Exercise 13, § 5

Soit A a ring (commutative), B an A-algebra (not necessarily commutative). It is assumed that the a-module B is faithful and generated by a finite number of elements $b_i (1 \leq i \leq n)$; recall that $B^0$ denotes the opposite algebra of B, and that one defines a canonical homomorphism of the A-algebra $B \otimes_A B^0$ into the A-algebra $\mathrm{End}_A(B)$, by making correspond to $b \otimes b'$ the endomorphism $z \to bz b'$.

a) Prove that the following two conditions are equivalent : α) The $b_i$ form a basis of the a-module B and the canonical homomorphism $B \otimes_A B^0 \to \mathrm{End}_A(B)$ is bijective. β) The matrix $U = (u_{ij})$ such that $u_{ij} = b_j b_i$ is invertible in $\mathbf{M}_n(B)$. b) Suppose in addition that A is a local ring, with maximal ideal $\mathfrak{m}$; denote by $\beta_i$ the canonical image of $b_i$ in $B/\mathfrak{m}B$. Prove that the conditions α) and β) are then equivalent to each of the following conditions : γ) The $\beta_i$ form a basis of the $A/\mathfrak{m}$-module $B/\mathfrak{m}B$, and $B/\mathfrak{m}B$ is a central simple $A/\mathfrak{m}$-algebra (Alg., chap. VIII, § 5, no 4). δ) The matrix $(\beta_j \beta_i)$ is invertible in $\mathbf{M}_n(B/\mathfrak{m}B)$. (To prove that δ) implies β), use the fact that $\mathfrak{m}B$ is contained in the radical of B, and exerc. 5 of Alg., chap. VIII, § 6).