Exercise 8, § 11

Let $A,B$, and C be rings and $f: A\rightarrow C$ and $g: B\rightarrow C$ ring homomorphisms; suppose that $f$ is surjective. Consider the subring D of $A\times B$ consisting of the pairs $(a, b)$ such that $f(a) =g(b)$. Let M be an A-module, N a B-module, and $u$ a C-isomorphism from $C\otimes_AM$ to $C\otimes_BN$. Denote by P the D-submodule of $M\times N$ consisting of the pairs $(m, n)$ such that $u(1\otimes m) = 1\otimes n$.

a) Suppose that the modules M and N admit finite bases $(e_i)_{i\in I}$ and $(f_j)_{j\in J}$ and that the matrix of $u$ in the bases $(1\otimes e_i)$ and $(1\otimes f_j)$ is the image of an invertible matrix $(a_{ji})$ with entries in A. Prove that the D-module P is free; more precisely, the elements $(e_i,\sum_ja_{ji}f_j)_{i\in I}$ form a basis of P.

b) Let $p$ and $q$ be integers and $X\in \mathbf{M}_{p,q}(C)$ and $Y\in \mathbf{M}_{q,p}(C)$ matrices such that XY = $I_p$ and YX = $I_q$. Prove that the square matrix $\begin{pmatrix} X & 0 \\ 0 & Y \end{pmatrix}$ is the image of an invertible matrix with entries in A, using the identity

=.

0 Y 0 $I_q-YI_q$ 0 $I_qI_q$ 0

c) Suppose that the modules M and N are free and finitely generated. Deduce from a) and b) that the D-module P is projective and finitely generated.

d) Suppose that M and N are projective and finitely generated. Construct an A-module $M'$ and a B-module $N'$ such that $M\oplus M'$ and $N\oplus N'$ are free and finitely generated and that the C-modules $C\otimes_AM'$ and $C\otimes_BN'$ are isomorphic. Deduce that the D-module P is projective and finitely generated.

e) Prove that the canonical homomorphisms $A\otimes_DP\rightarrow M$ and $B\otimes_DP\rightarrow N$ deduced from the two projections are bijective (reduce as above to the case when the conditions of a) are satisfied).

f ) Conversely, let $P'$ be a finitely generated projective D-module; set $M'= A\otimes_DP'$ and $N'= B\otimes_DP'$, and denote by $u': C\otimes_AM'\rightarrow C\otimes_BN'$ the canonical isomorphism. Prove that the canonical homomorphism from $P'$ to $M'\times N'$ induces a D-linear isomorphism from $P'$ to the D-module associated with $(M',N', u')$.

g) Denote by $\varphi_0: K_0(D)\rightarrow K_0(A)\times K_0(B)$ and $\psi_0: K_0(A)\times K_0(B)\rightarrow K_0(C)$ the homomorphisms such that

$\varphi_0([P]) = ([A\otimes_DP],[B\otimes_DP])$ and $\psi_0([M],[N]) = [C\otimes_AM]-[C\otimes_BN]$.

Prove that the sequence

is exact. Prove that if, moreover, there exists a ring homomorphism $s: C\rightarrow A$ such that $f\circ s=$ Id$_C$, then $\varphi_0$ is injective and $\psi_0$ is surjective.