Ext functor

Properties of Ext

Here are some of the basic properties and computations of ${\displaystyle \mathrm{Ext}}$ groups.

• ${\displaystyle {\mathrm{Ext}}_R^0(A,B)\cong {\mathrm{Hom}}_R(A,B)}$ for any ${\displaystyle R}$-modules ${\displaystyle A}$ and ${\displaystyle B}$.
• ${\displaystyle {\mathrm{Ext}}_R^i(A,B)=0}$ for all ${\displaystyle i>0}$ if the ${\displaystyle R}$-module ${\displaystyle A}$ is projective (for example, free) or if ${\displaystyle B}$ is injective.
• The converses also hold:
  • If Ext¹
    _R(A, B) = 0 for all B, then A is projective (and hence Extⁱ
    _R(A, B) = 0 for all i > 0).
  • If Ext¹
    _R(A, B) = 0 for all A, then B is injective (and hence Extⁱ
    _R(A, B) = 0 for all i > 0).
• ${\displaystyle {\mathrm{Ext}}_{\mathbb{Z}}^i(A,B)=0}$ for all ${\displaystyle i\ge 2}$ and all abelian groups ${\displaystyle A}$ and ${\displaystyle B}$.
• Generalizing the previous example, ${\displaystyle {\mathrm{Ext}}_R^i(A,B)=0}$ for all ${\displaystyle i\ge 2}$ if ${\displaystyle R}$ is a principal ideal domain.
• If ${\displaystyle R}$ is a commutative ring and ${\displaystyle u}$ in ${\displaystyle R}$ is not a zero divisor, then

$${\displaystyle {\mathrm{Ext}}_R^i(R/(u),B)\cong \{\begin{array}{ll}B[u]& i=0\\ B/uB& i=1\\ 0& \text{otherwise,}\end{array}}$$

for any ${\displaystyle R}$-module ${\displaystyle B}$. Here ${\displaystyle B[u]}$ denotes the ${\displaystyle u}$-torsion subgroup of ${\displaystyle B}$, ${\displaystyle \{x\in B:ux=0\}}$. Taking ${\displaystyle R}$ to be the ring ${\displaystyle \mathbb{Z}}$ of integers, this calculation can be used to compute ${\displaystyle {\mathrm{Ext}}_{\mathbb{Z}}^1(A,B)}$ for any finitely generated abelian group ${\displaystyle A}$.

• Generalizing the previous example, one can compute ${\displaystyle \mathrm{Ext}}$ groups when the first module is the quotient of a commutative ring by any regular sequence, using the Koszul complex. For example, if ${\displaystyle R}$ is the polynomial ring ${\displaystyle k[x_1,\dots ,x_n]}$ over a field ${\displaystyle k}$, then ${\displaystyle {\mathrm{Ext}}_R^{\ast }(k,k)}$ is the exterior algebra ${\displaystyle S}$ over ${\displaystyle k}$ on ${\displaystyle n}$ generators in ${\displaystyle {\mathrm{Ext}}^1}$. Moreover, ${\displaystyle {\mathrm{Ext}}_S^{\ast }(k,k)}$ is the polynomial ring ${\displaystyle R}$; this is an example of Koszul duality.
• By the general properties of derived functors, there are two basic exact sequences for ${\displaystyle \mathrm{Ext}}$. First, a short exact sequence ${\displaystyle 0\to K\to L\to M\to 0}$ of ${\displaystyle R}$-modules induces a long exact sequence of the form

${\displaystyle 0\to {\mathrm{H}\mathrm{o}\mathrm{m}}_R(A,K)\to {\mathrm{H}\mathrm{o}\mathrm{m}}_R(A,L)\to {\mathrm{H}\mathrm{o}\mathrm{m}}_R(A,M)\to {\mathrm{E}\mathrm{x}\mathrm{t}}_R^1(A,K)\to {\mathrm{E}\mathrm{x}\mathrm{t}}_R^1(A,L)\to \cdots ,}$

for any ${\displaystyle R}$-module ${\displaystyle A}$. Also, a short exact sequence ${\displaystyle 0\to K\to L\to M\to 0}$ induces a long exact sequence of the form

${\displaystyle 0\to {\mathrm{H}\mathrm{o}\mathrm{m}}_R(M,B)\to {\mathrm{H}\mathrm{o}\mathrm{m}}_R(L,B)\to {\mathrm{H}\mathrm{o}\mathrm{m}}_R(K,B)\to {\mathrm{E}\mathrm{x}\mathrm{t}}_R^1(M,B)\to {\mathrm{E}\mathrm{x}\mathrm{t}}_R^1(L,B)\to \cdots ,}$

for any ${\displaystyle R}$-module ${\displaystyle B}$.

