and Albert Einstein: Letters on Absolute Parallelism, 1929–1932 / original text in French & German, English trans. by Jules Leroy & Jim Ritter, ed. by Robert Debever, Princeton University Press, 1979
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passage: In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.
## History
Étale cohomology was introduced by , using some suggestions by Jean-Pierre Serre, and was motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as and SGA 4. Grothendieck used étale cohomology to prove some of the Weil conjectures (Bernard Dwork had already managed to prove the rationality part of the conjectures in 1960 using p-adic methods), and the remaining conjecture, the analogue of the Riemann hypothesis was proved by Pierre Deligne (1974) using ℓ-adic cohomology.
Further contact with classical theory was found in the shape of the Grothendieck version of the Brauer group; this was applied in short order to diophantine geometry, by Yuri Manin. The burden and success of the general theory was certainly both to integrate all this information, and to prove general results such as Poincaré duality and the Lefschetz fixed-point theorem in this context.
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passage: Further contact with classical theory was found in the shape of the Grothendieck version of the Brauer group; this was applied in short order to diophantine geometry, by Yuri Manin. The burden and success of the general theory was certainly both to integrate all this information, and to prove general results such as Poincaré duality and the Lefschetz fixed-point theorem in this context.
Grothendieck originally developed étale cohomology in an extremely general setting, working with concepts such as Grothendieck toposes and Grothendieck universes. With hindsight, much of this machinery proved unnecessary for most practical applications of the étale theory, and gave a simplified exposition of étale cohomology theory. Grothendieck's use of these universes (whose existence cannot be proved in Zermelo–Fraenkel set theory) led to some speculation that étale cohomology and its applications (such as the proof of Fermat's Last Theorem) require axioms beyond ZFC. However, in practice étale cohomology is used mainly in the case of constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with care the necessary objects can be constructed without using any uncountable sets, and this can be done in ZFC, and even in much weaker theories.
Étale cohomology quickly found other applications, for example Deligne and George Lusztig used it to construct representations of finite groups of Lie type; see Deligne–Lusztig theory.
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passage: However, in practice étale cohomology is used mainly in the case of constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with care the necessary objects can be constructed without using any uncountable sets, and this can be done in ZFC, and even in much weaker theories.
Étale cohomology quickly found other applications, for example Deligne and George Lusztig used it to construct representations of finite groups of Lie type; see Deligne–Lusztig theory.
## Motivation
For complex algebraic varieties, invariants from algebraic topology such as the fundamental group and cohomology groups are very useful, and one would like to have analogues of these for varieties over other fields, such as finite fields. (One reason for this is that Weil suggested that the Weil conjectures could be proved using such a cohomology theory.) In the case of cohomology of coherent sheaves, Serre showed that one could get a satisfactory theory just by using the Zariski topology of the algebraic variety, and in the case of complex varieties this gives the same cohomology groups (for coherent sheaves) as the much finer complex topology. However, for constant sheaves such as the sheaf of integers this does not work: the cohomology groups defined using the Zariski topology are badly behaved.
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passage: In the case of cohomology of coherent sheaves, Serre showed that one could get a satisfactory theory just by using the Zariski topology of the algebraic variety, and in the case of complex varieties this gives the same cohomology groups (for coherent sheaves) as the much finer complex topology. However, for constant sheaves such as the sheaf of integers this does not work: the cohomology groups defined using the Zariski topology are badly behaved. For example, Weil envisioned a cohomology theory for varieties over finite fields with similar power as the usual singular cohomology of topological spaces, but in fact, any constant sheaf on an irreducible variety has trivial cohomology (all higher cohomology groups vanish).
The reason that the Zariski topology does not work well is that it is too coarse: it has too few open sets. There seems to be no good way to fix this by using a finer topology on a general algebraic variety. Grothendieck's key insight was to realize that there is no reason why the more general open sets should be subsets of the algebraic variety: the definition of a sheaf works perfectly well for any category, not just the category of open subsets of a space. He defined étale cohomology by replacing the category of open subsets of a space by the category of étale mappings to a space: roughly speaking, these can be thought of as open subsets of finite unbranched covers of the space.
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passage: Grothendieck's key insight was to realize that there is no reason why the more general open sets should be subsets of the algebraic variety: the definition of a sheaf works perfectly well for any category, not just the category of open subsets of a space. He defined étale cohomology by replacing the category of open subsets of a space by the category of étale mappings to a space: roughly speaking, these can be thought of as open subsets of finite unbranched covers of the space. These turn out (after a lot of work) to give just enough extra open sets that one can get reasonable cohomology groups for some constant coefficients, in particular for coefficients Z/nZ when n is coprime to the characteristic of the field one is working over.
Some basic intuitions of the theory are these:
- The étale requirement is the condition that would allow one to apply the implicit function theorem if it were true in algebraic geometry (but it isn't — implicit algebraic functions are called algebroid in older literature).
- There are certain basic cases, of dimension 0 and 1, and for an abelian variety, where the answers with constant sheaves of coefficients can be predicted (via Galois cohomology and Tate modules).
## Definitions
For any scheme X the category Et(X) is the category of all étale morphisms from a scheme to X. It is an analogue of the category of open subsets of a topological space, and its objects can be thought of informally as "étale open subsets" of X.
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passage: ## Definitions
For any scheme X the category Et(X) is the category of all étale morphisms from a scheme to X. It is an analogue of the category of open subsets of a topological space, and its objects can be thought of informally as "étale open subsets" of X. The intersection of two open sets of a topological space corresponds to the pullback of two étale maps to X. There is a rather minor set-theoretical problem here, since Et(X) is a "large" category: its objects do not form a set.
A presheaf on a topological space X is a contravariant functor from the category of open subsets to sets. By analogy we define an étale presheaf on a scheme X to be a contravariant functor from Et(X) to sets.
A presheaf F on a topological space is called a sheaf if it satisfies the sheaf condition: whenever an open subset is covered by open subsets Ui, and we are given elements of F(Ui) for all i whose restrictions to Ui ∩ Uj agree for all i, j, then they are images of a unique element of F(U). By analogy, an étale presheaf is called a sheaf if it satisfies the same condition (with intersections of open sets replaced by pullbacks of étale morphisms, and where a set of étale maps to U is said to cover U if the topological space underlying U is the union of their images). More generally, one can define a sheaf for any Grothendieck topology on a category in a similar way.
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passage: By analogy, an étale presheaf is called a sheaf if it satisfies the same condition (with intersections of open sets replaced by pullbacks of étale morphisms, and where a set of étale maps to U is said to cover U if the topological space underlying U is the union of their images). More generally, one can define a sheaf for any Grothendieck topology on a category in a similar way.
The category of sheaves of abelian groups over a scheme has enough injective objects, so one can define right derived functors of left exact functors. The étale cohomology groups Hi(F) of the sheaf F of abelian groups are defined as the right derived functors of the functor of sections,
$$
F \to \Gamma(F)
$$
(where the space of sections Γ(F) of F is F(X)). The sections of a sheaf can be thought of as Hom(Z, F) where Z is the sheaf that returns the integers as an abelian group. The idea of derived functor here is that the functor of sections doesn't respect exact sequences as it is not right exact; according to general principles of homological algebra there will be a sequence of functors H 0, H 1, ... that represent the 'compensations' that must be made in order to restore some measure of exactness (long exact sequences arising from short ones). The H 0 functor coincides with the section functor Γ.
More generally, a morphism of schemes
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passage: The idea of derived functor here is that the functor of sections doesn't respect exact sequences as it is not right exact; according to general principles of homological algebra there will be a sequence of functors H 0, H 1, ... that represent the 'compensations' that must be made in order to restore some measure of exactness (long exact sequences arising from short ones). The H 0 functor coincides with the section functor Γ.
More generally, a morphism of schemes f : X → Y induces a map f∗ from étale sheaves over X to étale sheaves over Y, and its right derived functors are denoted by Rqf∗, for q a non-negative integer. In the special case when Y is the spectrum of an algebraically closed field (a point), Rqf∗(F ) is the same as Hq(F ).
Suppose that X is a Noetherian scheme. An abelian étale sheaf F over X is called finite locally constant if it is represented by an étale cover of X. It is called constructible if X can be covered by a finite family of subschemes on each of which the restriction of F is finite locally constant. It is called torsion if F(U) is a torsion group for all étale covers U of X. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves.
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passage: It is called torsion if F(U) is a torsion group for all étale covers U of X. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves.
