Exercise 7, § 7

Soient $A'$ a $k$-algebra (associative and unital) and $B = A \otimes_k A'$. a) Let M be a left A-module, Q a right A-module, $M'$ a left $A'$-module, Q’ a right $A'$-module. Let $(P, p)$ (resp. $(P', p')$, resp. $(R, r)$) be a projective resolution of the A-module M (resp. of the $A'$-module $M'$, resp. of the B-module $M \otimes_k M'$). Prove that there exists a morphism of B-complexes

unique up to homotopy, such that $r\circ u=p\otimes p'$. Define using $u$ a $k$-homomorphism graded of degree zero

Prove that $\mathrm{T}$ does not depend on the choice of the resolutions $P,P',R$ and of the morphism $u$. If $a\in\operatorname{Tor}^{A}(Q,M)$ and $a'\in\operatorname{Tor}^{A'}(Q',M')$, put $\mathrm{T}(a\otimes a')=a\mathrm{T}a'$.

b) Suppose in addition that the $k$-modules A and A’ are projective and that one has $\operatorname{Tor}^{k}_{n}(M,M')=0$ for $n>0$. Prove that $P\otimes_k P'$ is then a projective resolution of the B-module $M\otimes_k M'$; deduce, for every A-module N and every A’-module N’, a $k$-homomorphism graded of degree zero

Prove that $\vee$ is independent of the choice of the resolutions $P$ and $P'$. If $b\in\operatorname{Ext}_{A}(M,N)$ and $b'\in\operatorname{Ext}_{A'}(M',N')$, we note $\vee(b\otimes b')=b\vee b'$.

c) Let A’’ be a third $k$-algebra (associative and unital), M’’ and N’’ left A’’-modules, Q’’ a right A’’-module. We identify the $k$-algebras $A\otimes_k(A'\otimes_k A'')$ and $(A\otimes_k A')\otimes_k A''$, as well as the $k$-modules $M\otimes_k(M'\otimes_k M'')$ and $(M\otimes_k M')\otimes_k M''$, $N\otimes_k(N'\otimes_k N'')$ and $(N\otimes_k N')\otimes_k N''$, $Q\otimes_k(Q'\otimes_k Q'')$ and $(Q\otimes_k Q')\otimes_k Q''$. Prove the formulas

d) Let

and

the isomorphisms deduced from the commutation isomorphisms

Prove the formulas

e) Let $(\mathscr{E}):0\to M_2\to M_1\to M_0\to0$ be an exact sequence of A-modules, such that the sequence

be exact. We denote $\delta$ (resp. $\Delta$) the connecting homomorphism $\partial(Q,\mathscr{E})$ (resp. $\partial(Q\otimes_k Q',\mathscr{E}\otimes M')$. Now prove that one has $(\delta a)\mathrm{T}a'=\Delta(a\mathrm{T}a')$ for $a\in\operatorname{Tor}^{A}(Q,M_0)$, $a'\in\operatorname{Tor}^{A'}(Q',M')$. Exercise prove the analogous formulas for the product $\vee$.

f) If $k$ is a field, show that $\mathrm{T}$ is an isomorphism of graded $k$-modules; the same holds for $\vee$ if, in addition, A and A’ are left noetherian and if the A-module M and the A’-module M’ are of finite type.

g) If the modules $\operatorname{Tor}^{A}_{n}(Q,M)$ are identified with the sets $T_n(Q,M)$ defined in exercise 6, prove the formula