Exercise 1, § 6

Algebra, X, p. 189

Let $\rho : A \to B$ be a ring homomorphism, $M, N$ two left $B$ -modules, $P$ a right $B$ -module. a) Let $L_B$ (resp. $L_A$ ) be a projective resolution of the $B$ -module (resp. of the $A$ -module) $M$ . Prove that there exists an $A$ -linear morphism of resolutions $L_A \to L_B$ , unique up to homotopy; deduce graded $k$ -homomorphisms of degree zero

\alpha : \operatorname{Tor}^A(P, M) \to \operatorname{Tor}^B(P, M) \qquad \beta : \operatorname{Ext}_B(M, N) \to \operatorname{Ext}_A(M, N) .

b) Consider the canonical homomorphisms (X, p. 109)

\lambda : \operatorname{Tor}^A(P, M) \to \operatorname{Tor}^B(P \otimes_A B, M) \qquad \mu : \operatorname{Tor}^A(P, M) \to \operatorname{Tor}^B(P, B \otimes_A M)

\nu : \operatorname{Ext}_B(B \otimes_A M, N) \to \operatorname{Ext}_A(M, N) \qquad \rho : \operatorname{Ext}_B(M, \operatorname{Hom}_A(B, N)) \to \operatorname{Ext}_A(M, N) .

Let

m : B \otimes_A M \to M, p : P \otimes_A B \to B

and

n : N \to \operatorname{Hom}_A(B, N)

be the

B

-homomorphisms defined by

m(b \otimes x) = bx, p(y \otimes b) = yb

and

n(z)(b) = bz

for

b \in B, x \in M, y \in P, z \in N

. Prove the equalities

\alpha = \operatorname{Tor}(1_P, m) \circ \mu = \operatorname{Tor}(p, 1_M) \circ \lambda

\beta = \nu \circ \operatorname{Ext}(m, 1_N) = \rho \circ \operatorname{Ext}(1_M, n) .

Solution

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