## ℓ-adic cohomology groups
In applications to algebraic geometry over a finite field Fq with characteristic p, the main objective was to find a replacement for the singular cohomology groups with integer (or rational) coefficients, which are not available in the same way as for geometry of an algebraic variety over the complex number field. Étale cohomology works fine for coefficients Z/nZ for n co-prime to p, but gives unsatisfactory results for non-torsion coefficients. To get cohomology groups without torsion from étale cohomology one has to take an inverse limit of étale cohomology groups with certain torsion coefficients; this is called ℓ-adic cohomology, where ℓ stands for any prime number different from p. One considers, for schemes V, the cohomology groups
$$
H^i(V, \mathbf{Z}/\ell^k\mathbf{Z})
$$
and defines the ℓ-adic cohomology group
$$
H^i(V,\mathbf{Z}_\ell) = \varprojlim H^i(V, \mathbf{Z}/\ell^k\mathbf{Z})
$$
as their inverse limit.
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passage: Étale cohomology works fine for coefficients Z/nZ for n co-prime to p, but gives unsatisfactory results for non-torsion coefficients. To get cohomology groups without torsion from étale cohomology one has to take an inverse limit of étale cohomology groups with certain torsion coefficients; this is called ℓ-adic cohomology, where ℓ stands for any prime number different from p. One considers, for schemes V, the cohomology groups
$$
H^i(V, \mathbf{Z}/\ell^k\mathbf{Z})
$$
and defines the ℓ-adic cohomology group
$$
H^i(V,\mathbf{Z}_\ell) = \varprojlim H^i(V, \mathbf{Z}/\ell^k\mathbf{Z})
$$
as their inverse limit. Here Zℓ denotes the ℓ-adic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficients Z/ℓkZ. (There is a notorious trap here: cohomology does not commute with taking inverse limits, and the ℓ-adic cohomology group, defined as an inverse limit, is not the cohomology with coefficients in the étale sheaf Zℓ; the latter cohomology group exists but gives the "wrong" cohomology groups.)
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passage: To get cohomology groups without torsion from étale cohomology one has to take an inverse limit of étale cohomology groups with certain torsion coefficients; this is called ℓ-adic cohomology, where ℓ stands for any prime number different from p. One considers, for schemes V, the cohomology groups
$$
H^i(V, \mathbf{Z}/\ell^k\mathbf{Z})
$$
and defines the ℓ-adic cohomology group
$$
H^i(V,\mathbf{Z}_\ell) = \varprojlim H^i(V, \mathbf{Z}/\ell^k\mathbf{Z})
$$
as their inverse limit. Here Zℓ denotes the ℓ-adic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficients Z/ℓkZ. (There is a notorious trap here: cohomology does not commute with taking inverse limits, and the ℓ-adic cohomology group, defined as an inverse limit, is not the cohomology with coefficients in the étale sheaf Zℓ; the latter cohomology group exists but gives the "wrong" cohomology groups.)
More generally, if F is an inverse system of étale sheaves Fi, then the cohomology of F is defined to be the inverse limit of the cohomology of the sheaves Fi
$$
H^q(X, F) = \varprojlim H^q(X, F_i),
$$
and though there is a natural map
$$
H^q(X,\varprojlim F_i) \to \varprojlim H^q(X, F_i),
$$
this is not usually an isomorphism.
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passage: An ℓ-adic sheaf is a special sort of inverse system of étale sheaves Fi, where i runs through positive integers, and Fi is a module over Z/ℓi Z and the map from Fi+1 to Fi is just reduction mod Z/ℓi Z.
When V is a non-singular algebraic curve of genus g, H1 is a free Zℓ-module of rank 2g, dual to the Tate module of the Jacobian variety of V. Since the first Betti number of a Riemann surface of genus g is 2g, this is isomorphic to the usual singular cohomology with Zℓ coefficients for complex algebraic curves. It also shows one reason why the condition ℓ ≠ p is required: when ℓ = p the rank of the Tate module is at most g.
Torsion subgroups can occur, and were applied by Michael Artin and David Mumford to geometric questions. To remove any torsion subgroup from the ℓ-adic cohomology groups and get cohomology groups that are vector spaces over fields of characteristic 0 one defines
$$
H^i(V,\mathbf{Q}_\ell)=H^i(V,\mathbf{Z}_\ell)\otimes\mathbf{Q}_\ell.
$$
This notation is misleading: the symbol Qℓ on the left represents neither an étale sheaf nor an ℓ-adic sheaf.
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passage: It also shows one reason why the condition ℓ ≠ p is required: when ℓ = p the rank of the Tate module is at most g.
Torsion subgroups can occur, and were applied by Michael Artin and David Mumford to geometric questions. To remove any torsion subgroup from the ℓ-adic cohomology groups and get cohomology groups that are vector spaces over fields of characteristic 0 one defines
$$
H^i(V,\mathbf{Q}_\ell)=H^i(V,\mathbf{Z}_\ell)\otimes\mathbf{Q}_\ell.
$$
This notation is misleading: the symbol Qℓ on the left represents neither an étale sheaf nor an ℓ-adic sheaf. The etale cohomology with coefficients in the constant etale sheaf Qℓ does also exist but is quite different from
$$
H^i(V,\mathbf{Z}_\ell)\otimes\mathbf{Q}_\ell
$$
. Confusing these two groups is a common mistake.
## Properties
In general the ℓ-adic cohomology groups of a variety tend to have similar properties to the singular cohomology groups of complex varieties, except that they are modules over the ℓ-adic integers (or numbers) rather than the integers (or rationals). They satisfy a form of Poincaré duality on non-singular projective varieties, and the ℓ-adic cohomology groups of a "reduction mod p" of a complex variety tend to have the same rank as the singular cohomology groups. A Künneth formula also holds.
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passage: They satisfy a form of Poincaré duality on non-singular projective varieties, and the ℓ-adic cohomology groups of a "reduction mod p" of a complex variety tend to have the same rank as the singular cohomology groups. A Künneth formula also holds.
For example, the first cohomology group of a complex elliptic curve is a free module of rank 2 over the integers, while the first ℓ-adic cohomology group of an elliptic curve over a finite field is a free module of rank 2 over the ℓ-adic integers, provided ℓ is not the characteristic of the field concerned, and is dual to its Tate module.
There is one way in which ℓ-adic cohomology groups are better than singular cohomology groups: they tend to be acted on by Galois groups. For example, if a complex variety is defined over the rational numbers, its ℓ-adic cohomology groups are acted on by the absolute Galois group of the rational numbers: they afford Galois representations.
Elements of the Galois group of the rationals, other than the identity and complex conjugation, do not usually act continuously on a complex variety defined over the rationals, so do not act on the singular cohomology groups. This phenomenon of Galois representations is related to the fact that the fundamental group of a topological space acts on the singular cohomology groups, because Grothendieck showed that the Galois group can be regarded as a sort of fundamental group. (See also Grothendieck's Galois theory.)
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passage: This phenomenon of Galois representations is related to the fact that the fundamental group of a topological space acts on the singular cohomology groups, because Grothendieck showed that the Galois group can be regarded as a sort of fundamental group. (See also Grothendieck's Galois theory.)
## Calculation of étale cohomology groups for algebraic curves
The main initial step in calculating étale cohomology groups of a variety is to calculate them for complete connected smooth algebraic curves X over algebraically closed fields k. The étale cohomology groups of arbitrary varieties can then be controlled using analogues of the usual machinery of algebraic topology, such as the spectral sequence of a fibration. For curves the calculation takes several steps, as follows . Let Gm denote the sheaf of non-vanishing functions.
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passage: For curves the calculation takes several steps, as follows . Let Gm denote the sheaf of non-vanishing functions.
### Calculation of H1(X, Gm)
The exact sequence of étale sheaves
$$
1\to \mathbf{G}_m\to j_*\mathbf{G}_{m, K}\to \bigoplus_{x\in |X|}i_{x*}\mathbf{Z}\to 1
$$
gives a long exact sequence of cohomology groups
$$
\begin{align}
0 &\to H^0(\mathbf{G}_m)\to H^0(j_*\mathbf{G}_{m, K})\to \bigoplus\nolimits_{x\in |X|}H^0(i_{x*}\mathbf{Z}) \to \\
&\to H^1(\mathbf{G}_m)\to H^1(j_*\mathbf{G}_{m, K})\to \bigoplus\nolimits_{x\in |X|}H^1(i_{x*}\mathbf{Z}) \to \\
&\to \cdots
\end{align}
$$
Here j is the injection of the generic point, ix is the injection of a closed point x, Gm,K is the sheaf Gm on (the generic point of X), and Zx is a copy of Z for each closed point of X.
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passage: Let Gm denote the sheaf of non-vanishing functions.
