Subsequently there were further technical extensions (for example in Godement's book), and areas of application. For example, sheaves were applied to transformation groups; as an inspiration to homology theory in the form of Borel-Moore homology for locally compact spaces; to representation theory in the Borel-Bott-Weil theorem; as well as becoming standard in algebraic geometry and complex manifolds. The particular needs of étale cohomology were more about reinterpreting sheaf in sheaf cohomology, than cohomologySee also:

Sheaf cohomology, Sheaf cohomology - Definitions, Sheaf cohomology - Applications, Sheaf cohomology - Euler characteristics

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Locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces. Let X be locally ringed space with structure sheaf OX; we want to define the tangent space Tx at the point x ∈ X. Take the local ring (stalk) Rx at the point x, with maximal ideal mx. Then kx := Rx/mx is a field and mx/mx2 is a vector space over that field (the cotangent space). The tangent space RxSee also:

Ringed space, Ringed space - Definition, Ringed space - Examples, Ringed space - Morphisms, Ringed space - Tangent spaces, Ringed space - OX modules

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The abelian group, together with group homomorphisms, form a category, the prototype of an abelian category. In this encyclopedia, we denote this category Ab. See category of abelian groups for a list of its properties. Many large abelian groups carry a natural topology, turning them into topological groups. ...

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Abelian group, Abelian group - Notation, Abelian group - Examples, Abelian group - Multiplication table, Abelian group - Properties, Abelian group - Finite abelian groups, Abelian group - List of small abelian groups, Abelian group - Relation to other mathematical topics, Abelian group - A note on the typography

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Specifically, a left module over the ring R consists of an abelian group (M, +) and an operation R × M → M (called scalar multiplication, usually just written by juxtaposition, i.e. as rx for r in R and x in M) such that For all r,s in R, x,y in M, we have r(x+y) = rx+ry (r+s)x = rx+sx (rs)x = rSee also:

Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations

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An arbitrary topological space X can be considered a locally ringed space by taking OX to be the sheaf of real-valued (or complex-valued) continuous functions on open subsets of X (there may exist continuous functions over open subsets of X which are not the restriction of any continuous function over X). The stalk at a point x can be thought of as the set of all germs of continuous functions at x; this is a local ring with maximal ideal consisting of ...

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Ringed space, Ringed space - Definition, Ringed space - Examples, Ringed space - Morphisms, Ringed space - Tangent spaces, Ringed space - OX modules

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Formally, a ringed space is a topological space X together with a sheaf of commutative rings OX on X. The sheaf OX is called the structure sheaf of X. A locally ringed space is a ringed space (X, OX) such that all stalks of OX are local rings (i.e. they have unique maximal ideals). Note that it is not required that OX(U) be a local ring for every open set U — in fact, ...

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Ringed space, Ringed space - Definition, Ringed space - Examples, Ringed space - Morphisms, Ringed space - Tangent spaces, Ringed space - OX modules

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A morphism f : a → b is called a monomorphism (or monic) if fg1 = fg2 implies g1 = g2 for all morphisms g1, g2 : x → a. an epimorphism (or epic) if g1f = g2f implies g1 = g2 for all morphisms g1, g2 : b → x.See also:

Category mathematics, Category mathematics - Definition, Category mathematics - Examples, Category mathematics - Types of morphisms, Category mathematics - Types of categories

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Groups are used throughout mathematics and the sciences, often to capture the internal symmetry of other structures, in the form of automorphism groups. An internal symmetry of a structure is usually associated with an invariant property; the set of transformations that preserve this invariant property, together with the operation of composition of transformations, form a group called a symmetry group. In Galois theory, which is the historical origin of the group concept, one uses groups to describe the symmetries of the equations satisfied by the solutions to a polynomial equation. The solvable groups are ...

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Group theory, Group theory - History, Group theory - Elementary introduction, Group theory - Some useful theorems, Group theory - Generalizations, Group theory - Miscellany

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In abstract algebra, we get some related structures which are similar to groups by relaxing some of the axioms given at the top of the article. If we eliminate the requirement that every element have an inverse, then we get a monoid. If we additionally do not require an identity either, then we get a semigroup. Alternatively, if we relax the requirement that the operation be associative while still requiring the possibility of division, then we get a loop. If we additionally do not require an identity, then we get a quasigroup. If we don't require any axioms of the bina ...

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Group theory, Group theory - History, Group theory - Elementary introduction, Group theory - Some useful theorems, Group theory - Generalizations, Group theory - Miscellany

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Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or R-submodule, to be more explicit) if, for any n in N and any r in R, the product rn is in N (or nr for a right module). If M and N are left R-modules, then a map f : M → N is a homomorphism of R-modules if, for any m, n in M and r, s in R, f(rmSee also:

Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations

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Finitely generated. A module M is finitely generated if there exist finitely many elements x1,...,xn in M such that every element of M is a linear combination of those elements with coefficients from the scalar ring R. Free. A free module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the scalar ring R. These are the modules that behave very much like vector spaces. Projective. Projective modules are direct summands of fre ...

