Exercise 11, § 13

a) Let $f$ be an element of $C^2(A,P)$ (VIII, p. 239). Prove that we define the structure of an algebra on the K-module $A\oplus P$ by setting $(a, x)(a', x') =$ $(aa', ax'+xa'+f(a, a'))$. The canonical injection $i: P\rightarrow A\oplus P$ and the canonical surjection $p: A\oplus P\rightarrow A$ are ring homomorphisms, and the triple $(A\oplus P, i, p)$ is an extension of A by P (observe that we have $f(a,1) =af(1,1)$ and $f(1, a) =f(1,1)a$ for every $a\in A$).

b) Prove that the isomorphism class of this extension depends only on the class of $f$ in $H^2(A,P)$. So the construction in a) defines an isomorphism of K-modules from $H^2(A,P)$ to the K-submodule of Ex$_K(A,P)$ consisting of the extensions of A by P that are trivial as extensions of K-modules.

c) Suppose that the ring A is commutative; denote by $C^2(A,P)^s$ the K-submodule of $C^2(A,P)$ consisting of the elements $f$ such that $f(x, y) =f(y, x)$ for all $x, y$ in A and by $H^2(A,P)^s$ the image of $C^2(A,P)^s$ in $H^2(A,P)$. Prove that the isomorphism in b) induces an isomorphism from $H^2(A,P)^s$ to the K-submodule of Ex$^c_K(A,P)$ consisting of the commutative extensions of A by P that are trivial as extensions of K-modules.