# Extensions of Modules

# Introduction

A long exact sequence of RR-modules

0Bn1B1B0C0 0 \lra B_{n-1} \lra \ldots \lra B_{1} \lra B_{0} \lra C \lra 0

running from AA to CC though nn intermediate modules is called an nn-fold extension of AA by CC. These extensions, suitably classified by a congruence relation, are the elements of a group Ext(C,A)\Ext^{\bullet}(C,A). To calculate this group, we present CC as the quotient C=F0/S0C = F_{0}/S_{0} of a free module F0F_{0}; this process can be iterated as S0=F1/S1S_{0} = F_{1} / S_{1}, S1=F2/S2,S_{1} = F_{2}/S_{2}, \ldots, to give an exact sequence

FnFn1F1F0C0 \ldots \lra F_{n} \lra F_{n-1} \lra \ldots \lra F_{1} \lra F_{0} \lra C \lra 0

called a free resolution of CC. The complex Hom(Fn,A)\Hom(F_{n},A) then has cohomology Ext(C,A)\Ext^{\bullet}(C,A).

# Extensions and Ext

# Extensions of Modules

Definition

Let AA and CC be modules over a fixed ring RR. An extension of AA by CC is a short exact sequence E:ABCE : A \into B \onto C of RR-modules and RR-module homomorphisms. A morphism Γ:EE\Gamma : E \ra E^{\prime} of extensions is a triple Γ=(α,β,γ)\Gamma = (\alpha, \beta, \gamma) of module homomorphisms such that the diagram
(insert diagram) is commutative.

In particular, when A=AA^{\prime} = A and C=CC^{\prime} = C, then two extensions EE and EE^{\prime} of AA by CC are congruent (notation: EEE \equiv E^{\prime}) if there is a morphism (idA,β,idC):EE(\id_{A}, \beta, \id_{C}): E \ra E^{\prime}.

In the case that two extensions are congruent, the short Five Lemma shows that the middle homomorphism β\beta is an isomorphism. Thus congruence of extensions is a reflexive, symmetric, and transitive relation.

Definition

Let ExtR(C,A)\Ext_{R}(C,A) denote the set of all congruence classes of extensions of AA by CC.

Remark

An extension AA by CC is sometimes denoted by a pair (B,θ)(B,\theta), where AA is a submodule of BB, and θ\theta is an isomorphism B/ACB/A \cong C. Each such pair determines a short exact sequcen ABB/AA \into B \onto B/A and every extension of AA by CC is congruent to one so obtained.

Example

One extension of AA by CC is the direct sum AACCA \into A \oplus C \onto C. an extension is said to be split if it is congruent to this direct extension.

It can be shown that this is the case if and only if the map BCB \rightarrow C has a right-inverse μ:CB\mu : C \rightarrow B; or equivalently if the map ABA \rightarrow B has a left-inverse.

Example

Any extension by a projective module PP is split, to ExtR(P,A)\Ext_{R}(P,A) has only one element.

Proposition

For any abelian group AA and for Z/mZ\ZZ/m\ZZ the cyclic group of order mm, there is a one-to-one correspondence

η:ExtZ(Z/mZ,A)A/mA, \eta : \Ext_{\ZZ}\big(\ZZ/m\ZZ, A\big) \cong A/mA,

where mAmA is the subgroup of AA consisting on all mama for aAa \in A.

So here η\eta establishes a correspondence between a set ExtZ\Ext_{\ZZ} and an abelian group A/mAA/mA, hence ExtZ(Z/mZ,C)\Ext_{Z}(\ZZ/m\ZZ, C) is also an abelian group. We shall shortly show that ExtR(C,A)\Ext_{R}(C,A) is always an abelian group in fact.

# Ext Functor