### Calculation of H1(X, Gm)
The exact sequence of étale sheaves
$$
1\to \mathbf{G}_m\to j_*\mathbf{G}_{m, K}\to \bigoplus_{x\in |X|}i_{x*}\mathbf{Z}\to 1
$$
gives a long exact sequence of cohomology groups
$$
\begin{align}
0 &\to H^0(\mathbf{G}_m)\to H^0(j_*\mathbf{G}_{m, K})\to \bigoplus\nolimits_{x\in |X|}H^0(i_{x*}\mathbf{Z}) \to \\
&\to H^1(\mathbf{G}_m)\to H^1(j_*\mathbf{G}_{m, K})\to \bigoplus\nolimits_{x\in |X|}H^1(i_{x*}\mathbf{Z}) \to \\
&\to \cdots
\end{align}
$$
Here j is the injection of the generic point, ix is the injection of a closed point x, Gm,K is the sheaf Gm on (the generic point of X), and Zx is a copy of Z for each closed point of X. The groups H i(ix* Z) vanish if i > 0 (because ix* Z is a skyscraper sheaf) and for i = 0 they are Z
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passage: so their sum is just the divisor group of X. Moreover, the first cohomology group H 1(X, j∗Gm,K) is isomorphic to the Galois cohomology group H 1(K, K*) which vanishes by Hilbert's theorem 90. Therefore, the long exact sequence of étale cohomology groups gives an exact sequence
$$
K\to \operatorname{Div}(X)\to H^1(\mathbf{G}_m)\to 1
$$
where Div(X) is the group of divisors of X and K is its function field. In particular H 1(X, Gm) is the Picard group Pic(X) (and the first cohomology groups of Gm are the same for the étale and Zariski topologies). This step works for varieties X of any dimension (with points replaced by codimension 1 subvarieties), not just curves.
### Calculation of Hi(X, Gm)
The same long exact sequence above shows that if i ≥ 2 then the cohomology group H i(X, Gm) is isomorphic to H i(X, j*Gm,K), which is isomorphic to the Galois cohomology group H i(K, K*). Tsen's theorem implies that the Brauer group of a function field K in one variable over an algebraically closed field vanishes. This in turn implies that all the Galois cohomology groups H i(K, K*) vanish for i ≥ 1, so all the cohomology groups H i(X, Gm) vanish if i ≥ 2.
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passage: Tsen's theorem implies that the Brauer group of a function field K in one variable over an algebraically closed field vanishes. This in turn implies that all the Galois cohomology groups H i(K, K*) vanish for i ≥ 1, so all the cohomology groups H i(X, Gm) vanish if i ≥ 2.
### Calculation of Hi(X, μn)
If μn is the sheaf of n-th roots of unity and n and the characteristic of the field k are coprime integers, then:
$$
H^i (X, \mu_n) = \begin{cases} \mu_n(k) & i =0 \\ \operatorname{Pic}_n(X) & i = 1 \\ \mathbf{Z}/n\mathbf{Z} & i =2 \\ 0 & i \geqslant 3 \end{cases}
$$
where Picn(X) is group of n-torsion points of Pic(X).
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passage: ### Calculation of Hi(X, μn)
If μn is the sheaf of n-th roots of unity and n and the characteristic of the field k are coprime integers, then:
$$
H^i (X, \mu_n) = \begin{cases} \mu_n(k) & i =0 \\ \operatorname{Pic}_n(X) & i = 1 \\ \mathbf{Z}/n\mathbf{Z} & i =2 \\ 0 & i \geqslant 3 \end{cases}
$$
where Picn(X) is group of n-torsion points of Pic(X). This follows from the previous results using the long exact sequence
$$
\begin{align}
0 &\to H^0(X, \mu_n)\to H^0(X, \mathbf{G}_m)\to H^0(X, \mathbf{G}_m)\to \\
&\to H^1(X, \mu_n)\to H^1(X, \mathbf{G}_m)\to H^1(X, \mathbf{G}_m)\to \\
&\to H^2(X, \mu_n)\to H^2(X, \mathbf{G}_m)\to H^2(X, \mathbf{G}_m) \to \\
&\to \cdots
\end{align}
$$
of the Kummer exact sequence of étale sheaves
$$
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passage: &\to \cdots
\end{align}
$$
of the Kummer exact sequence of étale sheaves
$$
1 \to \mu_n \to \mathbf{G}_m \xrightarrow{(\cdot)^n} \mathbf{G}_m \to 1.
$$
and inserting the known values
$$
H^i (X, \mathbf{G}_m) = \begin{cases} k^* & i = 0 \\ \operatorname{Pic}(X) & i =1 \\ 0 &i \geqslant 2 \end{cases}
$$
In particular we get an exact sequence
$$
1\to H^1(X, \mu_n)\to \operatorname{Pic}(X)\xrightarrow{\times n} \operatorname{Pic}(X)\to H^2(X, \mu_n)\to 1.
$$
If n is divisible by p this argument breaks down because p-th roots of unity behave strangely over fields of characteristic p.
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passage: In the Zariski topology the Kummer sequence is not exact on the right, as a non-vanishing function does not usually have an n-th root locally for the Zariski topology, so this is one place where the use of the étale topology rather than the Zariski topology is essential.
### Calculation of H i(X, Z/nZ)
By fixing a primitive n-th root of unity we can identify the group Z/nZ with the group μn of n-th roots of unity. The étale group H i(X, Z/nZ) is then a free module over the ring Z/nZ and its rank is given by:
$$
\operatorname{rank}(H^i(X, \mathbf{Z}/n\mathbf{Z})) = \begin{cases} 1 & i =0 \\ 2g & i=1 \\1 & i = 2\\0 & i \geqslant 3 \end{cases}
$$
where g is the genus of the curve X. This follows from the previous result, using the fact that the Picard group of a curve is the points of its Jacobian variety, an abelian variety of dimension g, and if n is coprime to the characteristic then the points of order dividing n in an abelian variety of dimension g over an algebraically closed field form a group isomorphic to (Z/nZ)2g. These values for the étale group H i(X, Z/nZ) are the same as the corresponding singular cohomology groups when X is a complex curve.
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passage: This follows from the previous result, using the fact that the Picard group of a curve is the points of its Jacobian variety, an abelian variety of dimension g, and if n is coprime to the characteristic then the points of order dividing n in an abelian variety of dimension g over an algebraically closed field form a group isomorphic to (Z/nZ)2g. These values for the étale group H i(X, Z/nZ) are the same as the corresponding singular cohomology groups when X is a complex curve.
### Calculation of H i(X, Z/pZ)
It is possible to calculate étale cohomology groups with constant coefficients of order divisible by the characteristic in a similar way, using the Artin–Schreier sequence
$$
0\to \mathbf{Z}/p\mathbf{Z}\to K\ \xrightarrow{{}\atop x\mapsto x^p-x}\ K\to 0
$$
instead of the Kummer sequence. (For coefficients in Z/pnZ there is a similar sequence involving Witt vectors.) The resulting cohomology groups usually have ranks less than that of the corresponding groups in characteristic 0.
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passage: (For coefficients in Z/pnZ there is a similar sequence involving Witt vectors.) The resulting cohomology groups usually have ranks less than that of the corresponding groups in characteristic 0.
## Examples of étale cohomology groups
- If X is the spectrum of a field K with absolute Galois group G, then étale sheaves over X correspond to continuous sets (or abelian groups) acted on by the (profinite) group G, and étale cohomology of the sheaf is the same as the group cohomology of G, i.e. the Galois cohomology of K.
- If X is a complex variety, then étale cohomology with finite coefficients is isomorphic to singular cohomology with finite coefficients. (This does not hold for integer coefficients.) More generally the cohomology with coefficients in any constructible sheaf is the same.
- If F is a coherent sheaf (or Gm) then the étale cohomology of F is the same as Serre's coherent sheaf cohomology calculated with the Zariski topology (and if X is a complex variety this is the same as the sheaf cohomology calculated with the usual complex topology).
- For abelian varieties and curves there is an elementary description of ℓ-adic cohomology. For abelian varieties the first ℓ-adic cohomology group is the dual of the Tate module, and the higher cohomology groups are given by its exterior powers. For curves the first cohomology group is the first cohomology group of its Jacobian.
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passage: For abelian varieties the first ℓ-adic cohomology group is the dual of the Tate module, and the higher cohomology groups are given by its exterior powers. For curves the first cohomology group is the first cohomology group of its Jacobian. This explains why Weil was able to give a more elementary proof of the Weil conjectures in these two cases: in general one expects to find an elementary proof whenever there is an elementary description of the ℓ-adic cohomology.
## Poincaré duality and cohomology with compact support
The étale cohomology groups with compact support of a variety X are defined to be
$$
H_c^q(X, F) = H^q(Y, j_!F)
$$
where j is an open immersion of X into a proper variety Y and j! is the extension by 0 of the étale sheaf F to Y. This is independent of the immersion j. If X has dimension at most n and F is a torsion sheaf then these cohomology groups
$$
H_c^q(X, F)
$$
with compact support vanish if q > 2n, and if in addition X is affine of finite type over a separably closed field the cohomology groups
$$
H^q(X, F)
$$
vanish for q > n (for the last statement, see SGA 4, XIV, Cor.3.2).
