Exercise 2, § 7

We suppose that the $k$-algebra $A$ admits an augmentation $\pi:A\to k$ (X, p. 196, exercise 27) and that, moreover, $A$ is a projective $k$-module. We denote by $I$ the kernel of $\pi$. Let $M$ be a left $A$-module.

a) Let $(\mathscr{E}):0\to M_{(\pi)}\to B\xrightarrow{p}A\to0$ be an extension of $A$ by the $(A,A)$-bimodule $M_{(\pi)}$ (X, p. 196, exercise 27); we denote by $\overline{B}$ the ideal of $B$ formed by the elements $b\in B$ such that $p(b)\in I$. We have $i(m)b=0$ for $m\in M$ and $b\in\overline{B}$, so that $\overline{B}$ is endowed with a left $A$-module structure such that $p(\beta)b=\beta b$ for $\beta\in B$, $b\in\overline{B}$. Show that there exists an exact sequence of $A$-modules

we denote by $\delta(\mathscr{E})$ the class of this exact sequence in $Ext^1_A(I,M)$.

b) Show that $\delta$ defines a mapping from $Ex_k(A,M_{(\pi)})$ into $Ext^1_A(I,M)$. Prove the commutativity of the diagram:

where $\delta$ is the connecting homomorphism deduced from the exact sequence $0\to I\to A\xrightarrow{\pi}k\to0$, and where $\varphi$ is the isomorphism obtained by calculating with the standard resolution (X, p. 196, exercise 27, b)). Deduce that $\delta$ is a group isomorphism.