Cartan maps and projective modules

Let $R$ be a commutative ring, $π$ be a finite group, $Rπ$ be the group ring of $π$ over $R$. Theorem 1. If $R$ is a commutative artinian ring and $π$ is a finite group. Then the Cartan map $c:K_0(Rπ)\to G_0(Rπ)$ is injective. Theorem 2. Suppose that $R$ is a Dedekind domain with $\fn{char}R=p>0$ and $π$ is a $p$-group. Then every finitely generated projective $Rπ$-module is isomorphic to $F \oplus cA$ where $F$ is a free module and $cA$ is a projective ideal of $Rπ$. Moreover, $R$ is a principal ideal domain if and only if every finitely generated projective $Rπ$-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 $\{cM_1,\ldots,cM_n\}$ be the set of all maximal ideals of $R$ containing $I$. Assume that the Cartan map $c_i: K_0(A/cM_iA)\to G_0(A/cM_iA)$ is injective for all $1\le i\le n$. If $P$ and $Q$ are finitely generated $A$-projective modules with $KP\simeq KQ$, then $P/IP\simeq Q/IQ$.

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