Exercise 8, § 7

Algebra, X, p. 200

We keep the notations of Exercise 7; we suppose that the $k$ -modules A and A’ are projective.

(a) We suppose that one has $\operatorname{Tor}^{k}_{i}(M,M')=0$ for $i>0$ . Now prove that the mapping $b\mapsto b\vee1_{M'}$ defines a graded $k$ -homomorphism of degree zero from $\operatorname{Ext}_{A}(M,N)$ into $\operatorname{Ext}_{B}(M\otimes_k M',N\otimes_k M')$ , which induces in degree zero the canonical homomorphism $b\mapsto b\otimes1_{M'}$ .

(b) We suppose that the $k$ -module M’ is flat. Let

0\longrightarrow N\longrightarrow R_n\longrightarrow\cdots\longrightarrow R_1\longrightarrow M\longrightarrow0

be an exact sequence of A-modules, of class $b\in\operatorname{Ext}^{n}_{A}(M,N)$ . Now prove that the exact sequence of B-modules

0\longrightarrow N\otimes_k M'\longrightarrow R_n\otimes_k M'\longrightarrow\cdots\longrightarrow R_1\otimes_k M'\longrightarrow M\otimes_k M'\longrightarrow0

has for class $b\vee1_{M'}$ in $\operatorname{Ext}^{n}_{B}(M\otimes_k M',N\otimes_k N')$ .

b \vee b' = (b \vee 1_N) \circ (1_M \vee b') = (-1)^{pq} (1_N \vee b') \circ (b \vee 1_{M'})

for $b \in \mathrm{Ext}_A^p(M, N), b' \in \mathrm{Ext}_A^q(M', N')$ .

Solution

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