Exercise 12, § 13

Let A be a commutative ring, J an ideal of A, B the ring $A/J$, and P a B-module. a) Prove that the exact sequence $0\rightarrow J/J^2\rightarrow A/J^2\rightarrow B\rightarrow 0$ defines an extension of the A-algebra B by $J/J^2$ (Exercise 10).

b) Let P be a B-module. Prove that the homomorphism Hom$_B(J/J^2,P)\rightarrow$ Ex$^c_A(B,P)$ that sends $u$ to the extension deduced from the extension in a) via $u$ is an isomorphism.