nLab
Ext

under construction

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

In the context of homological algebra the right derived functor of the hom-functor is called the ExtExt-functor . It derives its name from the fact that the derived hom between abelian groups classifies abelian group extensions of AA by KK. (This is a special case of the general classificaton of principal ∞-bundles/∞-group extensions by general cohomology/group cohomology.)

Together with the Tor-functor it is one of the central objects of interest in homological algebra.

Given a abelian category 𝒜\mathcal{A} we may consider the hom-functor Hom 𝒜:𝒜 op×𝒜Hom_{\mathcal{A}} : \mathcal{A}^{op}\times \mathcal{A}\to Ab either as a functor in first or in second argument, and compute the corresponding right derived functors.

If they exist, the classical right derived functors of either functor agree and also agree with the homology of the mixed double complex obtained by taking simultaneously an injective resolution of the first contravariant argument and projective resolution of the second covariant argument. The last construction is called the balanced ExtExt.

Alternatively, one can consider the derived category D(𝒜)D(\mathcal{A}) and define

Ext p(X,A)Hom D(A)(X,A[p]) Ext^p(X,A) \coloneqq Hom_{D(A)}(X,A[p])

or define Ext iExt^i-groups as groups of abelian extensions of length ii, discussed below at Relation to extensions.

Definition

We give the definition following the discussion at derived functors in homological algebra.

Contravariant ExtExt on an ordinary object

Let 𝒜\mathcal{A} be an abelian category with enough projectives. And let A𝒜A \in \mathcal{A} be any object. Consider the contravariant hom-functor

Hom 𝒜(,A):𝒜 opAb. Hom_{\mathcal{A}}(-, A) : \mathcal{A}^{op} \to Ab \,.
Remark

This is a left exact functor.

Therefore to derive it by resolutions we need to consider injective resolutions in the opposite category 𝒜 op\mathcal{A}^{op}. But these are projective resolutions in 𝒜\mathcal{A} itself.

Definition

For X𝒜X \in \mathcal{A} any object and ((QX)X)Ch 0(𝒜)((Q X) \to X) \in Ch_{\bullet \geq 0}(\mathcal{A}) a projective resolution, and for nn \in \mathbb{N}, the nnth ExtExt-group of XX with coefficients in AA is the degree-nn cochain cohomology

Ext n(X,A)H (Hom 𝒜((QX) ,A)) Ext^n(X,A) \coloneqq H^\bullet ( Hom_{\mathcal{A}}((Q X)_\bullet, A))

of the cochain complex Hom((QX) ,A)Hom((Q X)_\bullet, A).

The following proposition expands a bit on the meaning of this definition. Write

[,]:Ch (𝒜) op×Ch (𝒜)Ch (Ab) [-,-] : Ch_{\bullet}(\mathcal{A})^{op} \times Ch_\bullet(\mathcal{A}) \to Ch_\bullet(Ab)

for the enriched hom of chain complexes.

Proposition

The nnth Ext-group is canonically identified with the 0-th homology of this enriched hom from the resolution QXQ X of XX to the nn-fold delooping/suspension chain complex of AA B nA=A[n]\mathbf{B}^n A = A[n] (concentrated on AA in degree nn):

Ext n(X,A)H 0[(QX),A[n]]; Ext^n(X,A) \simeq H_0 [(Q X), A[n] ] \,;

or equivalently, if we think of degree chain homology as the 0th homotopy group (under Dold-Kan correspondence) and write the nn-fold suspension/delooping of AA as B nA\mathbf{B}^n A:

Ext n(X,A)π 0[(QX),B nA]. Ext^n(X,A) \simeq \pi_0 [(Q X), \mathbf{B}^n A ] \,.
Proof

This is a special case of the general discussion at cochain cohomology.

By the discussion at internal hom of chain complexes, the 0-cycles of [(QX),B nA][(Q X), \mathbf{B}^n A ] are chain maps of the form

(QX) n+1 f n+1 0 n1 QX (QX) n f n A n1 QX (QX) n1 f n1 0 1 QX (QX) 1 f 1 0 0 QX (QX) 0 f 0 0. \array{ \vdots && \vdots \\ \downarrow && \downarrow \\ (Q X)_{n+1} &\stackrel{f_{n+1}}{\to}& 0 \\ \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} && \downarrow \\ (Q X)_{n} &\stackrel{f_{n}}{\to}& A \\ \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} && \downarrow \\ (Q X)_{n-1} &\stackrel{f_{n-1}}{\to}& 0 \\ \downarrow && \downarrow \\ \vdots && \vdots \\ \downarrow^{\mathrlap{\partial^{Q X}_1}} && \downarrow \\ (Q X)_1 &\stackrel{f_1}{\to}& 0 \\ \downarrow^{\mathrlap{\partial^{Q X}_0}} && \downarrow \\ (Q X)_0 &\stackrel{f_0}{\to}& 0 } \,.

