Exercise 7, § 6

Keep the previous assumptions. Suppose, moreover, given

– an A-module G(N) for every B-module N,

– an A-linear homomorphism $G(v) : G(N)\rightarrow G(N')$ for every homomorphism of B-modules $v: N\rightarrow N'$,

– an isomorphism $\alpha_M: M\rightarrow G(F(M))$ for every A-module M, and

– an isomorphism $\beta_N: N\rightarrow F(G(N))$ for every B-module N.

Suppose that the construction G has the properties analogous to (i) through (iii) above, that for every A-module homomorphism $u: M\rightarrow M'$, we have $G(F(u))\circ \alpha_M=$ $\alpha_{M'}\circ u$, and that for every B-module homomorphism $v: N\rightarrow N'$, we have $F(G(v))\circ \beta_N=\beta_{N'}\circ v$.

a) Let M and $M'$ be A-modules; prove that the homomorphism $u\mapsto F(u)$ is an isomorphism from Hom$_A(M,M')$ to Hom$_B(F(M),F(M'))$.

b) Let $f\in$ Hom$_A(M,M')$; then $F(f)$ is injective (resp. surjective) if and only if $f$ is (observe that $f$ is injective if and only if for every A-module E and every nonzero homomorphism $g: E\rightarrow M$, we have $f\circ g\not= 0$).

c) Let $0\rightarrow M'\rightarrow^iM\rightarrow^pM''\rightarrow 0$ be an exact sequence of A-modules; prove that the sequence $0\rightarrow F(M')\overset{F(i)}{\rightarrow}F(M)\overset{F(p)}{\rightarrow}F(M'')\rightarrow 0$ is exact.

d) Let $(M_{\alpha})_{\alpha\in I}$ be a family of A-modules, and let M be its direct sum. Construct a canonical isomorphism from $\oplus F(M_{\alpha})$ to F(M).

e) Let M be an A-module. Show that F(M) is projective (resp. generating, resp. finitely generated) if and only if M is (use Exercise 9 of VIII, p. 92).

f ) Prove that the $(B$, A)-bimodule $F(A_s)$ is invertible and that the $(A$, B)-bimodule $G(B_s)$ is an inverse.

g) Prove that for every A-module M, the homomorphism $\theta_M: F(A_s)\otimes_AM\rightarrow F(M)$ defined in Exercise 5, b) is bijective (use c) and d) by considering an exact sequence $A^{(J)}\rightarrow A^{(I)}\rightarrow M\rightarrow 0)$.