Exercise 17, § 1

Soit A an absolutely flat ring (Exercise 16).

a) Let P be a projective A-module. Now prove that every finitely generated submodule of P is a direct factor of P (reduce to the case where P is free of finite type). b) Now prove that every projective A-module is a direct sum of monogeneous submodules, isomorphic to monogeneous ideals of A. (Use Kaplansky's theorem (II, p. 183, exercise 2) to reduce to the case where P admits a countable family of generators, then use a).) c) Now prove that every ideal of A generated by a countable family of generators is projective. d) Give an example of an absolutely flat ring A and of a finitely generated nonprojective A-module. †18) Let M be an A-module. Now prove that the following conditions are equivalent: α) M is of finite type (resp. of finite presentation). β) For every inductive system (N_i, u_{ij}) of A-modules relative to a filtering preordered set I, the canonical homomorphism $\lim_{\longrightarrow i \in I} \mathrm{Hom}_A(M, N_i) \to \mathrm{Hom}_A(M, \lim_{\longrightarrow i \in I} N_i)$ is injective (resp. bijective). γ) For every family $(P_j)_{j \in J}$ of right A-modules, the canonical homomorphism

(II, p. 61) is surjective (resp. bijective). δ) For every set I, the canonical homomorphism $A_d^1 \otimes_A M \to M^I$ is surjective (resp. bijective). 19) Let A and B be two rings, $\varphi : A \to B$ a homomorphism, M a B-module. We equip $B \otimes_A M$ and $\mathrm{Hom}_A(B, M)$ with the structures of left B-module deduced from the B-module structure of B. We say that M is projective relatively to A, or $\varphi$-projective, if the canonical surjection $B \otimes_A M \to M$ admits a B-linear section; we say that M is injective relatively to A, or $\varphi$-injective, if the canonical injection admits a B-linear retraction. a) If M is projective (resp. injective) as an A-module and $\varphi$-projective (resp. $\varphi$-injective) then M is a projective (resp. injective) B-module. b) For every A-module N, the B-module $B \otimes_A N$ (resp. $\mathrm{Hom}_A(B, N)$) is $\varphi$-projective (resp. $\varphi$-injective). c) Suppose that $\varphi(A)$ is contained in the center of B. Let N be an A-module, P a $\varphi$-projective B-module. Now prove that the B-module $P \otimes_A N$ (resp. the right B-module $\mathrm{Hom}_A(P, N)$) is $\varphi$-projective (resp. $\varphi$-injective). 20) Let A be a principal ideal domain, a a nonzero element of A, $B = A/Aa$. Now prove that the B-module $B_s$ is injective. 21) Now prove that the following conditions are equivalent: α) A is noetherian. β) Every inductive limit A-module of injective A-modules is injective. γ) Every direct sum A-module of injective A-modules is injective. (To prove that γ) implies α), let $(a_n)_{n \geq 1}$ be an increasing sequence of left ideals of A, $a = \bigcup_n a_n$, $I_n$ an injective envelope of $a/a_n$. The natural homomorphism of a into $\bigoplus I_n$ is the restriction of a homomorphism $f : A \to \bigoplus I_n$; consider $f(1)$.) 22) Let A be a principal ideal domain, K its field of fractions, $\pi$ a prime element of A. Now prove that the $\pi$-primary component of the A-module $K/A$ is an injective envelope of the A-module $A/\pi^n A$ for all $n \geq 1$. †23) Let I be an injective A-module, E the ring $\mathrm{End}_A(I)$, r the set of elements $u$ of E such that I is an injective envelope of $\mathrm{Ker}(u)$. a) Now prove that r is a two-sided ideal and that the ring $E/r$ is absolutely flat (exercise 16). (Let $u$ be an element of E, N a submodule of I such that $N \cap \mathrm{Ker}\,u = 0$ and maximal for this property; now prove that there exists $v \in E$ such that $vu(x) = x$ for all $x \in N$, and deduce that $uvu - u \in r$.) b) Now prove that r is the radical of E (the inclusion $r(E) \subset r$ follows from a); now prove that for all $u \in r$, $1 - u$ is bijective). c) Suppose that I is a direct sum of a finite family of indecomposable injectives. Now prove that the ring $E/r$ is semisimple. †24) Let $i : A \to I$ be an injective envelope of the A-module $A_s$. We put $E = \mathrm{End}_A(I)$ and $B = \mathrm{End}_E(I)$, so that B is the bicommutant of the A-module I (VIII, § 3, def. 1). We consider A as a subring of B. We note $j : B \to I$ and $p : E \to I$ the mappings defined by $j(b) = b(i(1))$ and $p(u) = u(i(1))$ for $b \in B,\ u \in E$. a) Now prove that p is a surjection and j an injection.

