Exercise 54, § 1

Let $C$ be a ring and $A$ a $C$-algebra (not necessarily commutative). We denote by $\mathrm{Rep}_C(A)$ the additive set of classes of $A$-modules which are projective $C$-modules of finite type, and by $R_C(A)$ the grothendieck group $K(\mathrm{Rep}_C(A))$ (cf. A, VIII, § 10, No. 6). a) Let $\alpha : A' \to A$ be a homomorphism of $C$-algebras. If $E$ is an $A$-module of type $\mathrm{Rep}_C(A)$, then the module $\alpha_*E$ obtained by restriction of the ring of scalars $A$ to $A'$ (A, II, p. 30) is an $A'$-module of type $\mathrm{Rep}_C(A')$, and $[E] \mapsto [\alpha_*E]$ defines a homomorphism $\alpha_* : R_C(A) \to R_C(A')$.

EXERCISES

If $\alpha_1 : A_1 \to A'$ is a homomorphism of C-algebras, one has $(\alpha \circ \alpha_1)_* = \alpha_{1*} \circ \alpha_*$.

b) Put $K_0(C) = R_C(C)$. The structural homomorphism $\varepsilon : C \to A$ defines a homomorphism $\varepsilon_* : R_C(A) \to K_0(C)$. If there exists a homomorphism $\gamma : A \to C$ of C-algebras, then $\gamma_* : K_0(C) \to R_C(A)$ is a right inverse of $\varepsilon_*$.

c) Let $\gamma : C \to C'$ be a homomorphism of rings. If E is an A-module of type $\mathrm{Rep}_C(A)$, then $\gamma^*E = C' \otimes_C E$ is an $A_{C'}$-module of type $\mathrm{Rep}_{C'}(A_{C'})$, where $A_{C'} = C' \otimes_C A$, and $[E] \mapsto [\gamma^*E]$ defines a homomorphism $\gamma^* : R_C(A) \to R_{C'}(A_{C'})$. If $\gamma'_1 : C' \to C_1$ is another homomorphism of rings, one has $(\gamma_1 \circ \gamma)^* = \gamma_1^* \circ \gamma^*$.

d) Let $E \xrightarrow{f} F \xrightarrow{g} G$ be homomorphisms of A-modules, and put $h = g \circ f$. Construct an exact sequence of A-modules

Deduce from this that if $f$ and $g$ are injective, and if $\mathrm{Coker}(f)$ and $\mathrm{Coker}(g)$ are of type $\mathrm{Rep}_C(A)$, then $\mathrm{Coker}(h)$ is of type $\mathrm{Rep}_C(A)$, and one has

in $R_C(A)$.