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18 February 2025
 
  » arxiv » 1508.0095

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Cartan maps and projective modules
Ming-chang Kang ; Guangjun Zhu ;
Date 1 Aug 2015
AbstractLet $R$ be a commutative ring, $pi$ be a finite group, $Rpi$ 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 $c:K_0(Rpi) o G_0(Rpi)$ 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 $Rpi$-module is isomorphic to $F oplus c{A}$ where $F$ is a free module and $c{A}$ is a projective ideal of $Rpi$. Moreover, $R$ is a principal ideal domain if and only if every finitely generated projective $Rpi$-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 $c_i: K_0(A/c{M}_iA) o G_0(A/c{M}_iA)$ is injective for all $1le ile n$. If $P$ and $Q$ are finitely generated $A$-projective modules with $KPsimeq KQ$, then $P/IPsimeq Q/IQ$.
Source arXiv, 1508.0095
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