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passage: ## Poincaré duality and cohomology with compact support
The étale cohomology groups with compact support of a variety X are defined to be
$$
H_c^q(X, F) = H^q(Y, j_!F)
$$
where j is an open immersion of X into a proper variety Y and j! is the extension by 0 of the étale sheaf F to Y. This is independent of the immersion j. If X has dimension at most n and F is a torsion sheaf then these cohomology groups
$$
H_c^q(X, F)
$$
with compact support vanish if q > 2n, and if in addition X is affine of finite type over a separably closed field the cohomology groups
$$
H^q(X, F)
$$
vanish for q > n (for the last statement, see SGA 4, XIV, Cor.3.2).
More generally if f is a separated morphism of finite type from X to S (with X and S Noetherian) then the higher direct images with compact support Rqf! are defined by
$$
R^qf_!(F)=R^qg_*(j_!F)
$$
for any torsion sheaf F. Here j is any open immersion of X into a scheme Y with a proper morphism g to S (with f = gj), and as before the definition does not depend on the choice of j and Y. Cohomology with compact support is the special case of this with S a point. If f is a separated morphism of finite type then Rqf! takes constructible sheaves on X to constructible sheaves on S.
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passage: More generally if f is a separated morphism of finite type from X to S (with X and S Noetherian) then the higher direct images with compact support Rqf! are defined by
$$
R^qf_!(F)=R^qg_*(j_!F)
$$
for any torsion sheaf F. Here j is any open immersion of X into a scheme Y with a proper morphism g to S (with f = gj), and as before the definition does not depend on the choice of j and Y. Cohomology with compact support is the special case of this with S a point. If f is a separated morphism of finite type then Rqf! takes constructible sheaves on X to constructible sheaves on S. If in addition the fibers of f have dimension at most n then Rqf! vanishes on torsion sheaves for q > 2n. If X is a complex variety then Rqf! is the same as the usual higher direct image with compact support (for the complex topology) for torsion sheaves.
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passage: If in addition the fibers of f have dimension at most n then Rqf! vanishes on torsion sheaves for q > 2n. If X is a complex variety then Rqf! is the same as the usual higher direct image with compact support (for the complex topology) for torsion sheaves.
If X is a smooth algebraic variety of dimension N and n is coprime to the characteristic then there is a trace map
$$
\operatorname{Tr}: H_c^{2N}(X, \mu_n^N) \rightarrow \mathbf{Z}/n\mathbf{Z}
$$
and the bilinear form Tr(a ∪ b) with values in Z/nZ identifies each of the groups
$$
H^i_c(X,\mu_n^N)
$$
and
$$
H^{2N-i}(X,\mathbf{Z}/n\mathbf{Z})
$$
with the dual of the other. This is the analogue of Poincaré duality for étale cohomology.
## An application to curves
This is how the theory could be applied to the local zeta-function of an algebraic curve.
Theorem. Let be a curve of genus defined over , the finite field with elements. Then for
$$
\#X \left (\mathbf F_{p^n} \right ) = p^n + 1 -\sum_{i=1}^{2g} \alpha_i^n,
$$
where are certain algebraic numbers satisfying .
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passage: Let be a curve of genus defined over , the finite field with elements. Then for
$$
\#X \left (\mathbf F_{p^n} \right ) = p^n + 1 -\sum_{i=1}^{2g} \alpha_i^n,
$$
where are certain algebraic numbers satisfying .
This agrees with being a curve of genus with points. It also shows that the number of points on any curve is rather close (within ) to that of the projective line; in particular, it generalizes Hasse's theorem on elliptic curves.
### Idea of proof
According to the Lefschetz fixed-point theorem, the number of fixed points of any morphism is equal to the sum
$$
\sum_{i=0}^{2 \dim(X)} (-1)^i \operatorname{Tr} \left (f|_{H^i(X)} \right ).
$$
This formula is valid for ordinary topological varieties and ordinary topology, but it is wrong for most algebraic topologies. However, this formula does hold for étale cohomology (though this is not so simple to prove).
The points of that are defined over are those fixed by , where is the Frobenius automorphism in characteristic .
The étale cohomology Betti numbers of in dimensions 0, 1, 2 are 1, 2g, and 1 respectively.
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passage: The points of that are defined over are those fixed by , where is the Frobenius automorphism in characteristic .
The étale cohomology Betti numbers of in dimensions 0, 1, 2 are 1, 2g, and 1 respectively.
According to all of these,
$$
\#X \left (\mathbf F_{p^n} \right ) = \operatorname{Tr} \left (F^n|_{H^0(X)} \right )- \operatorname{Tr} \left (F^n|_{H^1(X)} \right ) + \operatorname{Tr} \left (F^n|_{H^2(X)} \right ).
$$
This gives the general form of the theorem.
The assertion on the absolute values of the is the 1-dimensional Riemann Hypothesis of the Weil Conjectures.
The whole idea fits into the framework of motives: formally [X] = [point] + [line] + [1-part], and [1-part] has something like points.
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passage: In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.
## Motivation
Let X be a topological space, and let
$$
\mathcal{U}
$$
be an open cover of X. Let
$$
N(\mathcal{U})
$$
denote the nerve of the covering. The idea of Čech cohomology is that, for an open cover
$$
\mathcal{U}
$$
consisting of sufficiently small open sets, the resulting simplicial complex
$$
N(\mathcal{U})
$$
should be a good combinatorial model for the space X. For such a cover, the Čech cohomology of X is defined to be the simplicial cohomology of the nerve. This idea can be formalized by the notion of a good cover. However, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of X, ordered by refinement. This is the approach adopted below.
## Construction
Let X be a topological space, and let
$$
\mathcal{F}
$$
be a presheaf of abelian groups on X. Let
$$
\mathcal{U}
$$
be an open cover of X.
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passage: This is the approach adopted below.
## Construction
Let X be a topological space, and let
$$
\mathcal{F}
$$
be a presheaf of abelian groups on X. Let
$$
\mathcal{U}
$$
be an open cover of X.
### Simplex
A q-simplex σ of
$$
\mathcal{U}
$$
is an ordered collection of q+1 sets chosen from
$$
\mathcal{U}
$$
, such that the intersection of all these sets is non-empty. This intersection is called the support of σ and is denoted |σ|.
Now let
$$
\sigma = (U_i)_{i \in \{ 0 , \ldots , q \}}
$$
be such a q-simplex. The j-th partial boundary of σ is defined to be the (q−1)-simplex obtained by removing the j-th set from σ, that is:
$$
\partial_j \sigma := (U_i)_{i \in \{ 0 , \ldots , q \} \setminus \{j\}}.
$$
The boundary of σ is defined as the alternating sum of the partial boundaries:
$$
\partial \sigma := \sum_{j=0}^q (-1)^{j+1} \partial_j \sigma
$$
viewed as an element of the free abelian group spanned by the simplices of
$$
\mathcal{U}
$$
.
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passage: Now let
$$
\sigma = (U_i)_{i \in \{ 0 , \ldots , q \}}
$$
be such a q-simplex. The j-th partial boundary of σ is defined to be the (q−1)-simplex obtained by removing the j-th set from σ, that is:
$$
\partial_j \sigma := (U_i)_{i \in \{ 0 , \ldots , q \} \setminus \{j\}}.
$$
The boundary of σ is defined as the alternating sum of the partial boundaries:
$$
\partial \sigma := \sum_{j=0}^q (-1)^{j+1} \partial_j \sigma
$$
viewed as an element of the free abelian group spanned by the simplices of
$$
\mathcal{U}
$$
.
### Cochain
A q-cochain of
$$
\mathcal{U}
$$
with coefficients in
$$
\mathcal{F}
$$
is a map which associates with each q-simplex σ an element of
$$
\mathcal{F}(|\sigma|)
$$
, and we denote the set of all q-cochains of
$$
\mathcal{U}
$$
with coefficients in
$$
\mathcal{F}
$$
by
$$
C^q(\mathcal U, \mathcal F)
$$
.
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passage: The j-th partial boundary of σ is defined to be the (q−1)-simplex obtained by removing the j-th set from σ, that is:
$$
\partial_j \sigma := (U_i)_{i \in \{ 0 , \ldots , q \} \setminus \{j\}}.
$$
The boundary of σ is defined as the alternating sum of the partial boundaries:
$$
\partial \sigma := \sum_{j=0}^q (-1)^{j+1} \partial_j \sigma
$$
viewed as an element of the free abelian group spanned by the simplices of
$$
\mathcal{U}
$$
.
### Cochain
A q-cochain of
$$
\mathcal{U}
$$
with coefficients in
$$
\mathcal{F}
$$
is a map which associates with each q-simplex σ an element of
$$
\mathcal{F}(|\sigma|)
$$
, and we denote the set of all q-cochains of
$$
\mathcal{U}
$$
with coefficients in
$$
\mathcal{F}
$$
by
$$
C^q(\mathcal U, \mathcal F)
$$
.
$$
C^q(\mathcal U, \mathcal F)
$$
is an abelian group by pointwise addition.
