Exercise 22, § 8

Show that the following conditions are equivalent:

$\alpha$) Every $A$-module of finite presentation and of finite projective dimension is projective.

$\beta$) Every injective $A$-homomorphism $A^p \to A^q$ admits a retraction.

$\gamma$) For every right ideal of finite type $\mathfrak{a}$ distinct from $A$, there exists a nonzero element $x$ of $A$ such that $x\mathfrak{a} = 0$.

δ) Every simple right A-module of finite presentation is isomorphic to a (minimal) right ideal of A. (Observe that condition γ) is equivalent to condition β) in the case $p = 1$.)

T 23) Show that the following conditions are equivalent: α) Every A-module of finite projective dimension is projective. β) The ring A satisfies the equivalent conditions of exercise 21, and for every right ideal of finite type a distinct from A, there exists a nonzero element $x \neq 0$ of A such that $xa = 0$. γ) The ring A satisfies the equivalent conditions of exercise 21, and every simple A-module is a quotient of an injective A-module. (Show that α) implies that every decreasing sequence of monogenic right ideals of A is stationary by adapting the proof of exercise 21. Deduce then from exercise 22 the equivalence of α) and β). To see that α) implies γ), consider a projective covering P of the simple module M, an injective module I containing P and a projective covering Q of I; define an A-homomorphism of Q into P, and deduce from it a surjection of I onto M. To show that γ) implies α), show that an A-module M such that $\mathrm{dp}_A(M) \leq 1$ is projective by considering a projective covering $p : P \to M$ and a surjection of $\ker p$ onto a simple module.)

24) We denote by dhf (A) (resp. dif (A), resp. tdf (A)) the supremum of the projective (resp. injective, resp. flat) dimensions of the A-modules for which this dimension is finite. If A is of finite homological dimension, one has dhf(A) = dif(A) = dh(A) and tdf(A) = td(A). If td(A) < ∞, one has tdf(A) = td(A). a) Show that one has tdf (A) ≤ dif (A°) ; there is equality if A is left noetherian (use X, p. 189, exercise 3). b) Suppose A is left noetherian. Show that one has dhf(A) ≤ di_A(A_s). If di_A(A_s) and dif(A) are finite, show that they are equal (use exercise 2, p. 202). c) Let G be a finite group not reduced to the identity element, and A the ring $\mathbf{Z}^{(G)}$. Show that one has

(cf. exercise 20, p. 193). d) Suppose the ring A autoinjective (cf. X, p. 171, exercise 26), non-semisimple. Show that every nonprojective A-module is of infinite projective dimension and infinite injective dimension. Deduce that one has $\mathrm{dh}(A) = + \infty , \mathrm{dhf}(A) = \mathrm{dif}(A) = \mathrm{tdf}(A) = \mathrm{di}_A(A_s) = 0$.

25) We denote by $A^e$ the k-algebra $A \otimes_k A^*$, and we consider A as a left $A^e$-module. We put

If n is an integer $\geq 0$, the inequality $\mathrm{da}_k(A) \leq n$ is therefore equivalent to the vanishing of $H^{n+1}(A, M)$ for every (A, A)-bimodule M (X, p. 195, exercise 26). One has $\mathrm{da}_k(A) = \mathrm{da}_k(A^o)$. a) Show that one has $\mathrm{da}_k(M_n(k)) = 0$ and $\mathrm{da}_k(k[X_1, ..., X_n]) = n$ for every $n \geq 0$. b) If k is a field, one has $\mathrm{da}_k(A) = 0$ if and only if the k-algebra A is absolutely semisimple (cf. VIII, § 11, no 3, th. 2). c) Let M be a free k-module. Prove that one has $\mathrm{da}_k(T_k(M)) = 1$ (if $(e_i)_{i \in I}$ is a basis of M, prove that the elements $(e_i \otimes 1 - 1 \otimes e_i)$ form a basis of the left ideal of $A^e$ which is the kernel of the multiplication $A \otimes_k A \to A$, with $A = T_k(M)$). d) Suppose now that the k-module A is projective. Let $\rho : k \to k'$ be a homomorphism of commutative rings; show that one has $\mathrm{da}_k(A \otimes_k k') \leq \mathrm{da}_k(A)$. If $\rho$ is injective and if $\rho(k)$ is a direct factor of the k-module $k'$, one has equality (use X, p. 195, exercise 26, c)). e) Let B be a second k-algebra (associative and unital), projective over k. Show that If moreover k is a field and A and B are finite-dimensional over k, there is equality (use X, p. 195, exercise 26, d)). f) Suppose that k is a field. Prove the inequalities $\mathrm{dh}(A) \leq \mathrm{da}_k(A)$ and $\mathrm{dh}(A^*) \leq \mathrm{da}_k(A)$ (cf. X, p. 196, exercise 26, e)).

