Exercise 19, § 3

We suppose that the ring $A$ is commutative.

a) Let $(S, d_S)$ be a differential graded $A$-algebra (exercise 18), $M$ an $A$-module, $u : M \to Z_n(S)$ an $A$-homomorphism. Now prove that there exists on $S \otimes_A T(M)$ a structure of differential graded $A$-algebra such that $d(s \otimes 1) = d_S s \otimes 1$ for $s \in S$ and $d(1 \otimes m) = u(m) \otimes 1$ for $m \in M$.

b) Let B be an A-algebra. Now prove that there exists a differential graded A-algebra S and a homomorphism of A-algebras $p:S_0\to B$ such that $(S,p)$ is a free resolution of the A-module B. (Construct by recurrence on $n$, with the help of a), a differential graded algebra $S(n)$, free in each degree, such that $H_i(S(n))=0$ for $0<i<n$ and $H_0(S(n))=B$.)

c) If the algebra B is commutative, now prove that one can find an A-algebra S alternating (III, p. 53) satisfying the conditions of b). If moreover A is noetherian and if B is a quotient of A by an ideal, now prove that one can find S in such a way that $S_0=A$ and that $S_n$ is a finitely generated A-module for every $n$.

☡20) Suppose that the ring A is an algebra over a commutative ring $k$. Let $\eta:k\to A$ be the homomorphism defined by $\eta(\lambda)=\lambda1_A$ for $\lambda\in k$, and let $\bar A$ be its cokernel. If $a$ is an element of A, we shall denote by $\bar a$ its class in $\bar A$. We put

for $n\geq0$, $N_n(A)=0$ for $n<0$, and

a) Now prove that there exists a $(A,A)$-endomorphism $d$ of $N(A)$, graded of degree $(-1)$, such that

for $a,a_1,\ldots,a_n,b\in A$.

b) Now prove that $d\mathbin{\circ}d=0$, so that $N(A)$ is a complex of $(A,A)$-bimodules. If M is a left A-module, we denote by $N(A,M)$ the A-complex $N(A)\otimes_A M$.

c) Let $B(A,M)$ be the standard resolution of M (X, p. 58), $p:B(A,M)\to N(A,M)$ the A-homomorphism defined by

for $a_0,\ldots,a_n\in A$, $m\in M$. Now prove that $p$ is a homotopy equivalence of A-complexes (construct by recurrence an inverse mapping up to homotopy).

*21) Let $k$ be a commutative ring, $\mathfrak g$ a $k$-algebra of Lie, U its enveloping algebra (cf. LIE, I, § 2). Suppose that $\mathfrak g$ is a free $k$-module.

(a) Now prove that there exists for every $n\geq0$ an injective U-homomorphism

such that for $u\in U,x_1,\ldots,x_n\in\mathfrak g$.

One thus defines a graded U-homomorphism of degree 0 $j:U\otimes_k\Lambda(\mathfrak g)\to B(U,k)$, where $B(U,k)$ is the standard resolution of the left U-module $k$ (the structure of U-module of $k$ being deduced from the natural homomorphism $U\to k$). Prove that $U\otimes_k\Lambda(\mathfrak g)$ is identified by $j$ with a subcomplex of $B(U,k)$, the differential being given by the formula:

for $u\in U,x_1,\ldots,x_n\in\mathfrak g$ (where the sign $\hat{\ }$ on a letter means that it is to be omitted). We denote by $V(\mathfrak g)$ the U-complex thus defined.

(b) Prove that $V(\mathfrak g)$ defines a free resolution of the U-module $k$ (let $(U_r)_{r\geq0}$ be the natural filtration of U, $F_rV(\mathfrak g)$ the sub-$k$-module of $V(\mathfrak g)$ generated by the elements $u_p\otimes x_q$ for $u_p\in U_p$, $x_q\in\Lambda^q(\mathfrak g)$, $p+q\leq r$). Prove that $F_rV(\mathfrak g)$ is a subcomplex of $V(\mathfrak g)$, and that the complex

is isomorphic to the complex $S(\mathfrak g)\otimes\Lambda(\mathfrak g)$ defined in X, p. 151). \*