Exercise 5, § 4

Let $A$ be a Henselian local ring, $B$ a (not necessarily commutative) $A$-algebra which is finitely generated $A$-module, $b$ a two-sided ideal of $B$ and let $\overline{B} = B/b$.

(a) Show that every idempotent $\varepsilon$ in $\overline{B}$ is the canonical image of an idempotent in $B$ (reduce it to the commutative case, considering the subalgebra of $B$ generated by a single element).

(b) Let $(\varepsilon_n)_{n \geq 1}$ be an infinite sequence of elements of $\overline{B}$ such that

for every ordered pair of indices $(i, j)$. Show that there exists in $B$ an orthogonal sequence $(e_n)_{n \geq 1}$ of idempotents such that $\varepsilon_n$ is the canonical image of $e_n$ for all $n$. (Argue by induction as in Algebra, Chapter VIII, § 6, Exercise 10; observe that, if $e, e'$ are two idempotents of $B$ such that $ee' = 0, e' - e'e = e''$ is an idempotent such that $ee'' = e''e = 0$.)

(c) Suppose now that b is the Jacobson radical of B. Let n be an integer and $(\varepsilon_{ij})\ (1 \leq i \leq n,\ 1 \leq j \leq n)$ a family of elements of $\overline{B}$ such that $\varepsilon_{ij}\varepsilon_{hk} = \delta_{jh}\varepsilon_{ik}$ and $1 = \sum_{i=1}^{n} \varepsilon_{ii}$. Show that there exists in B a family $(e_{ij})\ (1 \leq i \leq n,\ 1 \leq j \leq n)$ such that $e_{ij}e_{hk} = \delta_{jh}e_{ik}$ and $1 = \sum_{i=1}^{n} e_{ii}$ and that $\varepsilon_{ij}$ is the canonical image of $e_{ij}$ for every ordered pair of indices. (Use (b), Exercise 11 of Algebra, Chapter VIII, § 6, and Exercise 9 of Algebra, Chapter VIII, § 1.) Deduce that, if $\overline{B}$ is isomorphic to a matrix ring $M_r(\overline{D})$, where $\overline{D}$ is a not necessarily commutative field, then B is isomorphic to a matrix ring $M_r(D)$, where D is an A-algebra which is a finitely generated A-module and whose Jacobson radical $\mathfrak{d}$ is such that D/b is isomorphic to $\overline{D}$ (cf. Algebra, Chapter VIII, § 1, Exercises 9 and 3). Show that, if further B is an Azumaya algebra over A (Chapter 11, §5, Exercise 14), so is D.