Cartan maps and projective modules

TL;DR

This paper investigates the properties of Cartan maps and projective modules over group rings of finite groups, establishing conditions for injectivity and module decompositions over various classes of rings.

Contribution

It proves the injectivity of the Cartan map for group rings over artinian rings and characterizes finitely generated projective modules over Dedekind domains with positive characteristic.

Findings

Cartan map is injective over artinian rings

Finitely generated projective modules over Dedekind domains decompose into free modules and projective ideals

Isomorphism of modules over total quotient ring implies isomorphism modulo an ideal

Abstract

Let $R$ be a commutative ring, $\pi$ be a finite group, $R\pi$ be the group ring of $\pi$ over $R$. Theorem 1. If $R$ is a commutative artinian ring and $\pi$ is a finite group. Then the Cartan map $c:K_0(R\pi)\to G_0(R\pi)$ is injective. Theorem 2. Suppose that $R$ is a Dedekind domain with $\fn{char}R=p>0$ and $\pi$ is a $p$-group. Then every finitely generated projective $R\pi$-module is isomorphic to $F \oplus \c{A}$ where $F$ is a free module and $\c{A}$ is a projective ideal of $R\pi$. Moreover, $R$ is a principal ideal domain if and only if every finitely generated projective $R\pi$-module is isomorphic to a free module. Theorem 3. Let $R$ be a commutative noetherian ring with total quotient ring $K$, $A$ be an $R$-algebra which is a finitely generated $R$-projective module. Suppose that $I$ is an ideal of $R$ such that $R/I$ is artinian. Let $\{\c{M}_1,\ldots,\c{M}_n\}$ be the set of all maximal ideals of $R$ containing $I$. Assume that the Cartan map $c_i: K_0(A/\c{M}_iA)\to G_0(A/\c{M}_iA)$ is injective for all $1\le i\le n$. If $P$ and $Q$ are finitely generated $A$-projective modules with $KP\simeq KQ$, then $P/IP\simeq Q/IQ$.