By the definition of chain maps this are precisely those morphisms f n:(QX) nAf_n : (Q X)_n \to A such that

d nf nf n n QX=0 d^n f_n \coloneqq f_n \circ \partial^{Q X}_n = 0

which exhibits f nf_n as a degree-nn cochain in the cochain complex Hom((QX) ,A)Hom((Q X)_\bullet, A).

Similarly, the 00-boundaries in [(QX),B nA][(Q X), \mathbf{B}^n A] come from chain homotopies λ:0f\lambda : 0 \Rightarrow f:

(QX) n+1 f n+1 0 n1 QX λ n+1 (QX) n f n A n1 QX λ n (QX) n1 f n1 0 1 QX λ 2 (QX) 1 f 1 0 0 QX λ 1 (QX) 0 f 0 0. \array{ \vdots && \vdots \\ \downarrow && \downarrow \\ (Q X)_{n+1} &\stackrel{f_{n+1}}{\to}& 0 \\ \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} &\nearrow_{\mathrlap{\lambda_{n+1}}}& \downarrow \\ (Q X)_{n} &\stackrel{f_{n}}{\to}& A \\ \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} &\nearrow_{\mathrlap{\lambda_{n}}}& \downarrow \\ (Q X)_{n-1} &\stackrel{f_{n-1}}{\to}& 0 \\ \downarrow && \downarrow \\ \vdots && \vdots \\ \downarrow^{\mathrlap{\partial^{Q X}_1}} &\nearrow_{\mathrlap{\lambda_{2}}}& \downarrow \\ (Q X)_1 &\stackrel{f_1}{\to}& 0 \\ \downarrow^{\mathrlap{\partial^{Q X}_0}} &\nearrow_{\mathrlap{\lambda_{1}}}& \downarrow \\ (Q X)_0 &\stackrel{f_0}{\to}& 0 } \,.

in that

f n=λ n n1 QX. f_n = \lambda_n \circ \partial^{Q X}_{n-1} \,.

This are precisely the degree-nn coboundaries in Hom((QX) ,A)Hom((Q X)_\bullet, A).

Remark

This perspective on the Ext nExt^n-group as being the homotopy classes of maps out of (a resolution of) XX to B nA\mathbf{B}^n A is made more manifest in the discussion in terms of derived categories below. It connects ExtExt-groups and their relation to extensions to the general context of cohomology and ∞-group extensions. See at abelian sheaf cohomology for more on this.

In terms of derived categories

(…)

(Kashiwara-Shapira)

(…)

Properties

Relation to extensions

We discuss how the group Ext n(X,A)Ext^n(X,A) is identified with the group of extensions of XX by B n1A=A[n1]\mathbf{B}^{n-1} A = A[n-1]. In particular for n=1n = 1 and 𝒜=\mathcal{A} = Ab this means that Ext 1(X,A)Ext^1(X,A) classified ordinary group extensions of XX by AA.

This is the relation that the name ”ExtExt” derives from. At infinity-group extension is discussed how this relation is a special case of the more general relation that identifies derived hom-spaces H(X,B n+1A)\mathbf{H}(X,\mathbf{B}^{n+1} A) with B nA\mathbf{B}^n A-principal ∞-bundles over XX.

1-Extensions over single objects

Definition

For X,A𝒜X,A \in \mathcal{A} two objects, an extension of XX by AA is a short exact sequence

0APX0. 0 \to A \to P \to X \to 0 \,.

An homomorphism of two such extensions P 1P_1 and P 2P_2 is a morphism P 1P 2P_1 \to P_2 in 𝒜\mathcal{A} fitting into a commuting diagram of the form

P 1 A X P 2. \array{ && P_1 \\ & \nearrow & & \searrow \\ A && \downarrow && X \\ & \searrow & & \nearrow \\ && P_2 } \,.
Remark

All these homomophisms are necessarily isomorphisms, by the short five lemma.

Definition

Write Ext(X,A)Ext(X,A) for the set of isomorphism classes of such extensions.

Proposition

Under Baer sum Ext(X,A)Ext(X,A) becomes an abelian group.

Definition

Define a morphism

ExtractCocycle:Extensions(X,A)Ext 1(X,A) ExtractCocycle : Extensions(X,A) \to Ext^1(X,A)

by the following construction:

choose a projective presentation NQXN \hookrightarrow Q \to X of XX. Then for APXA \to P \to X an extension consider the diagram

N Q X σ N σ = A P X, \array{ N &\to& Q &\to& X \\ \downarrow^{\mathrlap{\sigma|_N}} && \downarrow^{\mathrlap{\sigma}} && \downarrow^{\mathrlap{=}} \\ A &\to& P &\to& X } \,,

where

  • σ\sigma is any choice of lifts of QAQ \to A through PAP \to A, which exists by definition since PP is a projective object,

  • σ N\sigma|_N is the induces morphism on the fibers, which exists by the exactness of the two sequences.

