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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Ext functors are the
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
s of the
Hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...
. Along with the
Tor functor In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to const ...
, Ext is one of the core concepts of
homological algebra Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...
, in which ideas from
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
are used to define invariants of algebraic structures. The cohomology of groups,
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s, and
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplicat ...
s can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies
extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...
of one module by another. In the special case of
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s, Ext was introduced by
Reinhold Baer Reinhold Baer (22 July 1902 – 22 October 1979) was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings and Baer groups. Biography Baer studied mechanical engineering f ...
(1934). It was named by
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to a ...
and Saunders MacLane (1942), and applied to topology (the universal coefficient theorem for cohomology). For modules over any ring, Ext was defined by
Henri Cartan Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of c ...
and Eilenberg in their 1956 book ''Homological Algebra''.


Definition

Let ''R'' be a ring and let ''R''-Mod be the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
of modules over ''R''. (One can take this to mean either left ''R''-modules or right ''R''-modules.) For a fixed ''R''-module ''A'', let ''T''(''B'') = Hom''R''(''A'', ''B'') for ''B'' in ''R''-Mod. (Here Hom''R''(''A'', ''B'') is the abelian group of ''R''-linear maps from ''A'' to ''B''; this is an ''R''-module if ''R'' is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
.) This is a
left exact functor In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much o ...
from ''R''-Mod to the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab is ...
Ab, and so it has right
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
s ''RiT''. The Ext groups are the abelian groups defined by :\operatorname_R^i(A,B)=(R^iT)(B), for an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''i''. By definition, this means: take any
injective resolution In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to de ...
:0 \to B \to I^0 \to I^1 \to \cdots, remove the term ''B'', and form the
cochain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
: :0 \to \operatorname_R(A,I^0) \to \operatorname_R(A,I^1) \to \cdots. For each integer ''i'', Ext(''A'', ''B'') is the
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
of this complex at position ''i''. It is zero for ''i'' negative. For example, Ext(''A'', ''B'') is the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of the map Hom''R''(''A'', ''I''0) → Hom''R''(''A'', ''I''1), which is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to Hom''R''(''A'', ''B''). An alternative definition uses the functor ''G''(''A'')=Hom''R''(''A'', ''B''), for a fixed ''R''-module ''B''. This is a contravariant functor, which can be viewed as a left exact functor from the
opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
(''R''-Mod)op to Ab. The Ext groups are defined as the right derived functors ''RiG'': :\operatorname_R^i(A,B)=(R^iG)(A). That is, choose any
projective resolution In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to d ...
:\cdots \to P_1 \to P_0 \to A \to 0, remove the term ''A'', and form the cochain complex: :0\to \operatorname_R(P_0,B)\to \operatorname_R(P_1,B) \to \cdots. Then Ext(''A'', ''B'') is the cohomology of this complex at position ''i''. Cartan and Eilenberg showed that these constructions are independent of the choice of projective or injective resolution, and that both constructions yield the same Ext groups. Moreover, for a fixed ring ''R'', Ext is a functor in each variable (contravariant in ''A'', covariant in ''B''). For a commutative ring ''R'' and ''R''-modules ''A'' and ''B'', Ext(''A'', ''B'') is an ''R''-module (using that Hom''R''(''A'', ''B'') is an ''R''-module in this case). For a non-commutative ring ''R'', Ext(''A'', ''B'') is only an abelian group, in general. If ''R'' is an
algebra over a ring In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
''S'' (which means in particular that ''S'' is commutative), then Ext(''A'', ''B'') is at least an ''S''-module.