• Ext takes direct sums (possibly infinite) in the first variable and products in the second variable to products. That is:

${\displaystyle \begin{array}{cc}{\mathrm{Ext}}_R^i\left(\underset{\alpha }{⨁}{M}_{\alpha },N\right)& \cong \prod _{\alpha }{\mathrm{Ext}}_R^i(M_{\alpha },N)\\ {\mathrm{Ext}}_R^i\left(M,\prod _{\alpha }{N}_{\alpha }\right)& \cong \prod _{\alpha }{\mathrm{Ext}}_R^i(M,N_{\alpha })\end{array}}$

• Let A be a finitely generated module over a commutative Noetherian ring R. Then Ext commutes with localization, in the sense that for every multiplicatively closed set S in R, every R-module B, and every integer i,

${\displaystyle S^{-1}{\mathrm{Ext}}_R^i(A,B)\cong {\mathrm{Ext}}_{S^{-1}R}^i\left(S^{-1}A,S^{-1}B\right).}$

Ext and extensions

Equivalence of extensions

The ${\displaystyle \mathrm{Ext}}$ groups derive their name from their relation to extensions of modules. Given ${\displaystyle R}$-modules ${\displaystyle A}$ and ${\displaystyle B}$, an extension of A by B is a short exact sequence of ${\displaystyle R}$-modules

${\displaystyle 0\to B\to E\to A\to 0.}$

Two extensions

${\displaystyle 0\to B\to E\to A\to 0}$

${\displaystyle 0\to B\to E^{\prime }\to A\to 0}$

are said to be equivalent (as extensions of ${\displaystyle A}$ by ${\displaystyle B}$) if there is a commutative diagram:

Note that the Five lemma implies that the middle arrow is an isomorphism. An extension of ${\displaystyle A}$ by ${\displaystyle B}$ is called split if it is equivalent to the trivial extension

${\displaystyle 0\to B\to A\oplus B\to A\to 0.}$

There is a one-to-one correspondence between equivalence classes of extensions of ${\displaystyle A}$ by ${\displaystyle B}$ and elements of ${\displaystyle {\mathrm{Ext}}_R^1(A,B)}$. This can be made precise as follows.

Proof

Fix a short exact sequence

${\displaystyle 0\to M\to P\to A\to 0}$

where ${\displaystyle P}$ is projective. Applying ${\displaystyle \mathrm{Hom}(-,B)}$ yields the long exact sequence

${\displaystyle \mathrm{Hom}(P,B)\to \mathrm{Hom}(M,B)\stackrel{\delta }{\to }\mathrm{Ext}(A,B)\to 0.}$

Given ${\displaystyle x\in \mathrm{Ext}(A,B)}$, choose ${\displaystyle \beta \in \mathrm{Hom}(M,B)}$ such that ${\displaystyle \delta (\beta )=x}$. Consider the pushout of ${\displaystyle j:M\to P}$ along ${\displaystyle \beta }$, given by the cokernel of the map

${\displaystyle M\to P\oplus B,m\mapsto (j(m),-\beta (m)).}$

Define ${\displaystyle X}$ as this pushout object. This yields the commutative diagram:

Here, ${\displaystyle X\to A}$ is induced by the map ${\displaystyle P\to A}$. The bottom row is an extension of ${\displaystyle A}$ by ${\displaystyle B}$, denoted ${\displaystyle \xi }$, and the connecting map ${\displaystyle \delta }$ ensures that ${\displaystyle \delta (\xi )=x}$, proving surjectivity.

To show well-definedness on equivalence classes, suppose ${\displaystyle {\beta }^{\prime }}$ is another lift of ${\displaystyle x}$. Then there exists ${\displaystyle f\in \mathrm{Hom}(P,B)}$ such that ${\displaystyle {\beta }^{\prime }=\beta +f\circ j}$. If ${\displaystyle X^{\prime }}$ is the pushout of ${\displaystyle j}$ and ${\displaystyle {\beta }^{\prime }}$, then an isomorphism ${\displaystyle X\cong X^{\prime }}$ is induced, making the extensions equivalent.

Conversely, given an extension

${\displaystyle 0\to B\to X\to A\to 0}$,

the lifting property of ${\displaystyle P}$ gives a map ${\displaystyle \tau :P\to X}$ fitting into the diagram

Here ${\displaystyle X}$ is the pushout of ${\displaystyle j}$ and ${\displaystyle \gamma }$. This shows that the map is injective.

Thus, the set of equivalence classes of extensions of ${\displaystyle A}$ by ${\displaystyle B}$ is naturally isomorphic to ${\displaystyle \mathrm{Ext}(A,B)}$.

