nLab Ext

under construction

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

Contents

Idea

In the context of homological algebra the right derived functor of the hom-functor is called the $Ext$-functor . It derives its name from the fact that the derived hom between abelian groups classifies abelian group extensions of $A$ by $K$. (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 $\mathcal{A}$ we may consider the hom-functor $Hom_{\mathcal{A}} : \mathcal{A}^{op}\times \mathcal{A}\to$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 $Ext$.

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

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

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

Definition

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

Contravariant $Ext$ on an ordinary object

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

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

This is a left exact functor.

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

Definition

For $X \in \mathcal{A}$ any object and $((Q X) \to X) \in Ch_{\bullet \geq 0}(\mathcal{A})$ a projective resolution, and for $n \in \mathbb{N}$, the $n$th $Ext$-group of $X$ with coefficients in $A$ is the degree-$n$ cochain cohomology

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

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

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

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

for the enriched hom of chain complexes.

Proposition

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

$Ext^n(X,A) \simeq H_0 [(Q X), A[n] ] \,;$

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

$Ext^n(X,A) \simeq \pi_0 [(Q X), \mathbf{B}^n A ] \,.$
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 $[(Q X), \mathbf{B}^n A ]$ are chain maps of the form

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

By the definition of chain maps this are precisely those morphisms $f_n : (Q X)_n \to A$ such that

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

which exhibits $f_n$ as a degree-$n$ cochain in the cochain complex $Hom((Q X)_\bullet, A)$.

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

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

in that

$f_n = \lambda_n \circ \partial^{Q X}_{n-1} \,.$

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

Remark

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

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Properties

Relation to extensions

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

This is the relation that the name “$Ext$” 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 $\mathbf{H}(X,\mathbf{B}^{n+1} A)$ with $\mathbf{B}^n A$-principal ∞-bundles over $X$.

1-Extensions over single objects

Definition

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

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

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

$\array{ && P_1 \\ & \nearrow & & \searrow \\ A && \downarrow && X \\ & \searrow & & \nearrow \\ && P_2 } \,.$
Remark

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

Definition

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

Proposition

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

Definition

Define a morphism

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

by the following construction:

choose a projective presentation $N \hookrightarrow Q \to X$ of $X$. Then for $A \to P \to X$ an extension consider the diagram

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

where

• $\sigma$ is any choice of lifts of $Q \to A$ through $P \to A$, which exists by definition since $P$ is a projective object,

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

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

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

This definition is independent of the choice of $P$ and $\sigma$ involved.

Proposition

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

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

Higher extensions over general chain complexes

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

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

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

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Relation to group cohomology

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

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

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Techniques for constructing $Ext^n$

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

Proposition

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

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

is the zero-functor for all $n \geq 1$.

Proof

The covariant hom-functor $Hom(X,-)$ is generally a left exact functor. By the construction of $Ext^n$ via projective resolutions, def. 1, it is sufficient to show that it is also a right exact functor if $P$ 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 $X, A \in \mathcal{A}$ two objects, and

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

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

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

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

Applications in cohomology

Derived hom-functorssuch as the $Ext$ on chain compelxes compute general notions of cohomology (see the discussion there). Here we list some specific incarnations of the $Ext$-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 $Ext^1$-groups.

Various notions of cohomology expressed by $Ext$

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

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

$Ext^\bullet_{\mathbb{Z}[G]Mod}(\mathbb{Z}, N) \,.$
• For $A$ an associative algebra over some field $k$ and $N$ an $A$-bimodule, hence an $A \otimes A^{op}$-module,

$Ext^\bullet_{(A \otimes A^{op})Mod}(A, N)$

is the Hochschild cohomology of $A$ with coefficients in $N$.

• For $\mathfrak{g}$ a Lie algebra with universal enveloping algebra $\mathcal{U}(\mathcal{g})$ and $N$ a Lie algebra module, hence an $\mathcal{U}(\mathfrak{g})$-module, the Lie algebra cohomology of $\mathfrak{g}$ with coefficients in $N$ is

$Ext^\bullet_{\mathcal{U}(\mathfrak{g}) Mod}(\mathcal{U}(\mathfrak{g}), N) \,.$
homotopycohomologyhomology
$[S^n,-]$$[-,A]$$(-) \otimes A$
category theorycovariant homcontravariant homtensor product
homological algebraExtExtTor
enriched category theoryendendcoend
homotopy theoryderived hom space $\mathbb{R}Hom(S^n,-)$cocycles $\mathbb{R}Hom(-,A)$derived tensor product $(-) \otimes^{\mathbb{L}} A$

References

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)