Properties of Ext

Here are some of the basic properties and computations of Ext groups. *Ext(''A'', ''B'') ≅ Hom''R''(''A'', ''B'') for any ''R''-modules ''A'' and ''B''. *Ext(''A'', ''B'') = 0 for all ''i'' > 0 if the ''R''-module ''A'' is projective (for example,
free Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procur ...
) or if ''B'' is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
. *The converses also hold: **If Ext(''A'', ''B'') = 0 for all ''B'', then ''A'' is projective (and hence Ext(''A'', ''B'') = 0 for all ''i'' > 0). **If Ext(''A'', ''B'') = 0 for all ''A'', then ''B'' is injective (and hence Ext(''A'', ''B'') = 0 for all ''i'' > 0). *\operatorname^i_(A,B) = 0 for all ''i'' ≥ 2 and all abelian groups ''A'' and ''B''. *If ''R'' is a commutative ring and ''u'' in ''R'' is not a
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
, then ::\operatorname_R^i(R/(u),B)\cong\begin B & i=0\\ B/uB & i=1\\ 0 &\text\end :for any ''R''-module ''B''. Here ''B'' 'u''denotes the ''u''-torsion subgroup of ''B'', . Taking ''R'' to be the ring \Z of integers, this calculation can be used to compute \operatorname^1_(A,B) for any
finitely generated abelian group In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
''A''. *Generalizing the previous example, one can compute Ext groups when the first module is the quotient of a commutative ring by any
regular sequence In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection. Definitions F ...
, using the
Koszul complex In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its h ...
. For example, if ''R'' is the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variab ...
''k'' 'x''1,...,''x''''n''over a field ''k'', then Ext(''k'',''k'') is the
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
''S'' over ''k'' on ''n'' generators in Ext1. Moreover, Ext(''k'',''k'') is the polynomial ring ''R''; this is an example of Koszul duality. *By the general properties of derived functors, there are two basic
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the conte ...
s for Ext. First, a
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the contex ...
0 → ''K'' → ''L'' → ''M'' → 0 of ''R''-modules induces a long exact sequence of the form ::0 \to \mathrm_R(A,K) \to \mathrm_R(A,L) \to \mathrm_R(A,M) \to \mathrm^1_R(A,K) \to \mathrm^1_R(A,L) \to \cdots, :for any ''R''-module ''A''. Also, a short exact sequence 0 → ''K'' → ''L'' → ''M'' → 0 induces a long exact sequence of the form ::0 \to \mathrm_R(M,B) \to \mathrm_R(L,B) \to \mathrm_R(K,B) \to \mathrm^1_R(M,B) \to \mathrm^1_R(L,B) \to \cdots, :for any ''R''-module ''B''. *Ext takes direct sums (possibly infinite) in the first variable and products in the second variable to products. That is: ::\begin \operatorname^i_R \left(\bigoplus_\alpha M_\alpha,N \right) &\cong\prod_\alpha \operatorname^i_R (M_\alpha,N) \\ \operatorname^i_R \left(M,\prod_\alpha N_\alpha \right ) &\cong\prod_\alpha \operatorname^i_R (M,N_\alpha) \end * Let ''A'' be a finitely generated module over a commutative
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noeth ...
''R''. Then Ext commutes with localization, in the sense that for every
multiplicatively closed set In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold: * 1 \in S, * xy \in S for all x, y \in S. In other words, ''S'' is closed under taking finite ...
''S'' in ''R'', every ''R''-module ''B'', and every integer ''i'', ::S^ \operatorname_R^i(A, B) \cong \operatorname_^i \left (S^ A, S^ B \right ).


Ext and extensions


Equivalence of extensions

The Ext groups derive their name from their relation to extensions of modules. Given ''R''-modules ''A'' and ''B'', an extension of ''A'' by ''B'' is a short exact sequence of ''R''-modules :0\to B\to E\to A\to 0. Two extensions :0\to B\to E\to A\to 0 :0\to B\to E' \to A\to 0 are said to be equivalent (as extensions of ''A'' by ''B'') if there is a
commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
: : Note that the
Five lemma In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also wo ...
implies that the middle arrow is an isomorphism. An extension of ''A'' by ''B'' is called split if it is equivalent to the trivial extension :0\to B\to A\oplus B\to A\to 0. There is a one-to-one correspondence between
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of extensions of ''A'' by ''B'' and elements of Ext(''A'', ''B''). The trivial extension corresponds to the zero element of Ext(''A'', ''B'').


The Baer sum of extensions

The Baer sum is an explicit description of the abelian group structure on Ext(''A'', ''B''), viewed as the set of equivalence classes of extensions of ''A'' by ''B''. Namely, given two extensions :0\to B\xrightarrow E \xrightarrow A\to 0 and :0\to B\xrightarrow 'E'\xrightarrow 'A\to 0, first form the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
over A, :\Gamma = \left\. Then form the
quotient module In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by ...
:Y = \Gamma / \. The Baer sum of ''E'' and ''E′'' is the extension :0\to B\to Y\to A\to 0, where the first map is b \mapsto f(b), 0)= 0, f'(b))/math> and the second is (e, e') \mapsto g(e) = g'(e').
Up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
equivalence of extensions, the Baer sum is commutative and has the trivial extension as identity element. The negative of an extension 0 → ''B'' → ''E'' → ''A'' → 0 is the extension involving the same module ''E'', but with the homomorphism ''B'' → ''E'' replaced by its negative.


Construction of Ext in abelian categories

Nobuo Yoneda defined the abelian groups Ext(''A'', ''B'') for objects ''A'' and ''B'' in any
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abel ...
C; this agrees with the definition in terms of resolutions if C has enough projectives or
enough injectives In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. ...
. First, Ext(''A'',''B'') = HomC(''A'', ''B''). Next, Ext(''A'', ''B'') is the set of equivalence classes of extensions of ''A'' by ''B'', forming an abelian group under the Baer sum. Finally, the higher Ext groups Ext(''A'', ''B'') are defined as equivalence classes of ''n-extensions'', which are exact sequences :0\to B\to X_n\to\cdots\to X_1\to A\to 0, under the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
generated by the relation that identifies two extensions :\begin \xi : 0 &\to B\to X_n\to\cdots\to X_1\to A\to 0 \\ \xi': 0 &\to B\to X'_n\to\cdots\to X'_1\to A\to 0 \end if there are maps X_m \to X'_m for all ''m'' in so that every resulting square commutes, that is, if there is a
chain map A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. ...
ξ → ξ' which is the identity on ''A'' and ''B''. The Baer sum of two ''n''-extensions as above is formed by letting X''_1 be the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
of X_1 and X'_1 over ''A'', and X''_n be the pushout of X_n and X'_n under ''B''. Then the Baer sum of the extensions is :0\to B\to X''_n\to X_\oplus X'_\to\cdots\to X_2\oplus X'_2\to X''_1\to A\to 0.