EXERCISES

b) Prove that the following conditions are equivalent: α) The mapping $p$ is bijective. β) The mapping $j$ is bijective. γ) The A-module B is injective. If these conditions are satisfied, the mapping $j^{-1} \circ p$ is an isomorphism of the ring E onto the opposite ring of B. c) Let $\mathfrak{F}$ be the set of left ideals $a$ of A such that $a \cap b \neq (0)$ for every nonzero left ideal $b$. We denote by $s(A)$ the set of elements $a$ of A for which there exists $a \in \mathfrak{F}$ such that $aa = 0$. Prove that $s(A)$ is a two-sided ideal of A. d) Prove that the following conditions are equivalent: α) $s(A) = 0$. β) The ring E is without radical. γ) The ring B is absolutely flat. If these conditions are satisfied, the equivalent properties of b) are satisfied. (Let $s(I)$ be the set of elements $x$ of I for which there exists $a \in \mathfrak{F}$ such that $ax = 0$. Using exercise 23, prove that $p^{-1}(s(I))$ is equal to the radical r of E, whence the implication $\beta$) $\Rightarrow \alpha)$. Prove next that $\alpha)$ implies $r = \mathrm{Ker}\,(p) = 0$, which entails $\beta)$ and $\gamma)$ (in view of exercise 23) as well as condition $\alpha)$ of b). Finally prove that $\gamma)$ implies $\alpha)$.) e) If A is left noetherian, the ideal $s(A)$ is nilpotent (first prove that every element $a$ of $s(A)$ is nilpotent, by considering the left annihilators of the $a^n$; then use exercise 8, b) of VIII, § 9). † 25) Suppose that the ring A is left noetherian and contains no nilpotent two-sided ideals. a) Prove that the ring B defined in exercise 24 is semisimple (« Goldie theorem »; use exercise 23, c) and exercise 24, d) and e)). If in addition (0) is a prime two-sided ideal in A (VIII, § 8, exercise 19), prove that B is simple. b) We identify A with a subring of B. Prove the following properties: (i) Every element of A which is left cancellable is invertible in B (therefore cancellable in A). (ii) Every element $b$ of B can be written $s^{-1} a$, with $s \in A$, $a \in A$ and $s$ cancellable, It is sometimes said that B is the total ring of left fractions of A. (To prove (ii), reduce to the case where B is simple, Prove that then the ideal (0) of A is prime, Let $a$ be an ideal of $\mathfrak{F}$, $a$ an element of $a$ for which the right ideal $aB$ is maximal among the ideals $a'$ B ($a' \in a$), $x$ an element of B such that $axa = a$. Prove that $B(1 - xa) \subset aB$, whence