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passage: ### Cochain
A q-cochain of
$$
\mathcal{U}
$$
with coefficients in
$$
\mathcal{F}
$$
is a map which associates with each q-simplex σ an element of
$$
\mathcal{F}(|\sigma|)
$$
, and we denote the set of all q-cochains of
$$
\mathcal{U}
$$
with coefficients in
$$
\mathcal{F}
$$
by
$$
C^q(\mathcal U, \mathcal F)
$$
.
$$
C^q(\mathcal U, \mathcal F)
$$
is an abelian group by pointwise addition.
### Differential
The cochain groups can be made into a cochain complex
$$
(C^{\bullet}(\mathcal U, \mathcal F), \delta)
$$
by defining the coboundary operator
$$
\delta_q : C^q(\mathcal U, \mathcal F) \to C^{q+1}(\mathcal{U}, \mathcal{F})
$$
by:
$$
\quad (\delta_q f)(\sigma) := \sum_{j=0}^{q+1} (-1)^j \mathrm{res}^{|\partial_j \sigma|}_{|\sigma|} f (\partial_j \sigma),
$$
where
$$
\mathrm{res}^{|\partial_j \sigma|}_{|\sigma|}
$$
is the restriction morphism from
$$
\mathcal F(|\partial_j \sigma|)
$$
to
$$
\mathcal F(|\sigma|).
$$
(Notice that ∂jσ ⊆ σ, but σ ⊆ ∂jσ.)
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passage: A calculation shows that
$$
\delta_{q+1} \circ \delta_q = 0.
$$
The coboundary operator is analogous to the exterior derivative of De Rham cohomology, so it sometimes called
the differential of the cochain complex.
#### Cocycle
A q-cochain is called a q-cocycle if it is in the kernel of
$$
\delta
$$
, hence
$$
Z^q(\mathcal{U}, \mathcal{F}) := \ker ( \delta_q) \subseteq C^q(\mathcal U, \mathcal F)
$$
is the set of all q-cocycles.
Thus a (q−1)-cochain
$$
f
$$
is a cocycle if for all q-simplices
$$
\sigma
$$
the cocycle condition
$$
\sum_{j=0}^{q} (-1)^j \mathrm{res}^{|\partial_j \sigma|}_{|\sigma|} f (\partial_j \sigma) = 0
$$
holds.
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passage: #### Cocycle
A q-cochain is called a q-cocycle if it is in the kernel of
$$
\delta
$$
, hence
$$
Z^q(\mathcal{U}, \mathcal{F}) := \ker ( \delta_q) \subseteq C^q(\mathcal U, \mathcal F)
$$
is the set of all q-cocycles.
Thus a (q−1)-cochain
$$
f
$$
is a cocycle if for all q-simplices
$$
\sigma
$$
the cocycle condition
$$
\sum_{j=0}^{q} (-1)^j \mathrm{res}^{|\partial_j \sigma|}_{|\sigma|} f (\partial_j \sigma) = 0
$$
holds.
A 0-cocycle
$$
f
$$
is a collection of local sections of
$$
\mathcal{F}
$$
satisfying a compatibility relation on every intersecting
$$
A,B\in \mathcal{U}
$$
$$
f(A)|_{A \cap B} = f(B)|_{A \cap B}
$$
A 1-cocycle
$$
f
$$
satisfies for every non-empty
$$
U = A\cap B \cap C
$$
with
$$
A,B,C \in \mathcal{U}
$$
$$
f(B \cap C)|_U - f(A \cap C)|_U + f(A \cap B)|_U = 0
$$
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passage: Thus a (q−1)-cochain
$$
f
$$
is a cocycle if for all q-simplices
$$
\sigma
$$
the cocycle condition
$$
\sum_{j=0}^{q} (-1)^j \mathrm{res}^{|\partial_j \sigma|}_{|\sigma|} f (\partial_j \sigma) = 0
$$
holds.
A 0-cocycle
$$
f
$$
is a collection of local sections of
$$
\mathcal{F}
$$
satisfying a compatibility relation on every intersecting
$$
A,B\in \mathcal{U}
$$
$$
f(A)|_{A \cap B} = f(B)|_{A \cap B}
$$
A 1-cocycle
$$
f
$$
satisfies for every non-empty
$$
U = A\cap B \cap C
$$
with
$$
A,B,C \in \mathcal{U}
$$
$$
f(B \cap C)|_U - f(A \cap C)|_U + f(A \cap B)|_U = 0
$$
#### Coboundary
A q-cochain is called a q-coboundary if it is in the image of
$$
\delta
$$
and
$$
B^q(\mathcal{U}, \mathcal{F}) := \mathrm{Im} ( \delta_{q-1}) \subseteq C^{q}(\mathcal{U}, \mathcal{F})
$$
is the set of all q-coboundaries.
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passage: For example, a 1-cochain
$$
f
$$
is a 1-coboundary if there exists a 0-cochain
$$
h
$$
such that for every intersecting
$$
A,B\in \mathcal{U}
$$
$$
f(A \cap B) = h(A)|_{A \cap B} - h(B)|_{A \cap B}
$$
### Cohomology
The Čech cohomology of
$$
\mathcal{U}
$$
with values in
$$
\mathcal{F}
$$
is defined to be the cohomology of the cochain complex
$$
(C^{\bullet}(\mathcal{U}, \mathcal{F}), \delta)
$$
. Thus the qth Čech cohomology is given by
$$
\check{H}^q(\mathcal{U}, \mathcal{F}) := H^q((C^{\bullet}(\mathcal U, \mathcal F), \delta)) = Z^q(\mathcal{U}, \mathcal{F}) / B^q(\mathcal{U}, \mathcal{F})
$$
.
The Čech cohomology of X is defined by considering refinements of open covers.
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passage: Thus the qth Čech cohomology is given by
$$
\check{H}^q(\mathcal{U}, \mathcal{F}) := H^q((C^{\bullet}(\mathcal U, \mathcal F), \delta)) = Z^q(\mathcal{U}, \mathcal{F}) / B^q(\mathcal{U}, \mathcal{F})
$$
.
The Čech cohomology of X is defined by considering refinements of open covers. If
$$
\mathcal{V}
$$
is a refinement of
$$
\mathcal{U}
$$
then there is a map in cohomology
$$
\check{H}^*(\mathcal U,\mathcal F) \to \check{H}^*(\mathcal V,\mathcal F).
$$
The open covers of X form a directed set under refinement, so the above map leads to a direct system of abelian groups. The Čech cohomology of X with values in is defined as the direct limit
$$
\check{H}(X,\mathcal F) := \varinjlim_{\mathcal U} \check{H}(\mathcal U,\mathcal F)
$$
of this system.
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passage: The open covers of X form a directed set under refinement, so the above map leads to a direct system of abelian groups. The Čech cohomology of X with values in is defined as the direct limit
$$
\check{H}(X,\mathcal F) := \varinjlim_{\mathcal U} \check{H}(\mathcal U,\mathcal F)
$$
of this system.
The Čech cohomology of X with coefficients in a fixed abelian group A, denoted
$$
\check{H}(X;A)
$$
, is defined as
$$
\check{H}(X,\mathcal{F}_A)
$$
where
$$
\mathcal{F}_A
$$
is the constant sheaf on X determined by A.
A variant of Čech cohomology, called numerable Čech cohomology, is defined as above, except that all open covers considered are required to be numerable: that is, there is a partition of unity {ρi} such that each support
$$
\{x\mid\rho_i(x)>0\}
$$
is contained in some element of the cover. If X is paracompact and Hausdorff, then numerable Čech cohomology agrees with the usual Čech cohomology.
## Relation to other cohomology theories
If X is homotopy equivalent to a CW complex, then the Čech cohomology
$$
\check{H}^{*}(X;A)
$$
is naturally isomorphic to the singular cohomology
$$
H^*(X;A) \,
$$
.
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passage: ## Relation to other cohomology theories
If X is homotopy equivalent to a CW complex, then the Čech cohomology
$$
\check{H}^{*}(X;A)
$$
is naturally isomorphic to the singular cohomology
$$
H^*(X;A) \,
$$
. If X is a differentiable manifold, then
$$
\check{H}^*(X;\R)
$$
is also naturally isomorphic to the de Rham cohomology; the article on de Rham cohomology provides a brief review of this isomorphism. For less well-behaved spaces, Čech cohomology differs from singular cohomology. For example if X is the closed topologist's sine curve, then
$$
\check{H}^1(X;\Z)=\Z,
$$
whereas
$$
H^1(X;\Z)=0.
$$
If X is a differentiable manifold and the cover
$$
\mathcal{U}
$$
of X is a "good cover" (i.e. all the sets Uα are contractible to a point, and all finite intersections of sets in
$$
\mathcal{U}
$$
are either empty or contractible to a point), then
$$
\check{H}^{*}(\mathcal U;\R)
$$
is isomorphic to the de Rham cohomology.