The derived category and the Yoneda product

An important point is that Ext groups in an abelian category C can be viewed as sets of morphisms in a category associated to C, the
derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proc ...
''D''(C). The objects of the derived category are complexes of objects in C. Specifically, one has :\operatorname^i_(A,B) = \operatorname_(A,B , where an object of C is viewed as a complex concentrated in degree zero, and 'i''means shifting a complex ''i'' steps to the left. From this interpretation, there is a
bilinear map In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, ...
, sometimes called the Yoneda product: :\operatorname^i_(A,B) \times \operatorname^j_(B,C) \to \operatorname^_(A,C), which is simply the composition of morphisms in the derived category. The Yoneda product can also be described in more elementary terms. For ''i'' = ''j'' = 0, the product is the composition of maps in the category C. In general, the product can be defined by splicing together two Yoneda extensions. Alternatively, the Yoneda product can be defined in terms of resolutions. (This is close to the definition of the derived category.) For example, let ''R'' be a ring, with ''R''-modules ''A'', ''B'', ''C'', and let ''P'', ''Q'', and ''T'' be projective resolutions of ''A'', ''B'', ''C''. Then Ext(''A'',''B'') can be identified with the group of chain homotopy classes of chain maps ''P'' → ''Q'' 'i'' The Yoneda product is given by composing chain maps: :P\to Q to T +j By any of these interpretations, the Yoneda product is associative. As a result, \operatorname^*_R(A,A) is a
graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
, for any ''R''-module ''A''. For example, this gives the ring structure on
group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology lo ...
H^*(G, \Z), since this can be viewed as \operatorname^*_(\Z,\Z). Also by associativity of the Yoneda product: for any ''R''-modules ''A'' and ''B'', \operatorname^*_R(A,B) is a module over \operatorname^*_R(A,A).


Important special cases

*
Group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology lo ...
is defined by H^*(G,M)=\operatorname_^*(\Z, M), where ''G'' is a group, ''M'' is a
representation Representation may refer to: Law and politics * Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
of ''G'' over the integers, and \Z /math> is the
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...
of ''G''. *For an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
''A'' over a field ''k'' and an ''A''-
bimodule In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in ...
''M'', Hochschild cohomology is defined by ::HH^*(A,M)=\operatorname^*_ (A, M). *
Lie algebra cohomology In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to prope ...
is defined by H^*(\mathfrak g,M)=\operatorname^*_(k,M), where \mathfrak g is a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
over a commutative ring ''k'', ''M'' is a \mathfrak g-module, and U\mathfrak g is the
universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representat ...
. *For a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
''X'',
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when ...
can be defined as H^*(X, A) = \operatorname^*(\Z_X, A). Here Ext is taken in the abelian category of sheaves of abelian groups on ''X'', and \Z_X is the sheaf of
locally constant In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function. ...
\Z-valued functions. *For a commutative Noetherian
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic nu ...
''R'' with residue field ''k'', \operatorname^*_R(k,k) is the universal enveloping algebra of a graded Lie algebra π*(''R'') over ''k'', known as the homotopy Lie algebra of ''R''. (To be precise, when ''k'' has characteristic 2, π*(''R'') has to be viewed as an "adjusted Lie algebra".) There is a natural homomorphism of graded Lie algebras from the André–Quillen cohomology ''D''*(''k''/''R'',''k'') to π*(''R''), which is an isomorphism if ''k'' has characteristic zero.Avramov (2010), section 10.2.


See also

*
global dimension In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring ''A'' denoted gl dim ''A'', is a non-negative integer or infinity which is a homological invariant ...
* bar resolution *
Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic i ...
*
Grothendieck local duality In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves. Statement Suppose that ''R'' is a Cohen–Macaulay local ring of dimension ''d'' ...


Notes


References

* * * * * * * *{{Citation, author1-last=Weibel , author1-first=Charles A. , author1-link=Charles Weibel , chapter=History of homological algebra , title=History of topology , pages=797–836 , publisher=North-Holland , location=Amsterdam , year=1999 , mr=1721123 , isbn=9780444823755 , chapter-url= http://sites.math.rutgers.edu/~weibel/HA-history.pdf Homological algebra Binary operations