26) Let A be an augmented k-algebra (X, p. 196, exercise 27); suppose that there exists a homomorphism $\rho : A \to A^e$ which satisfies the conditions of X, p. 196, exercise 27, c). Show that $\mathrm{da}_k(A) = \mathrm{dp}_A(k)$; if moreover k is a field, one has $\mathrm{da}_k(A) = \mathrm{dp}_A(k) = \mathrm{dh}(A) = \mathrm{dh}(A^*)$. (Use exercise 27, b) and c), p. 196, and exercise 26, e), p. 196.) In particular, if V is a nonzero k-vector space, one has $\mathrm{dh}\ T_k(V) = 1$.

\* 27) Suppose that $k$ is a field. Let $g$ be a Lie algebra over $k$, of dimension $n$, and $U$ its enveloping algebra. Show that $\mathrm{dh}U = n$. (Use exercise 26, exercise 27, d), p. 196, and exercise 28, c), p. 197.) \*

**28) Let $G$ be a group. The cohomological dimension of $G$, denoted by $\mathrm{dc}(G)$, is called the projective dimension of the $\mathbf{Z}^{(G)}$-module $\mathbf{Z}$. If $n$ is an integer, one therefore has $\mathrm{dc}(G) \leq n$ if and only if $H^q(G, M) = 0$ for every $\mathbf{Z}^{(G)}$-module $M$ and every $q > n$.

a) Prove that one has $\mathrm{da}_\mathbf{Z}(\mathbf{Z}^{(G)}) = \mathrm{dc}(G)$ (use Exercise 26 and Exercise 27, d), p. 196).

b) If $G$ is a free group not reduced to the identity element, one has $\mathrm{dc}(G) = 1$.

c) Let $H$ be a subgroup of $G$; prove the inequality $\mathrm{dc}(H) \leq \mathrm{dc}(G)$. If $\mathrm{dc}(G) < \infty$ and if $H$ is of finite index in $G$, show that $\mathrm{dc}(H) = \mathrm{dc}(G)$ (show that for $q = \mathrm{dc}(G)$, the homomorphism

of Exercise 14, p. 191, is surjective).

d) Show that the cohomological dimension of a finite group not reduced to the identity element is infinite. Deduce that if $\mathrm{dc}(G) < \infty$, every element of $G$ distinct from the identity element is of infinite order (one then says that $G$ is torsion-free).

e) If $G$ is torsion-free and if $H$ is a subgroup of finite index of $G$, show that one has $\mathrm{dc}(H) = \mathrm{dc}(G)$. (If $(P, p)$ is a projective resolution of the $\mathbf{Z}^{(H)}$-module $\mathbf{Z}$ and if $(G : H) = n$, define on the graded $\mathbf{Z}^{(H)}$-module $T^n_\mathbf{Z}(P)$ a structure of $\mathbf{Z}^{(G)}$-complex in such a way that $(T^n_\mathbf{Z}(P), p^{\otimes n})$ is a projective resolution of the $\mathbf{Z}^{(G)}$-module $\mathbf{Z}$.)

f) If $H$ is a normal subgroup of $G$, show that one has