Exercise 12, § 4

Let $A$ be a regular local ring, $M$ a finitely generated $A$-module, such that the $A$-module $\mathrm{End}_A(M)$ is free. We propose to prove that the following conditions are equivalent: (i) the $A$-module $M$ is free ; (ii) the $A$-module $M$ is reflexive ; (iii) $\mathrm{Ext}_A^1(M, M) = 0$. a) Prove the implication (iii) $\Rightarrow$ (ii) (use Exercise 2 of VII, $§ 4$).

b) Prove the opposite implication (use Exercise 4 if dim(A) $\leq 2$, then reason by induction on dim(A) with the aid of Exercise 18 of § 1).

c) Suppose that dim(A) $\leq 3$. Prove the implication (ii) $\Rightarrow$ (i) (use Exercise 4 if dim(A) $\leq 2$; if dim(A) = 3, observe that (ii) implies dp_A(M) $\leq 1$, then that (iii) and Remark 2 of No. 3 imply that M is projective).

d) Suppose that M is reflexive. Let x be an element of $m_A - m_A^2$; denote by B the ring $A/xA$, by N the B-module $M/xM$ and by $N^*$ its dual. Prove with the aid of b) that the B-module End_B(N) is free, then that End_B(N*) is free (deduce from VII, § 4, No. 2 that End_B(N*) identifies with the bidual of End_B(N)).

e) Prove that (ii) implies (i) (reason by induction on the dimension of A, assumed $\geq 4$; with the notations of d), the induction hypothesis implies that the B-module N* is free ; deduce with the aid of Exercise 18 of § 1 and Proposition 7 of No. 4 that one has prof_A(Ext^1_A(M, B)) $\geq 1$, then that Ext^1_A(M, A), which is of finite length by the induction hypothesis, is zero ; conclude that the B-module $M^*/xM^*$ identifies with N*, hence is free, then that the A-module M* is free and finally that M is free.)

f) Deduce from these results another proof of the fact that the ring A is factorial.

12) Let A be a ring, $(x_1, \ldots, x_n)$ a family of elements of A, generating an ideal I. A family $(x_1, \ldots, x_n)$ is said to be A-independent if the classes of the $x_i$ in $I/I^2$ form a basis of this A/I-module.

a) Every completely transversal family for A is A-independent.

b) Let $(x_1, \ldots, x_n)$ be an A-independent family, with $x_1 = x'_1 x''_1$. Prove that the families $(x'_1, \ldots, x_n)$ and $(x''_1, \ldots, x_n)$ are A-independent, and that one has

(observe that the quotient $(x'_1, \ldots, x_n)/(x_1, \ldots, x_n)$ is isomorphic to $A/(x''_1, \ldots, x_n)$).