deduce from this that $a$ is cancellable. Conclude by taking for $a$ the set of the $a \in A$ such that $ab \in A$.) c) Suppose that every nonzero element of A is cancellable. Prove that B is a field, which is a left fraction field for A (cf. I, p. 155, exercise 15). † 26) a) Prove that the following conditions are equivalent: α) The ring A is right noetherian, and the A-module $A_s$ is injective. $\alpha^0$ A is left noetherian and the A-module $A_d$ is injective, β) The ring A is autoinjective (VIII, § 2, exercise 11). (Prove that $\alpha$ implies that every right ideal r satisfies $r = r^0$. Prove then that $r(A)$ is nilpotent, and deduce that A is right artinian by using exercise 23. Prove next that the dual of a simple left A-module is simple. Deduce that A is left noetherian and finally autoinjective. b) Prove that the ring A is autoinjective if and only if every injective A-module is projective and every projective A-module is injective. † 27) Suppose the ring A is left noetherian. a) Let I be an indecomposable injective A-module. Prove that the set of annihilator ideals of a nonzero submodule of I admits a greatest element $\mathscr{U}(I)$, which is a prime two-sided ideal (VIII, § 8, exercise 19). b) Prove that for every prime two-sided ideal $p$ of A, there exists an indecomposable injective A-module I such that $\mathscr{U}(I) = p$ (consider an indecomposable direct factor of the injective envelope of $A/p$). c) Suppose A is commutative. Prove that every indecomposable injective A-module I is the injective envelope of $A/\mathscr{U}(I)$. Deduce that the mapping $I \mapsto \mathscr{U}(I)$ defines a bijection of the set $\mathscr{J}$ of classes of indecomposable injective A-modules onto the set of prime ideals of A. The inverse bijection associates with the ideal $p$ the injective envelope of $A/p$. d) Prove that in order that the mapping $I \mapsto \mathscr{U}(I)$ of $\mathscr{J}$ onto the set of prime two-sided ideals of A be bijective, it is sufficient that A satisfy the following condition (H):

(H) For every left ideal $a$ of $A$, there exist elements $x_1, \ldots, x_k$ of $A/a$ such that

