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.

Bibliographie

[1] Timoshenko E. A., “Base fields of csp-rings”, Algebra and Logic, 49:4 (2010), 378–385 | DOI | MR | Zbl

[2] Timoshenko E. A., “Purely transcendental extensions of the field of rational numbers as base fields of csp-rings”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika - Tomsk State University Journal of Mathematics and Mechanics, 2013, no. 5(25), 30–39

[3] Timoshenko E. A., “Base fields of csp-rings. II”, J. Math. Sci. (New York), 230:3 (2018), 451–456 | DOI | MR | Zbl | Zbl

[4] Fomin A. A., “Some mixed abelian groups as modules over the ring of pseudo-rational numbers”, Abelian Groups and Modules, Birkhauser, Basel et al., 1999, 87–100 | DOI | MR | Zbl

[5] Krylov P. A., Pakhomova E. G., Podberezina E. I., “On a class of mixed abelian groups”, Vestnik Tomskogo gosudarstvennogo universiteta - Tomsk State University Journal, 2000, no. 269, 47–51

[6] Rosenberg J., Algebraic K-theory and its Applications, Springer, New York, 1994 | DOI | MR | Zbl

[7] Zinoviev E. G., “On a generalization of rings of pseudorational numbers”, Vestnik Tomskogo gosudarstvennogo universiteta - Tomsk State University Journal, 2006, no. 290, 46–47

[8] Timoshenko E. A., “Projective modules over csp-rings”, Zhurnal Sibirskogo federal'nogo universiteta. Matematika i fizika - Journal of Siberian Federal University. Mathematics and Physics, 5:4 (2012), 581–585

[9] Tsarev A. V., “Projective and generating modules over the ring of pseudorational numbers”, Math. Notes, 80:3 (2006), 417–427 | DOI | DOI | MR | Zbl

[10] Timoshenko E. A., “Projective modules over the ring of pseudorational numbers”, Zhurnal Sibirskogo federal'nogo universiteta. Matematika i fizika - Journal of Siberian Federal University. Mathematics and Physics, 4:4 (2011), 541–550

[11] Bass H., Algebraic K-theory, W.A. Benjamin, New York–Amsterdam, 1968 | MR | Zbl