The trivial extension corresponds to the zero element of ${\displaystyle {\mathrm{Ext}}_R^1(A,B)}$.

The Baer sum of extensions

The Baer sum is an explicit description of the abelian group structure on ${\displaystyle {\mathrm{Ext}}_R^1(A,B)}$, viewed as the set of equivalence classes of extensions of ${\displaystyle A}$ by ${\displaystyle B}$. Namely, given two extensions

$${\displaystyle 0\to B\stackrel{f}{\to }E\stackrel{g}{\to }A\to 0}$$

and

$${\displaystyle 0\to B\stackrel{f^{\prime }}{\to }{E}^{\prime }\stackrel{g^{\prime }}{\to }A\to 0,}$$

first form the pullback over ${\displaystyle A}$,

$${\displaystyle \mathrm{\Gamma }=\left\{(e,e^{\prime })\in E\oplus E^{\prime }|g(e)=g^{\prime }(e^{\prime })\right\}.}$$

Then form the quotient module

$${\displaystyle Y=\mathrm{\Gamma }/\{(f(b),-f^{\prime }(b))|b\in B\}.}$$

The Baer sum of ${\displaystyle E}$ and ${\displaystyle E^{\prime }}$ is the extension

$${\displaystyle 0\to B\to Y\to A\to 0,}$$

where the first map is ${\displaystyle b\mapsto [(f(b),0)]=[(0,f^{\prime }(b))]}$ and the second is ${\displaystyle (e,e^{\prime })\mapsto g(e)=g^{\prime }(e^{\prime })}$.

Up to equivalence of extensions, the Baer sum is commutative and has the trivial extension as identity element. The negative of an extension ${\displaystyle 0\to B\to E\to A\to 0}$ is the extension involving the same module ${\displaystyle E}$, but with the homomorphism ${\displaystyle B\to E}$ replaced by its negative.

Construction of Ext in abelian categories

Nobuo Yoneda defined the abelian groups Extⁿ
_C(A, B) for objects A and B in any abelian category C; this agrees with the definition in terms of resolutions if C has enough projectives or enough injectives. First, Ext⁰
_C(A,B) = Hom_C(A, B). Next, Ext¹
_C(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 Extⁿ
_C(A, B) are defined as equivalence classes of n-extensions, which are exact sequences

${\displaystyle 0\to B\to X_n\to \cdots \to X_1\to A\to 0,}$

under the equivalence relation generated by the relation that identifies two extensions

${\displaystyle \begin{array}{cc}\xi :0& \to B\to X_n\to \cdots \to X_1\to A\to 0\\ {\xi }^{\prime }:0& \to B\to X_n^{\prime }\to \cdots \to X_1^{\prime }\to A\to 0\end{array}}$

if there are maps ${\displaystyle X_m\to X_m^{\prime }}$ for all m in {1, 2, ..., n} so that every resulting square commutes $${\displaystyle \begin{array}{ccccccccccccc}0& ⟶& B& ⟶& X_n& ⟶& \dots & ⟶& X_1& ⟶& A& ⟶& 0\\ & & \Vert & & \downarrow {\iota }_n& & & & \downarrow {\iota }_1& & \Vert & & \\ 0& ⟶& B& ⟶& X_n^{\prime }& ⟶& \dots & ⟶& X_1^{\prime }& ⟶& A& ⟶& 0\end{array}}$$ that is, if there is a chain map ${\displaystyle \iota :\xi \to {\xi }^{\prime }}$ which is the identity on A and B.

The Baer sum of two n-extensions as above is formed by letting ${\displaystyle X_1^{″}}$ be the pullback of ${\displaystyle X_1}$ and ${\displaystyle X_1^{\prime }}$ over A, and ${\displaystyle X_n^{″}}$ be the pushout of ${\displaystyle X_n}$ and ${\displaystyle X_n^{\prime }}$ under B. Then the Baer sum of the extensions is

${\displaystyle 0\to B\to X_n^{″}\to X_{n-1}\oplus X_{n-1}^{\prime }\to \cdots \to X_2\oplus X_2^{\prime }\to X_1^{″}\to A\to 0.}$

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 D(C). The objects of the derived category are complexes of objects in C. Specifically, one has

${\displaystyle {\mathrm{Ext}}_{\mathbf{C}}^i(A,B)={\mathrm{Hom}}_{D(\mathbf{C})}(A,B[i]),}$

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, sometimes called the Yoneda product:

