The grothendieck group $K_0$ of an arbitrary csp-ring

E. A. Timoshenko

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 55 (2018), pp. 38-44

Résumé

Fix an infinite set $L$ of primes. For every $p\in L$, let $R_p$ be either the ring of $p$-adic integers or the residue class ring $\mathbf{Z}/p^k\mathbf{Z}$ (the number $k>0$ may depend on $p$). Define $$ K=\prod_{p\in L} R_p\text{ and } T=\bigoplus_{p\in L} R_p\subset K; $$ it is clear that $T$ is an ideal of the ring $K$. By a csp-ring we mean any subring $R$ of the ring $K$ such that $T\subset R$ and the quotient ring $R/T$ is a field. The symbol $K_0(R)$ denotes the Grothendieck group of the monoid of isomorphism classes of finitely generated projective modules over $R$ (with direct sum as the operation). We find necessary and sufficient conditions for a module over $R$ to be a finitely generated projective module. These conditions enable us to prove the following theorem. Theorem 7. For every csp-ring $R$, the Grothendieck group $K_0(R)$ is a free group of countable rank. If we have two csp-rings $R$ and $S$, then every ring homomorphism $R\to S$ induces a group homomorphism $K_0(R)\to K_0(S)$. We describe this group homomorphism for arbitrary csp-rings $R$ and $S$.

Détail

Keywords: csp-ring, projective module, Grothendieck group.