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Cartan maps and projective modules | Ming-chang Kang
; Guangjun Zhu
; | Date: |
1 Aug 2015 | Abstract: | Let $R$ 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 | Services: | Forum | Review | PDF | Favorites |
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