If X is compact Hausdorff, then Čech cohomology (with coefficients in a discrete group) is isomorphic to Alexander-Spanier cohomology.
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passage: For example if X is the closed topologist's sine curve, then
$$
\check{H}^1(X;\Z)=\Z,
$$
whereas
$$
H^1(X;\Z)=0.
$$
If X is a differentiable manifold and the cover
$$
\mathcal{U}
$$
of X is a "good cover" (i.e. all the sets Uα are contractible to a point, and all finite intersections of sets in
$$
\mathcal{U}
$$
are either empty or contractible to a point), then
$$
\check{H}^{*}(\mathcal U;\R)
$$
is isomorphic to the de Rham cohomology.
If X is compact Hausdorff, then Čech cohomology (with coefficients in a discrete group) is isomorphic to Alexander-Spanier cohomology.
For a presheaf
$$
\mathcal{F}
$$
on X, let
$$
\mathcal{F}^+
$$
denote its sheafification. Then we have a natural comparison map
$$
\chi: \check{H}^*(X,\mathcal{F}) \to H^*(X,\mathcal{F}^+)
$$
from Čech cohomology to sheaf cohomology. If X is paracompact Hausdorff, then
$$
\chi
$$
is an isomorphism. More generally,
$$
\chi
$$
is an isomorphism whenever the Čech cohomology of all presheaves on X with zero sheafification vanishes.
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passage: If X is paracompact Hausdorff, then
$$
\chi
$$
is an isomorphism. More generally,
$$
\chi
$$
is an isomorphism whenever the Čech cohomology of all presheaves on X with zero sheafification vanishes.
## In algebraic geometry
Čech cohomology can be defined more generally for objects in a site C endowed with a topology. This applies, for example, to the Zariski site or the etale site of a scheme X. The Čech cohomology with values in some sheaf
$$
\mathcal{F}
$$
is defined as
$$
\check H^n (X, \mathcal{F}) := \varinjlim_{\mathcal U} \check H^n(\mathcal U, \mathcal{F}).
$$
where the colimit runs over all coverings (with respect to the chosen topology) of X. Here
$$
\check H^n(\mathcal U, \mathcal F)
$$
is defined as above, except that the r-fold intersections of open subsets inside the ambient topological space are replaced by the r-fold fiber product
$$
\mathcal U^{\times^r_X} := \mathcal U \times_X \dots \times_X \mathcal U.
$$
As in the classical situation of topological spaces, there is always a map
$$
\check H^n(X, \mathcal F) \rightarrow H^n(X, \mathcal F)
$$
from Čech cohomology to sheaf cohomology.
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passage: ## In algebraic geometry
Čech cohomology can be defined more generally for objects in a site C endowed with a topology. This applies, for example, to the Zariski site or the etale site of a scheme X. The Čech cohomology with values in some sheaf
$$
\mathcal{F}
$$
is defined as
$$
\check H^n (X, \mathcal{F}) := \varinjlim_{\mathcal U} \check H^n(\mathcal U, \mathcal{F}).
$$
where the colimit runs over all coverings (with respect to the chosen topology) of X. Here
$$
\check H^n(\mathcal U, \mathcal F)
$$
is defined as above, except that the r-fold intersections of open subsets inside the ambient topological space are replaced by the r-fold fiber product
$$
\mathcal U^{\times^r_X} := \mathcal U \times_X \dots \times_X \mathcal U.
$$
As in the classical situation of topological spaces, there is always a map
$$
\check H^n(X, \mathcal F) \rightarrow H^n(X, \mathcal F)
$$
from Čech cohomology to sheaf cohomology. It is always an isomorphism in degrees n = 0 and 1, but may fail to be so in general. For the Zariski topology on a Noetherian separated scheme, Čech and sheaf cohomology agree for any quasi-coherent sheaf.
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passage: It is always an isomorphism in degrees n = 0 and 1, but may fail to be so in general. For the Zariski topology on a Noetherian separated scheme, Čech and sheaf cohomology agree for any quasi-coherent sheaf. For the étale topology, the two cohomologies agree for any étale sheaf on X, provided that any finite set of points of X are contained in some open affine subscheme. This is satisfied, for example, if X is quasi-projective over an affine scheme.
The possible difference between Čech cohomology and sheaf cohomology is a motivation for the use of hypercoverings: these are more general objects than the Čech nerve
$$
N_X \mathcal U : \dots \to \mathcal U \times_X \mathcal U \times_X \mathcal U \to \mathcal U \times_X \mathcal U \to \mathcal U.
$$
A hypercovering K∗ of X is a certain simplicial object in C, i.e., a collection of objects Kn together with boundary and degeneracy maps. Applying a sheaf
$$
\mathcal{F}
$$
to K∗ yields a simplicial abelian group
$$
\mathcal{F}(K_\ast)
$$
whose n-th cohomology group is denoted
$$
H^n(\mathcal F (K_\ast))
$$
.
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passage: The possible difference between Čech cohomology and sheaf cohomology is a motivation for the use of hypercoverings: these are more general objects than the Čech nerve
$$
N_X \mathcal U : \dots \to \mathcal U \times_X \mathcal U \times_X \mathcal U \to \mathcal U \times_X \mathcal U \to \mathcal U.
$$
A hypercovering K∗ of X is a certain simplicial object in C, i.e., a collection of objects Kn together with boundary and degeneracy maps. Applying a sheaf
$$
\mathcal{F}
$$
to K∗ yields a simplicial abelian group
$$
\mathcal{F}(K_\ast)
$$
whose n-th cohomology group is denoted
$$
H^n(\mathcal F (K_\ast))
$$
. (This group is the same as
$$
\check H^n(\mathcal U, \mathcal F)
$$
in case K∗ equals
$$
N_X \mathcal U
$$
.) Then, it can be shown that there is a canonical isomorphism
$$
H^n (X, \mathcal F) \cong \varinjlim_{K_*} H^n(\mathcal F(K_*)),
$$
where the colimit now runs over all hypercoverings.
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passage: (This group is the same as
$$
\check H^n(\mathcal U, \mathcal F)
$$
in case K∗ equals
$$
N_X \mathcal U
$$
.) Then, it can be shown that there is a canonical isomorphism
$$
H^n (X, \mathcal F) \cong \varinjlim_{K_*} H^n(\mathcal F(K_*)),
$$
where the colimit now runs over all hypercoverings.
### Examples
The most basic example of Čech cohomology is given by the case where the presheaf
$$
\mathcal{F}
$$
is a constant sheaf, e.g.
$$
\mathcal{F}=\mathbb{R}
$$
. In such cases, each
$$
q
$$
-cochain
$$
f
$$
is simply a function which maps every
$$
q
$$
-simplex to
$$
\mathbb{R}
$$
. For example, we calculate the first Čech cohomology with values in
$$
\mathbb{R}
$$
of the unit circle
$$
X=S^1
$$
. Dividing
$$
X
$$
into three arcs and choosing sufficiently small open neighborhoods, we obtain an open cover
$$
\mathcal{U}=\{U_0,U_1,U_2\}
$$
where
$$
U_i \cap U_j \ne \emptyset
$$
but
$$
U_0 \cap U_1 \cap U_2 = \emptyset
$$
.
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passage: For example, we calculate the first Čech cohomology with values in
$$
\mathbb{R}
$$
of the unit circle
$$
X=S^1
$$
. Dividing
$$
X
$$
into three arcs and choosing sufficiently small open neighborhoods, we obtain an open cover
$$
\mathcal{U}=\{U_0,U_1,U_2\}
$$
where
$$
U_i \cap U_j \ne \emptyset
$$
but
$$
U_0 \cap U_1 \cap U_2 = \emptyset
$$
.
Given any 1-cocycle
$$
f
$$
,
$$
\delta f
$$
is a 2-cochain which takes inputs of the form
$$
(U_i,U_i,U_i),(U_i,U_i,U_j),(U_j,U_i,U_i),(U_i,U_j,U_i)
$$
where
$$
i \ne j
$$
(since
$$
U_0 \cap U_1 \cap U_2 = \emptyset
$$
and hence
$$
(U_i,U_j,U_k)
$$
is not a 2-simplex for any permutation
$$
\{i,j,k\}=\{1,2,3\}
$$
).
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passage: Dividing
$$
X
$$
into three arcs and choosing sufficiently small open neighborhoods, we obtain an open cover
$$
\mathcal{U}=\{U_0,U_1,U_2\}
$$
where
$$
U_i \cap U_j \ne \emptyset
$$
but
$$
U_0 \cap U_1 \cap U_2 = \emptyset
$$
.
