NationMaster - Encyclopedia: Ext functor

In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ... In mathematics, specifically in category theory, Hom-sets, i. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...

1 Definition and computation
2 Properties of Ext
3 Ext and extensions
- 3.1 Ext in abelian categories
4 Ring structure and module structure on specific Exts
5 Interesting examples
6 Reference

Definition and computation

Let $R$ be a ring and let $Mod R$ be the category of modules over R. Let $B$ be in $Mod R$ and set $T(B) = operatorname{Hom}_{mathrm{Mod}_R}(A,B)$ , for fixed $A$ in $Mod R$ . (This is a left exact functor and thus has right derived functors $R n T$ ). To this end, define In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In homological algebra, an exact functor is one which preserves exact sequences. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...

$operatorname{Ext}_R^n(A,B)=(R^nT)(B),$

i.e., take an injective resolution In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. ...

$J(B)leftarrow Bleftarrow 0,$

compute

$operatorname{Hom}_{mathrm{Mod}_R}(A,J(B))leftarrowoperatorname{Hom}_{mathrm{Mod}_R}(A,B)leftarrow0,$

and take the cohomology of this complex. In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ...

Similarly, we can view the functor $G(A)=operatorname{Hom}_{mathrm{Mod}_R}(A,B)$ for a fixed module B as a contravariant left exact functor, and thus we also have right derived functors $R n G$ by instead of the injective resolution used above choosing a projective resolution $P (B)$ , and proceeding dually by calculating from Contravariant is a mathematical term with a precise definition in tensor analysis. ... In homological algebra, an exact functor is one which preserves exact sequences. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ... In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). ...

$P(A)rightarrow Arightarrow 0,$

compute

$operatorname{Hom}_{mathrm{Mod}_R}(P(A),B)leftarrowoperatorname{Hom}_{mathrm{Mod}_R}(A,B)leftarrow0,$

and then take the cohomology.

These two constructions turn out to yield isomorphic results, and so both may be used for calculation of Ext. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

Properties of Ext

The Ext functor exhibits some convenient properties, useful in computations.

$operatorname{Ext}^i_{mathrm{Mod}_R}(A,B)=0$ for $i > 0$ if either $B$ is injective or $A$ is projective.

The inverse also holds: if $operatorname{Ext}^1_{mathrm{Mod}_R}(A,B)=0$ for all $A$ , then $operatorname{Ext}^i_{mathrm{Mod}_R}(A,B)=0$ for all $A$ and $B$ is injective, and if $operatorname{Ext}^1_{mathrm{Mod}_R}(A,B)=0$ for all $B$ , then $operatorname{Ext}^i_{mathrm{Mod}_R}(A,B)=0$ for all $B$ and $A$ is projective.

$operatorname{Ext}^n_{mathrm{Mod}_R}(bigoplus_alpha A_alpha,B)congprod_alphaoperatorname{Ext}^n_{mathrm{Mod}_R}(A_alpha,B)$

$operatorname{Ext}^n_{mathrm{Mod}_R}(A,prod_beta B_beta)congprod_betaoperatorname{Ext}^n_{mathrm{Mod}_R}(A,B_beta)$

Ext and extensions

Ext functors derive their name from the relationship to extensions. Given $R$ -modules $A$ and $B$ , there is a bijective correspondence between equivalence classes of extensions In mathematics, for G a group or algebra over a field, or other algebraic structure, G′ is an extension of G if there is an exact sequence . See also central extension, extension problem, field extension. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

$0rightarrow Brightarrow Crightarrow Arightarrow 0$

of $A$ by $B$ and elements of

$operatorname{Ext}_R^1(A,B).$

Given two extensions

$0rightarrow Brightarrow Crightarrow Arightarrow 0$ and

$0rightarrow Brightarrow C'rightarrow Arightarrow 0$

we can construct the Baer sum, by forming the pullback $Γ$ of $Crightarrow A$ and $C'rightarrow A$ . We form the quotient $Y = Γ / Δ$ , with $Delta={(-b,b):bin B}$ . The extension This article needs to be cleaned up to conform to a higher standard of quality. ... In category theory, there is a general definition of subobject extending the idea of subset and subgroup. ...

$0rightarrow Brightarrow Yrightarrow Arightarrow 0$

thus formed is called the Baer sum of the extensions $C$ and $C'$ .

The Baer sum ends up being an abelian group operation on the set of equivalence classes, with the extension In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...

$0rightarrow Brightarrow Aoplus Brightarrow Arightarrow 0$

acting as the identity.

Ext in abelian categories

This identification enables us to define $operatorname{Ext}^1_{mathcal{C}}(A,B)$ even for abelian categories $mathcal{C}$ without reference to projectives and injectives. We simply take $operatorname{Ext}^1_{mathcal{C}}(A,B)$ to be the set of equivalence classes of extensions of $A$ by $B$ , forming an abelian group under the Baer sum. Similarily, we can define higher Ext groups $operatorname{Ext}^n_{mathcal{C}}(A,B)$ as equivalence classes of n-extensions In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ... In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). ... In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. ...

$0rightarrow Brightarrow X_nrightarrowcdotsrightarrow X_1rightarrow Arightarrow0$

under the equivalence relation generated by the relation that identifies two extensions In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...

