Exercise 1, § 7

Let $k$ be a commutative ring, $A$ a $k$-algebra associative and unital, $M$ an $(A,A)$-bimodule. We call extension of $A$ by $M$ the data of a $k$-algebra $B$ associative and unital and of an exact sequence of $k$-modules

where $p$ is an homomorphism of algebras and where

Two extensions $0\longrightarrow M\xrightarrow{i}B\xrightarrow{p}A\longrightarrow0$ and $0\longrightarrow M\xrightarrow{i'}B'\xrightarrow{p'}A\longrightarrow0$ are said to be equivalent if there exists an homomorphism of algebras $u:B\longrightarrow B'$ such that $p'\circ u=p$ and $u\circ i=i'$; the homomorphism $u$ is then bijective.

(a) Suppose that the exact sequence of $k$-modules $0\longrightarrow M\xrightarrow{i}B\xrightarrow{p}A\longrightarrow0$ is split; let $s:A\longrightarrow B$ be a $k$-linear section of $p$. Now prove that there exists an element $f\in C^2(A,M)$ (X, p. 195, exercise 26, b)) such that

for $a,a'\in A$. Now prove that $f$ is a $2$-cocycle and that its class in $H^2(A,M)$ is independent of the choice of $s$; we denote it by $(\mathscr{E})$.

(b) Let $Ex_k(A,M)$ be the set of equivalence classes of extensions of $A$ by $M$ which are trivial as extensions of $k$-modules. Now prove that $c$ defines a bijection from $Ex_k(A,M)$ onto $H^2(A,M)$.

c) Let $\mathscr{E}:0\to M \xrightarrow{i} B \xrightarrow{p} A\to0$ and $\mathscr{E}':0\to M \xrightarrow{i'} B' \xrightarrow{p'} A\to0$ be two extensions of $A$ by $M$. Let $C$ denote the subalgebra of $B\times B'$ formed by the elements $(b,b')$ such that $p(b)=p'(b')$, $c$ the ideal of $C$ formed by the elements $(i(m),-i'(m))$ for $m\in M$, $\overline{B}$ the algebra $C/c$, $\pi:C\to\overline{B}$ the homomorphism of passing to the quotient. Let $i'':M\to\overline{B}$ and $p'':\overline{B}\to A$ be the homomorphisms such that $i''(m)=\pi((m,0))$ for $m\in M$ and $p''\pi((b,b'))=p(b)$ for $b\in B,b'\in B'$. Prove that the sequence $0\to M\to\overline{B}\to A\to0$ is an extension of $A$ by $M$, called the sum extension of $(\mathscr{E})$ and $(\mathscr{E}')$, and that one defines in this way a commutative group law on the set of equivalence classes of extensions of $A$ by $M$. In particular, the set $Ex_k(A,M)$ is endowed in this way with a group structure, and the mapping $c$ is a group isomorphism.