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Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations

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Every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. In particular, the integers Z form an abelian group under addition, as do the integers modulo n Z/nZ. The real numbers form an abelian group under addition, as do the n ...

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Abelian group, Abelian group - Notation, Abelian group - Examples, Abelian group - Multiplication table, Abelian group - Properties, Abelian group - Finite abelian groups, Abelian group - List of small abelian groups, Abelian group - Relation to other mathematical topics, Abelian group - A note on the typography

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There are two main notational conventions for abelian groups -- additive and multiplicative. The multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules. When studying abelian groups in their own right, the additive notation is usually used. ...

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Abelian group, Abelian group - Notation, Abelian group - Examples, Abelian group - Multiplication table, Abelian group - Properties, Abelian group - Finite abelian groups, Abelian group - List of small abelian groups, Abelian group - Relation to other mathematical topics, Abelian group - A note on the typography

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Formally, given two categories C and D, an equivalence of categories consists of a functor F : C -> D, a functor G : D -> C, and two natural isomorphisms η: FG->ID and ε: IC->GF. Here FG: D->D and GF: C->C, denote the respective compositions of F and G, and IC< ...

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Equivalence of categories, Equivalence of categories - Definition, Equivalence of categories - Equivalent characterizations, Equivalence of categories - Examples, Equivalence of categories - Properties

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To verify that a certain finite group is indeed abelian, a table (matrix) can be drawn up in the similar fashion to a multiplication table, where, if the group is G = {g1 = e, g2, ..., gn} under the operation ⋅, the (i, j)'th entry of this table contains the product gi ⋅ gj. The group is abelian if and only if this table is symmetric about the mai ...

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Abelian group, Abelian group - Notation, Abelian group - Examples, Abelian group - Multiplication table, Abelian group - Properties, Abelian group - Finite abelian groups, Abelian group - List of small abelian groups, Abelian group - Relation to other mathematical topics, Abelian group - A note on the typography

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The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order. This is a special application of the fundamental theorem of finitely generated abelian groups in the case when G has torsion-free rank equal to 0. Zmn is isomorphic to the direct product of Zm and Zn if and only if m and n are coprime. Therefore we can write any finite abelian group GSee also:

Abelian group, Abelian group - Notation, Abelian group - Examples, Abelian group - Multiplication table, Abelian group - Properties, Abelian group - Finite abelian groups, Abelian group - List of small abelian groups, Abelian group - Relation to other mathematical topics, Abelian group - A note on the typography

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If M is a left R-module, then the action of an element r in R is defined to be the map M → M that sends each x to rx (or xr in the case of a right module), and is necessarily a group endomorphism of the abelian group (M,+). The set of all group endomorphisms of M is denoted EndZ(M) and forms a ring under addition and composition, and sending a ring element r of R to its action actually define ...

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Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations

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Any ring R can be viewed as a preadditive category with a single object. With this understanding, a left R-module is nothing but a (covariant) additive functor from R to the category Ab of abelian groups. Right R-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C; these functors form a functor category C-Mod which is the natural gene ...

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Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations

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Extracted from the list of small groups is the following table of small abelian groups. Note that e.g. "3 × Z2" means that there are 3 subgroups of type Z2, while elsewhere the cross means direct product. ...

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Abelian group, Abelian group - Notation, Abelian group - Examples, Abelian group - Multiplication table, Abelian group - Properties, Abelian group - Finite abelian groups, Abelian group - List of small abelian groups, Abelian group - Relation to other mathematical topics, Abelian group - A note on the typography

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The first origins of sheaf theory are hard to pin down — they may be co-extensive with the idea of analytic continuation. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology. 1936 Eduard Čech introduces the nerve construction, for associating a simplicial complex to an open covering. 1938 Hassler Whitney gives a 'modern' definition of cohomology, summarizing the work since J. W. Alexander and Kolmogorov first defined cochainsSee also:

Sheaf mathematics, Sheaf mathematics - Introduction, Sheaf mathematics - The formal definition, Sheaf mathematics - Definition of a presheaf, Sheaf mathematics - The gluing axiom, Sheaf mathematics - Examples, Sheaf mathematics - Morphisms of sheaves, Sheaf mathematics - Stalks of a sheaf at a point and germs of functions, Sheaf mathematics - The étale space of a sheaf, Sheaf mathematics - Generalizations, Sheaf mathematics - History

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