$0rightarrow Brightarrow X_nrightarrowcdotsrightarrow X_1rightarrow Arightarrow0$ and

$0rightarrow Brightarrow X'_nrightarrowcdotsrightarrow X'_1rightarrow Arightarrow0$

if there are maps $X_mrightarrow X'_m$ for all $m$ in $1,2,.., n$ so that every resulting square commutes. In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

The Baer sum of the two n-extensions above is formed by letting $X'' 1$ be the pullback of $X 1$ and $X' 1$ over $A$ , and $Y n$ be the quotient of the pushout of $X n$ and $X' n$ under $B$ by the skew diagonal, as above. Then we define the Baer sum of the extensions to be This article needs to be cleaned up to conform to a higher standard of quality. ...

$0rightarrow Brightarrow Y_nrightarrow X_{n-1}oplus X'_{n-1}rightarrowcdotsrightarrow X_2oplus X'_2rightarrow X''_1rightarrow Arightarrow0.$

Ring structure and module structure on specific Exts

One more very useful way to view the Ext functor is this: when an element of $operatorname{Ext}^n_{mathrm{Mod}_R}(A,B)$ is considered as an equivalence class of maps $f: P_nrightarrow B$ for a projective resolution $P *$ of $A$ ; so, then we can pick a long exact sequence $Q *$ ending with $B$ and lift the map $f$ using the projectivity of the modules $P m$ to a chain map $f_*: P_*rightarrow Q_*$ of degree -n. It turns out that homotopy classes of such chain maps correspond precisely to the equivalence classes in the definition of Ext above. In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). ... In mathematics, a chain complex is a construct originally used in the field of algebraic topology. ... In mathematics, a chain complex is a construct originally used in the field of algebraic topology. ...

Under sufficiently nice circumstances, such as when the ring $R$ is a group ring, or a k-algebra, for a field k or even a noetherian ring k, we can impose a ring structure on $operatorname{Ext}^*_{mathrm{Mod}_R}(k,k)$ . The multiplication has quite a few equivalent interpretations, corresponding to different interpretations of the elements of $operatorname{Ext}^*_{mathrm{Mod}_R}(k,k)$ . Look up ring in Wiktionary, the free dictionary. ... In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ...

One interpretation is in terms of these homotopy classes of chain maps. Then the product of two elements is precisely the composition of the corresponding representatives. We can choose a single resolution of $k$ , and do all the calculations inside $operatorname{Hom}_{mathrm{Mod}_R}(P_*,P_*)$ , which is a differential graded algebra, with homology precisely $operatorname{Ext}_{mathrm{Mod}_R}(k,k)$ .

Another interpretation, not in fact relying on the existence of projective or injective modules is that of Yoneda splices. Then we take the viewpoint above that an element of $operatorname{Ext}^n_{mathrm{Mod}_R}(A,B)$ is an exact sequence starting in $A$ and ending in $B$ . This is then spliced with an element in $operatorname{Ext}^m_{mathrm{Mod}_R}(B,C)$ , by replacing

$rightarrow X_1rightarrow Brightarrow 0$ and $0rightarrow Brightarrow Y_nrightarrow$

with

$rightarrow X_1rightarrow Y_nrightarrow$

where the middle arrow is the composition of the functions $X_1rightarrow B$ and $Brightarrow Y_n$ .

These viewpoints turn out to be equivalent whenever both make sense.

Using similar interpretations, we find that $operatorname{Ext}_{mathrm{Mod}_R}^*(k,M)$ is a module over $operatorname{Ext}^*_{mathrm{Mod}_R}(k,k)$ , again for sufficiently nice situations. Look up module in Wiktionary, the free dictionary. ...

Interesting examples

If $mathbb ZG$ is the integral group ring for a group $G$ , then $operatorname{Ext}^*_{mathrm{Mod}_{mathbb ZG}}(mathbb Z,M)$ is the group cohomology $H * (G, M)$ with coefficients in $M$ . In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described... Look up group in Wiktionary, the free dictionary. ... In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. ...

For $mathbb F_p$ the finite field on $p$ elements, we also have that $H^*(G,M)=operatorname{Ext}^*_{mathrm{Mod}_{mathbb F_pG}}(mathbb F_p,M)$ , and it turns out that the group cohomology doesn't depend on the base ring chosen.

If $A$ is a $k$ -algebra, then $operatorname{Ext}^*_{mathrm{Mod}_{Aotimes_k A^{op}}}(A,M)$ is the Hochschild cohomology $operatorname{HH}^*(A,M)$ with coefficients in the module M. In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, Hochschild homology is a homology theory for associative algebras over rings. ...

If $A$ is chosen to be the universal enveloping algebra for a Lie algebra $mathfrak g$ , then $operatorname{Ext}^*_{mathrm{Mod}_R}(A,M)$ is the Lie algebra cohomology $operatorname{H}^*(mathfrak g,M)$ with coefficients in the module M. In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... Lie algebra cohomology is a cohomology theory for Lie algebras. ...

Reference

An introduction to homological algebra by Charles A. Weibel, ISBN 0-521-55987-1

Categories: Homological algebra | Binary operations

Results from FactBites:

Ext functor - Wikipedia, the free encyclopedia (142 words)
	In mathematics, the Ext functors of homological algebra are derived functors of Hom functors.
	Ext functors take their name from their relationship to extensions.
	Given R-modules A and B, there is a bijective correspondence between equivalence classes of extensions of A by B and elements of

NationMaster - Encyclopedia: Derived functor (1358 words)
	The functor which assigns to each such sheaf L the group L(X) of global sections is left exact, and the right derived functors are the sheaf cohomology functors, usually written as H Derived functors and the long exact sequences are "natural" in several technical senses.
	Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps.
	Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.

More results at FactBites »

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