e) Show that condition (H) is verified in the following cases: $\alpha$ All the left ideals of $A$ are two-sided, $\beta$ $A$ is artinian. $\gamma$ The center $Z$ of $A$ is a noetherian ring and $A$ is a $Z$-module of finitely generated type. f) Let $k$ be a field, $\mathbf{Q}^+$ the additive monoid of rational numbers $\geqslant 0$, $A$ the algebra $k^{(\mathbf{Q}^+)}$, $a$ a nonzero element of $A$, $I$ the injective envelope of $A/Aa$. Show that the $A$-module $I$ is indecomposable, but does not contain a submodule isomorphic to $A/p$ for $p$ prime (note that the set of ideals of $A$ is totally ordered and that $A$ contains only two prime ideals). 28) Let $A$ be a noetherian ring, $M$ an $A$-module, $I$ an injective envelope of $M$, direct sum of a family $(I_\alpha)_{\alpha \in J}$ of indecomposable submodules. We denote by $\operatorname{Ass}(M)$ the set of two-sided prime ideals $\mathfrak{U}(I_\alpha)$, for $\alpha \in J$ (exercise 27); if $p \in \operatorname{Ass}(M)$, we say that $p$ is associated with $M$. a) One has $\operatorname{Ass}(M) \neq \varnothing$ if $M \neq 0$; if $M$ is of finitely generated type, the set $\operatorname{Ass}(M)$ is finite. b) Show that a two-sided prime ideal $p$ is associated with $M$ if and only if $M$ contains a submodule $N$ such that $\operatorname{Ann}(N') = p$ for every nonzero submodule $N'$ of $N$. If $A$ is commutative, $p$ is associated with $M$ if and only if $M$ contains a submodule isomorphic to $A/p$, c) Let $M'$ be a submodule of $M$. Show that one has d) Put $J_p = \sum_{\mathfrak{U}(I_\alpha) = p} I_\alpha$ and $Q(p) = \left( \sum_{q \neq p} J_q \right) \cap M$. Show that one has $\bigcap_{p \in \operatorname{Ass}(M)} Q(p) = 0$ and $\bigcap_{p \in E} Q(p) \neq 0$ for every subset $E$ of $\operatorname{Ass}(M)$ distinct from $\operatorname{Ass}(M)$. For every $p \in \operatorname{Ass}(M)$, one has $\operatorname{Ass}(M/Q(p)) = \{ p \}$. e) Let $a$ be an element of the center of $A$. Show that for the homothety of ratio $a$ to be injective in $M$, it is necessary and sufficient that $a$ belong to none of the prime ideals associated with $M$. \* 29) Suppose that the ring $A$ is commutative and noetherian. Let $a$ be an ideal of $A$. a) Let $H$ be an $A$-module such that every element of $H$ is annihilated by a power of $a$. Show that for $H$ to be injective, it is sufficient that for every integer $n \geqslant 1$, every $A$-module $M$ of finitely generated type annihilated by $a^n$ and every submodule $M'$ of $M$, the natural homomorphism $\operatorname{Hom}_A(M, H) \to \operatorname{Hom}_A(M', H)$ is surjective. (Use the fact that for every ideal $b$ of $A$, there exists an integer $k \geqslant 0$ such that $b \cap a^k \subset ba$ (AC, III, § 3, No. 1, cor. 2).) b) Let $M$ be an injective $A$-module. For $n \geqslant 1$, denote by $H_n$ the submodule of $M$ formed by the elements annihilated by $a^n$; put $H = \bigcup_n H_n$. Show that $H$ is an injective $A$-module. c) Show that every element of the injective envelope of $A/a$ is annihilated by a power of $a$, 30) Suppose that the ring $A$ is local, commutative and noetherian; denote by $m$ its maximal ideal, $k = A/m$. Let $I$ be an $A$-module such that every element of $I$ is annihilated by a power of $m$. For every $A$-module $M$, denote by $T(M)$ the $A$-module $\operatorname{Hom}_A(M, I)$ and $c_M : M \to T(T(M))$ the homomorphism defined by Show that the following properties are equivalent: $\alpha$ For every $A$-module $M$ of finite length, the $A$-module $T(M)$ is of finite length and the homomorphism $c_M : M \to T(T(M))$ is bijective. $\beta$ For every $A$-module $M$ of finite length, one has $\operatorname{long} T(M) = \operatorname{long} M$. $\gamma$ The $A$-module $I$ is injective and the $A$-modules $k$ and $T(k)$ are isomorphic. $\delta$ The $A$-module $I$ is isomorphic to the injective envelope of $k$. When these conditions are verified, one sometimes says that $I$ is a dualizing module for the ring $A$. (Show that the property $\gamma$ is equivalent to each of the other three: to see that $\alpha$ or $\beta$ implies $\gamma$, use exercise 29.) 31) With the hypotheses of the preceding exercise, let $B$ be a local, commutative and noetherian ring and let $\rho : A \to B$ be a ring homomorphism which makes $B$ an $A$-module of finitely generated type. If $I$ is a dualizing module for $A$, show that the $B$-module $\operatorname{Hom}_A(B, I)$ is dualizing for $B$. In particular for every ideal $a$ of $A$, the submodule of $I$ formed by the elements annihilated by $a$ is a dualizing module for $A/a$.

32) (a) Keep the hypotheses of Exercise 30; suppose in addition that A contains a field $k_0$ such that $k$ is a $k_0$-espace vectoriel de dimension finie. Now prove that the A-module $I = \lim_{\longrightarrow n} \mathrm{Hom}_{k_0}(A/m^n, k_0)$ is dualizing for A. (b) Let $k$ be a field of characteristic zero, B the ring of formal series $k[[T_1, ..., T_n]]$. Give the group $I = k[X_1, ..., X_n]$ a B-module structure by setting $T_i . P = \partial P / \partial X_i$ for every $P \in k[X_1, ..., X_n]$. Now prove that I is an injective envelope of $k$.