Given any 1-cocycle
$$
f
$$
,
$$
\delta f
$$
is a 2-cochain which takes inputs of the form
$$
(U_i,U_i,U_i),(U_i,U_i,U_j),(U_j,U_i,U_i),(U_i,U_j,U_i)
$$
where
$$
i \ne j
$$
(since
$$
U_0 \cap U_1 \cap U_2 = \emptyset
$$
and hence
$$
(U_i,U_j,U_k)
$$
is not a 2-simplex for any permutation
$$
\{i,j,k\}=\{1,2,3\}
$$
). The first three inputs give
$$
f(U_i,U_i)=0
$$
; the fourth gives
$$
\delta f(U_i,U_j,U_i)=f(U_j,U_i)-f(U_i,U_i)+f(U_i,U_j)=0 \implies f(U_j,U_i)=-f(U_i,U_j).
$$
Such a function is fully determined by the values of
$$
f(U_0,U_1),f(U_0,U_2),f(U_1,U_2)
$$
.
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passage: Thus,
$$
Z^1(\mathcal{U},\mathbb{R})=\{f \in C^1(\mathcal{U},\mathbb{R}) : f(U_i,U_i)=0, f(U_j,U_i)=-f(U_i,U_j)\} \cong \mathbb{R}^3.
$$
On the other hand, given any 1-coboundary
$$
f = \delta g
$$
, we have
$$
\begin{cases}
f(U_i,U_i)=g(U_i)-g(U_i)=0 & (i=0,1,2); \\
f(U_i,U_j)=g(U_j)-g(U_i)=-f(U_j,U_i) & (i \ne j)
\end{cases}
$$
However, upon closer inspection we see that
$$
f(U_0,U_1)+f(U_1,U_2)=f(U_0,U_2)
$$
and hence each 1-coboundary
$$
f
$$
is uniquely determined by
$$
f(U_0,U_1)
$$
and
$$
f(U_1,U_2)
$$
.
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passage: Thus,
$$
Z^1(\mathcal{U},\mathbb{R})=\{f \in C^1(\mathcal{U},\mathbb{R}) : f(U_i,U_i)=0, f(U_j,U_i)=-f(U_i,U_j)\} \cong \mathbb{R}^3.
$$
On the other hand, given any 1-coboundary
$$
f = \delta g
$$
, we have
$$
\begin{cases}
f(U_i,U_i)=g(U_i)-g(U_i)=0 & (i=0,1,2); \\
f(U_i,U_j)=g(U_j)-g(U_i)=-f(U_j,U_i) & (i \ne j)
\end{cases}
$$
However, upon closer inspection we see that
$$
f(U_0,U_1)+f(U_1,U_2)=f(U_0,U_2)
$$
and hence each 1-coboundary
$$
f
$$
is uniquely determined by
$$
f(U_0,U_1)
$$
and
$$
f(U_1,U_2)
$$
. This gives the set of 1-coboundaries:
$$
\begin{align}
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passage: $$
and
$$
f(U_1,U_2)
$$
. This gives the set of 1-coboundaries:
$$
\begin{align}
B^1(\mathcal{U},\mathbb{R})=\{f \in C^1(\mathcal{U},\mathbb{R}) : \ & f(U_i,U_i)=0, f(U_j,U_i)=-f(U_i,U_j), \\
&f(U_0,U_2)=f(U_0,U_1)+f(U_1,U_2)\} \cong \mathbb{R}^2.
\end{align}
$$
Therefore,
$$
\check{H}^1(\mathcal{U},\mathbb{R})=Z^1(\mathcal{U},\mathbb{R})/B^1(\mathcal{U},\mathbb{R}) \cong \mathbb{R}
$$
.
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passage: Since
$$
\mathcal{U}
$$
is a good cover of
$$
X
$$
, we have
$$
\check{H}^1(X,\mathbb{R}) \cong \mathbb{R}
$$
by Leray's theorem.
We may also compute the coherent sheaf cohomology of
$$
\Omega^1
$$
on the projective line
$$
\mathbb{P}^1_\mathbb{C}
$$
using the Čech complex. Using the cover
$$
\mathcal{U} = \{ U_1 = \text{Spec}(\Complex[y]), U_2 = \text{Spec}(\Complex[y^{-1}]) \}
$$
we have the following modules from the cotangent sheaf
$$
\begin{align}
&\Omega^1(U_1) = \Complex[y]dy \\
&\Omega^1(U_2) = \Complex \left [y^{-1} \right ]dy^{-1}
\end{align}
$$
If we take the conventions that
$$
dy^{-1} = -(1/y^2)dy
$$
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passage: We may also compute the coherent sheaf cohomology of
$$
\Omega^1
$$
on the projective line
$$
\mathbb{P}^1_\mathbb{C}
$$
using the Čech complex. Using the cover
$$
\mathcal{U} = \{ U_1 = \text{Spec}(\Complex[y]), U_2 = \text{Spec}(\Complex[y^{-1}]) \}
$$
we have the following modules from the cotangent sheaf
$$
\begin{align}
&\Omega^1(U_1) = \Complex[y]dy \\
&\Omega^1(U_2) = \Complex \left [y^{-1} \right ]dy^{-1}
\end{align}
$$
If we take the conventions that
$$
dy^{-1} = -(1/y^2)dy
$$
then we get the Čech complex
$$
0 \to \Complex[y]dy \oplus \Complex \left [y^{-1} \right ]dy^{-1} \xrightarrow{d^0} \Complex \left [y,y^{-1} \right ]dy \to 0
$$
Since
$$
d^0
$$
is injective and the only element not in the image of
$$
d^0
$$
is
$$
y^{-1}dy
$$
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passage: Using the cover
$$
\mathcal{U} = \{ U_1 = \text{Spec}(\Complex[y]), U_2 = \text{Spec}(\Complex[y^{-1}]) \}
$$
we have the following modules from the cotangent sheaf
$$
\begin{align}
&\Omega^1(U_1) = \Complex[y]dy \\
&\Omega^1(U_2) = \Complex \left [y^{-1} \right ]dy^{-1}
\end{align}
$$
If we take the conventions that
$$
dy^{-1} = -(1/y^2)dy
$$
then we get the Čech complex
$$
0 \to \Complex[y]dy \oplus \Complex \left [y^{-1} \right ]dy^{-1} \xrightarrow{d^0} \Complex \left [y,y^{-1} \right ]dy \to 0
$$
Since
$$
d^0
$$
is injective and the only element not in the image of
$$
d^0
$$
is
$$
y^{-1}dy
$$
we get that
$$
\begin{align}
&H^1(\mathbb{P}_{\Complex}^1,\Omega^1) \cong \Complex \\
&H^k(\mathbb{P}_{\Complex}^1,\Omega^1) \cong 0 \text{ for } k \neq 1
\end{align}
$$
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passage: then we get the Čech complex
$$
0 \to \Complex[y]dy \oplus \Complex \left [y^{-1} \right ]dy^{-1} \xrightarrow{d^0} \Complex \left [y,y^{-1} \right ]dy \to 0
$$
Since
$$
d^0
$$
is injective and the only element not in the image of
$$
d^0
$$
is
$$
y^{-1}dy
$$
we get that
$$
\begin{align}
&H^1(\mathbb{P}_{\Complex}^1,\Omega^1) \cong \Complex \\
&H^k(\mathbb{P}_{\Complex}^1,\Omega^1) \cong 0 \text{ for } k \neq 1
\end{align}
$$
## References
### Citation footnotes
### General references
-
-
-
Category: Cohomology theories
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|
passage: The μ-law algorithm (sometimes written mu-law, often abbreviated as u-law) is a companding algorithm, primarily used in 8-bit PCM digital telecommunications systems in North America and Japan. It is one of the two companding algorithms in the G.711 standard from ITU-T, the other being the similar A-law. A-law is used in regions where digital telecommunication signals are carried on E-1 circuits, e.g. Europe.
The terms PCMU, G711u or G711MU are used for G711 μ-law.
Companding algorithms reduce the dynamic range of an audio signal. In analog systems, this can increase the signal-to-noise ratio (SNR) achieved during transmission; in the digital domain, it can reduce the quantization error (hence increasing the signal-to-quantization-noise ratio). These SNR increases can be traded instead for reduced bandwidth for equivalent SNR.
At the cost of a reduced peak SNR, it can be mathematically shown that μ-law's non-linear quantization effectively increases dynamic range by 33 dB or bits over a linearly-quantized signal, hence 13.5 bits (which rounds up to 14 bits) is the most resolution required for an input digital signal to be compressed for 8-bit μ-law.
## Algorithm types
The μ-law algorithm may be described in an analog form and in a quantized digital form.
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https://en.wikipedia.org/wiki/%CE%9C-law_algorithm
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passage: At the cost of a reduced peak SNR, it can be mathematically shown that μ-law's non-linear quantization effectively increases dynamic range by 33 dB or bits over a linearly-quantized signal, hence 13.5 bits (which rounds up to 14 bits) is the most resolution required for an input digital signal to be compressed for 8-bit μ-law.
## Algorithm types
The μ-law algorithm may be described in an analog form and in a quantized digital form.