${\displaystyle {\mathrm{Ext}}_{\mathbf{C}}^i(A,B)\times {\mathrm{Ext}}_{\mathbf{C}}^j(B,C)\to {\mathrm{Ext}}_{\mathbf{C}}^{i+j}(A,C),}$

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 Extⁱ
_R(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:

${\displaystyle P\to Q[i]\to T[i+j].}$

By any of these interpretations, the Yoneda product is associative. As a result, ${\displaystyle {\mathrm{Ext}}_R^{\ast }(A,A)}$ is a graded ring, for any R-module A. For example, this gives the ring structure on group cohomology ${\displaystyle H^{\ast }(G,\mathbb{Z}),}$ since this can be viewed as ${\displaystyle {\mathrm{Ext}}_{\mathbb{Z}[G]}^{\ast }(\mathbb{Z},\mathbb{Z})}$. Also by associativity of the Yoneda product: for any R-modules A and B, ${\displaystyle {\mathrm{Ext}}_R^{\ast }(A,B)}$ is a module over ${\displaystyle {\mathrm{Ext}}_R^{\ast }(A,A)}$.

Important special cases

• Group cohomology is defined by

  ${\displaystyle H^{\ast }(G,M)={\mathrm{Ext}}_{\mathbb{Z}[G]}^{\ast }(\mathbb{Z},M)}$,

where ${\displaystyle G}$ is a group, ${\displaystyle M}$ is a representation of ${\displaystyle G}$ over the integers, and ${\displaystyle \mathbb{Z}[G]}$ is the group ring of ${\displaystyle G}$.

• For an algebra ${\displaystyle A}$ over a field ${\displaystyle k}$ and an ${\displaystyle A}$-bimodule ${\displaystyle M}$, Hochschild cohomology is defined by

  ${\displaystyle H{H}^{\ast }(A,M)={\mathrm{Ext}}_{A{\otimes }_k{A}^{\text{op}}}^{\ast }(A,M).}$

• Lie algebra cohomology is defined by ${\displaystyle H^{\ast }(\mathfrak{g},M)={\mathrm{Ext}}_{U\mathfrak{g}}^{\ast }(k,M)}$, where ${\displaystyle \mathfrak{g}}$ is a Lie algebra over a commutative ring ${\displaystyle k}$, ${\displaystyle M}$ is a ${\displaystyle \mathfrak{g}}$-module, and ${\displaystyle U\mathfrak{g}}$ is the universal enveloping algebra.

• For a topological space ${\displaystyle X}$, sheaf cohomology can be defined as ${\displaystyle H^{\ast }(X,A)={\mathrm{Ext}}^{\ast }({\mathbb{Z}}_X,A).}$ Here Ext is taken in the abelian category of sheaves of abelian groups on ${\displaystyle X}$, and ${\displaystyle {\mathbb{Z}}_X}$ is the sheaf of locally constant ${\displaystyle \mathbb{Z}}$-valued functions. Instead of ${\displaystyle {\mathbb{Z}}_X}$, one can consider any sheaf of rings ${\displaystyle {\mathcal{O}}_X}$ on ${\displaystyle X}$ and take Ext in the category of sheaves of ${\displaystyle {\mathcal{O}}_X}$-modules.

• For a sheaf of modules ${\displaystyle \mathcal{F}}$ on a ringed space ${\displaystyle (X,{\mathcal{O}}_X)}$, taking the right derived functors of the sheaf Hom ${\displaystyle {\mathcal{H}\mathcal{o}\mathcal{m}}_X(\mathcal{F},-)}$, the internal Hom in the category of ${\displaystyle {\mathcal{O}}_X}$-modules, gives the Ext sheaves ${\displaystyle {\mathcal{E}\mathcal{x}\mathcal{t}}_X^{\ast }(\mathcal{F},-)}$. They are related to the global Ext groups via the local-to-global Ext spectral sequence.

• For a commutative Noetherian local ring ${\displaystyle R}$ with residue field ${\displaystyle k}$, ${\displaystyle {\mathrm{Ext}}_R^{\ast }(k,k)}$ is the universal enveloping algebra of a graded Lie algebra ${\displaystyle {\pi }^{\ast }(R)}$ over ${\displaystyle k}$, known as the homotopy Lie algebra of ${\displaystyle R}$. (To be precise, when ${\displaystyle k}$ has characteristic 2, ${\displaystyle {\pi }^{\ast }(R)}$ has to be viewed as an "adjusted Lie algebra".) There is a natural homomorphism of graded Lie algebras from the André–Quillen cohomology ${\displaystyle D^{\ast }(k/R,k)}$ to ${\displaystyle {\pi }^{\ast }(R)}$, which is an isomorphism if ${\displaystyle k}$ has characteristic zero.