Ext Functor

DEFINITION AND COMPUTATION

Let

R

be a Ring and let

\mathrm{Mod}_R

be the Category of Modules over ''R''. Let

B

be in

\mathrm{Mod}_R

and set

T(B) = \operatorname{Hom}_{\mathrm{Mod}_R}(A,B)

, for fixed

A

\mathrm{Mod}_R

. (This is a Left Exact Functor and thus has right Derived Functor s

R^nT

). To this end, define

:

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

i.e., take an Injective Resolution

:

J(B)\leftarrow B\leftarrow 0,

compute

:

\operatorname{Hom}_{\mathrm{Mod}_R}(A,J(B))\leftarrow\operatorname{Hom}_{\mathrm{Mod}_R}(A,B)\leftarrow0,

and take the Cohomology of this 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 Functor s

R^nG

, but instead of the injective resolution used above, choose a Projective Resolution

P(A)

, and proceed dually by calculating from
:

P(A)
ightarrow A
ightarrow 0,

compute

:

\operatorname{Hom}_{\mathrm{Mod}_R}(P(A),B)\leftarrow\operatorname{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.

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.

$\operatorname{Ext}^n_{\mathrm{Mod}_R}(\bigoplus_\alpha A_\alpha,B)\cong\prod_\alpha\operatorname{Ext}^n_{\mathrm{Mod}_R}(A_\alpha,B)$

$\operatorname{Ext}^n_{\mathrm{Mod}_R}(A,\prod_\beta B_\beta)\cong\prod_\beta\operatorname{Ext}^n_{\mathrm{Mod}_R}(A,B_\beta)$

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 Class es of extensions
:

0
ightarrow B
ightarrow C
ightarrow A
ightarrow 0

A

B

and elements of
:

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

Given two extensions
:

0
ightarrow B
ightarrow C
ightarrow A
ightarrow 0

and
:

0
ightarrow B
ightarrow C'
ightarrow A
ightarrow 0

we can construct the Baer sum, by forming the Pullback

\Gamma

C
ightarrow A

and

C'
ightarrow A

. We form the Quotient

Y=\Gamma/\Delta

, with

\Delta=\{(-b,b):b\in B\}

. The extension
:

0
ightarrow B
ightarrow Y
ightarrow A
ightarrow 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
:

0
ightarrow B
ightarrow A\oplus B
ightarrow A
ightarrow 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

B

, forming an abelian group under the Baer sum. Similarly, we can define higher Ext groups

\operatorname{Ext}^n_{\mathcal{C}}(A,B)

as equivalence classes of ''n-extensions''

:

0
ightarrow B
ightarrow X_n
ightarrow\cdots
ightarrow X_1
ightarrow A
ightarrow0

under the Equivalence Relation generated by the relation that identifies two extensions

:

0
ightarrow B
ightarrow X_n
ightarrow\cdots
ightarrow X_1
ightarrow A
ightarrow0

and

:

0
ightarrow B
ightarrow X'_n
ightarrow\cdots
ightarrow X'_1
ightarrow A
ightarrow0

if there are maps

X_m
ightarrow X'_m

for all

m

1,2,..,n

so that every resulting Square Commutes .

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
:

0
ightarrow B
ightarrow Y_n
ightarrow X_{n-1}\oplus X'_{n-1}
ightarrow\cdots
ightarrow X_2\oplus X'_2
ightarrow X''_1
ightarrow A
ightarrow0.

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_n
ightarrow B

for a of such chain maps correspond precisely to the equivalence classes in the definition of Ext above.

_{\mathrm{Mod}_R}(k,k). The multiplication has quite a few equivalent interpretations, corresponding to different interpretations of the elements of $\operatorname{Ext}^---_{\mathrm{Mod}_R}(k,k)$ .

,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
:

ightarrow X_1
ightarrow B
ightarrow 0

and

0
ightarrow B
ightarrow Y_n
ightarrow

with
:

ightarrow X_1
ightarrow Y_n
ightarrow

where the middle arrow is the composition of the functions

X_1
ightarrow B

and

B
ightarrow Y_n

.

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

(k,M) is a Module over $\operatorname{Ext}^---_{\mathrm{Mod}_R}(k,k)$ , again for sufficiently nice situations.

INTERESTING EXAMPLES

_{\mathrm{Mod}_{\mathbb ZG}}(\mathbb Z,M) is the Group Cohomology $H^---(G,M)$ with coefficients in $M$ .

(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.

_{\mathrm{Mod}_{A\otimes_k A^{op}}}(A,M) is the Hochschild Cohomology $\operatorname{HH}^---(A,M)$ with coefficients in the module ''M''.

_{\mathrm{Mod}_R}(R,M) is the Lie Algebra Cohomology $\operatorname{H}^---(\mathfrak g,M)$ with coefficients in the module ''M''.

REFERENCE

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

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Ext Functor