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arXiv · 1210.4767

Syzygies and tensor product of modules

Olgur Celikbas·Greg Piepmeyer
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Abstract

We give an application of the New Intersection Theorem and prove the following: let $R$ be a local complete intersection ring of codimension $c$ and let $M$ and $N$ be nonzero finitely generated $R$-modules. Assume $n$ is a nonnegative integer and that the tensor product $M\tensor_{R}N$ is an $(n+c)$th syzygy of some finitely generated $R$-module. If Tor$^{R}_{>0}(M,N)=0$, then both $M$ and $N$ are $n$th syzygies of some finitely generated $R$-modules.

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Olgur Celikbas, Greg Piepmeyer. 2012-10-17. Syzygies and tensor product of modules. https://doi.org/10.1007/s00209-013-1208-z

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