### Continuous
For a given input , the equation for μ-law encoding is
$$
F(x) = \sgn(x) \dfrac{\ln(1 + \mu |x|)}{\ln(1 + \mu)}, \quad -1 \leq x \leq 1,
$$
where in the North American and Japanese standards, and is the sign function. The range of this function is −1 to 1.
μ-law expansion is then given by the inverse equation:
$$
F^{-1}(y) = \sgn(y) \dfrac{(1 + \mu)^{|y|} - 1}{\mu}, \quad -1 \leq y \leq 1.
$$
### Discrete
The discrete form is defined in ITU-T Recommendation G.711.
G.711 is unclear about how to code the values at the limit of a range (e.g. whether +31 codes to 0xEF or 0xF0).
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https://en.wikipedia.org/wiki/%CE%9C-law_algorithm
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passage: ### Discrete
The discrete form is defined in ITU-T Recommendation G.711.
G.711 is unclear about how to code the values at the limit of a range (e.g. whether +31 codes to 0xEF or 0xF0).
However, G.191 provides example code in the C language for a μ-law encoder. The difference between the positive and negative ranges, e.g. the negative range corresponding to +30 to +1 is −31 to −2. This is accounted for by the use of 1's complement (simple bit inversion) rather than 2's complement to convert a negative value to a positive value during encoding.
+ Quantized μ-law algorithm 14-bit binary linear input code 8-bit compressed code +8158 to +4063 in 16 intervals of 256 0x80 + interval number +4062 to +2015 in 16 intervals of 128 0x90 + interval number +2014 to +991 in 16 intervals of 64 0xA0 + interval number +990 to +479 in 16 intervals of 32 0xB0 + interval number +478 to +223 in 16 intervals of 16 0xC0 + interval number +222 to +95 in 16 intervals of 8 0xD0 + interval number +94 to +31 in 16 intervals of 4 0xE0 + interval number +30 to +1 in 15 intervals of 2 0xF0 + interval number 0 0xFF −1 0x7F
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https://en.wikipedia.org/wiki/%CE%9C-law_algorithm
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passage: The difference between the positive and negative ranges, e.g. the negative range corresponding to +30 to +1 is −31 to −2. This is accounted for by the use of 1's complement (simple bit inversion) rather than 2's complement to convert a negative value to a positive value during encoding.
+ Quantized μ-law algorithm 14-bit binary linear input code 8-bit compressed code +8158 to +4063 in 16 intervals of 256 0x80 + interval number +4062 to +2015 in 16 intervals of 128 0x90 + interval number +2014 to +991 in 16 intervals of 64 0xA0 + interval number +990 to +479 in 16 intervals of 32 0xB0 + interval number +478 to +223 in 16 intervals of 16 0xC0 + interval number +222 to +95 in 16 intervals of 8 0xD0 + interval number +94 to +31 in 16 intervals of 4 0xE0 + interval number +30 to +1 in 15 intervals of 2 0xF0 + interval number 0 0xFF −1 0x7F −31 to −2 in 15 intervals of 2 0x70 + interval number −95 to −32 in 16 intervals of 4 0x60 + interval number −223 to −96 in 16 intervals of 8 0x50 + interval number −479 to −224 in 16 intervals of 16 0x40 + interval number −991 to −480 in 16 intervals of 32 0x30 + interval number −2015 to −992 in 16 intervals of 64 0x20 + interval number −4063 to −2016 in 16 intervals of 128 0x10 + interval number −8159 to −4064 in 16 intervals of 256 0x00 + interval number
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https://en.wikipedia.org/wiki/%CE%9C-law_algorithm
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passage: This is accounted for by the use of 1's complement (simple bit inversion) rather than 2's complement to convert a negative value to a positive value during encoding.
+ Quantized μ-law algorithm 14-bit binary linear input code 8-bit compressed code +8158 to +4063 in 16 intervals of 256 0x80 + interval number +4062 to +2015 in 16 intervals of 128 0x90 + interval number +2014 to +991 in 16 intervals of 64 0xA0 + interval number +990 to +479 in 16 intervals of 32 0xB0 + interval number +478 to +223 in 16 intervals of 16 0xC0 + interval number +222 to +95 in 16 intervals of 8 0xD0 + interval number +94 to +31 in 16 intervals of 4 0xE0 + interval number +30 to +1 in 15 intervals of 2 0xF0 + interval number 0 0xFF −1 0x7F −31 to −2 in 15 intervals of 2 0x70 + interval number −95 to −32 in 16 intervals of 4 0x60 + interval number −223 to −96 in 16 intervals of 8 0x50 + interval number −479 to −224 in 16 intervals of 16 0x40 + interval number −991 to −480 in 16 intervals of 32 0x30 + interval number −2015 to −992 in 16 intervals of 64 0x20 + interval number −4063 to −2016 in 16 intervals of 128 0x10 + interval number −8159 to −4064 in 16 intervals of 256 0x00 + interval number
## Implementation
The μ-law algorithm may be implemented in several ways:
Analog Use an amplifier with non-linear gain to achieve companding entirely in the analog domain.
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https://en.wikipedia.org/wiki/%CE%9C-law_algorithm
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passage: Non-linear ADC Use an analog-to-digital converter with quantization levels which are unequally spaced to match the μ-law algorithm.
Digital Use the quantized digital version of the μ-law algorithm to convert data once it is in the digital domain.
Software/DSP
Use the continuous version of the μ-law algorithm to calculate the companded values.
## Usage justification
μ-law encoding is used because speech has a wide dynamic range. In analog signal transmission, in the presence of relatively constant background noise, the finer detail is lost. Given that the precision of the detail is compromised anyway, and assuming that the signal is to be perceived as audio by a human, one can take advantage of the fact that the perceived acoustic intensity level or loudness is logarithmic by compressing the signal using a logarithmic-response operational amplifier (Weber–Fechner law). In telecommunications circuits, most of the noise is injected on the lines, thus after the compressor, the intended signal is perceived as significantly louder than the static, compared to an uncompressed source. This became a common solution, and thus, prior to common digital usage, the μ-law specification was developed to define an interoperable standard.
This pre-existing algorithm had the effect of significantly lowering the amount of bits required to encode a recognizable human voice in digital systems. A sample could be effectively encoded using μ-law in as little as 8 bits, which conveniently matched the symbol size of the majority of common computers.
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https://en.wikipedia.org/wiki/%CE%9C-law_algorithm
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passage: This pre-existing algorithm had the effect of significantly lowering the amount of bits required to encode a recognizable human voice in digital systems. A sample could be effectively encoded using μ-law in as little as 8 bits, which conveniently matched the symbol size of the majority of common computers.
μ-law encoding effectively reduced the dynamic range of the signal, thereby increasing the coding efficiency while biasing the signal in a way that results in a signal-to-distortion ratio that is greater than that obtained by linear encoding for a given number of bits.
The μ-law algorithm is also used in the .au format, which dates back at least to the SPARCstation 1 by Sun Microsystems as the native method used by the /dev/audio interface, widely used as a de facto standard for sound on Unix systems. The au format is also used in various common audio APIs such as the classes in the sun.audio Java package in Java 1.1 and in some C# methods.
This plot illustrates how μ-law concentrates sampling in the smaller (softer) values. The horizontal axis represents the byte values 0-255 and the vertical axis is the 16-bit linear decoded value of μ-law encoding.
## Comparison with A-law
The μ-law algorithm provides a slightly larger dynamic range than the A-law at the cost of worse proportional distortions for small signals. By convention, A-law is used for an international connection if at least one country uses it.
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https://en.wikipedia.org/wiki/%CE%9C-law_algorithm
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passage: In computability theory, the μ-operator, minimization operator, or unbounded search operator searches for the least natural number with a given property. Adding the μ-operator to the primitive recursive functions makes it possible to define all computable functions.
## Definition
Suppose that R(y, x1, ..., xk) is a fixed (k+1)-ary relation on the natural numbers. The μ-operator "μy", in either the unbounded or bounded form, is a "number theoretic function" defined from the natural numbers to the natural numbers. However, "μy" contains a predicate over the natural numbers, which can be thought of as a condition that evaluates to true when the predicate is satisfied and false when it is not.
The bounded μ-operator appears earlier in Kleene (1952) Chapter IX Primitive Recursive Functions, §45 Predicates, prime factor representation as:
"
$$
\mu y_{y<z} R(y). \ \ \mbox{The least} \ y<z \ \mbox{such that} \ R(y), \ \mbox{if} \ (\exists y)_{y<z} R(y); \ \mbox{otherwise}, \ z.
$$
" (p. 225)
Stephen Kleene notes that any of the six inequality restrictions on the range of the variable y is permitted, i.e. y < z, y ≤ z, w < y < z, w < y ≤ z, w ≤ y < z and w ≤ y ≤ z.
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https://en.wikipedia.org/wiki/%CE%9C_operator
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