Exercise 8, § 6

Let $k$ be a commutative ring and A a $k$-algebra.

a) Prove that the set of isomorphism classes of invertible $(A,A)_k$-bimodules (No. 8), endowed with the multiplication law $[P][Q] = [P\otimes_AQ]$, is a group; we denote it by $\mathscr{P}_k(A)$.

b) Let G be the automorphism group of the $k$-algebra A; for $g\in G$, we denote by $A_g$ the left A-module $A_s$ endowed with the right A-module structure defined by $x\cdot a=x g(a)$. Show that the mapping $g\mapsto [A_g]$ is a group homomorphism from G to $\mathscr{P}_k$(A), with kernel the subgroup of inner automorphisms.

c) Let Z be the center of A; define an exact sequence

If the algebra A is commutative, the group $\mathscr{P}_k(A)$ is the semidirect product of G and $\mathscr{P}_A(A)$.