By prop. 5 σ N\sigma|_N represents an element in [σ N]Ext 1(X,A) [\sigma|_N] \in Ext^1(X,A). Let this be the image of the map to be defined:

ExtractCocycle:(APX)[σ N]. ExtractCocycle : (A \to P \to X) \mapsto [\sigma|_N] \,.

This definition is independent of the choice of PP and σ\sigma involved.

Proposition

The map from def. 4 is a natural isomorphism of abelian groups

Ext(X,A)Ext 1(X,A). Ext(X,A) \stackrel{\simeq}{\to} Ext^1(X,A) \,.

Higher extensions over general chain complexes

(…)

Given [g]Hom(X,A[n])[g] \in \mathbb{R}Hom(X, A[n]).

Let QXQ \to X be a projective resolution. Let g:QA[n]g : Q \to A[n] be a representative of [g][g].
Consider the pullback

P cone(0A[n]) Q g A[n] X \array{ P &\to& cone(0 \to A[n]) \\ \downarrow && \downarrow \\ Q &\stackrel{g}{\to}& A[n] \\ \downarrow \\ X }

(…)

Relation to group cohomology

For GG a discrete group with [G]\mathbb{Z}[G] its group ring, over the integers, and for NN a linear GG-representation, hence a [G]\mathbb{Z}[G]-module, the group cohomology of GG with coefficients in NN is

Ext [G]Mod (,N). Ext^\bullet_{\mathbb{Z}[G]Mod}(\mathbb{Z}, N) \,.

(…)

Localization

(…)

Techniques for constructing Ext nExt^n

We discuss some facts helpful for the construction of Ext nExt^n-groups in certain situations.

Proposition

If X𝒜X \in \mathcal{A} is a projective object, then

Ext n(X,)=0 Ext^n(X, -) = 0

is the zero-functor for all n1n \geq 1.

Proof

The covariant hom-functor Hom(X,)Hom(X,-) is generally a left exact functor. By the construction of Ext nExt^n via projective resolutions, def. 1, it is sufficient to show that it is also a right exact functor if PP is projective. In fact, this is one of the equivalent characterizations of projective objects (ee the section projective object – in abelian categories – equivalent characterizations for details).

Proposition

For X,A𝒜X, A \in \mathcal{A} two objects, and

0NiPpX0 0 \to N \stackrel{i}{\hookrightarrow} P \stackrel{p}{\to} X \to 0

a short exact sequence with PP a projective object, hence exhibiting a projective presentation Xcoker(NP)X \simeq coker(N \hookrightarrow P) of XX, there is an exact sequence

0Hom(X,A)Hom(p,A)Hom(P,A)Hom(i,A)Hom(N,A)Ext 1(X,A)0 0 \to Hom(X,A) \stackrel{Hom(p,A)}{\to} Hom(P, A) \stackrel{Hom(i,A)}{\to} Hom(N,A) \to Ext^1(X,A) \to 0

exhibiting Ext 1(X,A)Ext^1(X,A) as the cokernel of Hom(i,A)Hom(i,A).

Applications in cohomology

Derived hom-functorssuch as the ExtExt on chain compelxes compute general notions of cohomology (see the discussion there). Here we list some specific incarnations of the ExtExt-construction in the context of cohomology.

Universal coefficient theorem

The universal coefficient theorem identifies, under suitable conditions, cohomology to the dual of homology up to Ext 1Ext^1-groups.

Various notions of cohomology expressed by ExtExt

Various notions of cohomology groups in the context of algebra can be expressed as ExtExt-groups, for instance:

homotopycohomologyhomology
[S n,][S^n,-][,A][-,A]()A(-) \otimes A
category theorycovariant homcontravariant homtensor product
homological algebraExtExtTor
enriched category theoryendendcoend
homotopy theoryderived hom space Hom(S n,)\mathbb{R}Hom(S^n,-)cocycles Hom(,A)\mathbb{R}Hom(-,A)derived tensor product () 𝕃A(-) \otimes^{\mathbb{L}} A

References

Standard texbook accounts include (see also most references at homological algebra)

A systematic discussion from the point of view of derived categories is in

Lecture notes include

  • Kiyoshi Igusa, 25 The Ext Functor (pdf)

section 4 of

as well as

  • Patrick Morandi, Ext Groups and Ext Functors, (pdf)

    (warning: the last section on resolutions for cocycles for group (abelian) exensions is not correct)

Original articles include

  • Saunders MacLane, Group Extensions by primary abelian groups, ransactions of the American Mathematical Society Vol. 95, No. 1 (Apr., 1960), pp. 1-16 (JSTOR)

See also

Revised on October 14, 2012 15:42:58 by Urs Schreiber (89.204.130.129)