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Tensoring with the Frobenius endomorphism
- Source :
- Homology, Homotopy and Applications, vol. 20(1), 2018, pp.251-257
- Publication Year :
- 2017
-
Abstract
- Let $R$ be a commutative Noetherian Cohen-Macaulay local ring that has positive dimension and prime characteristic. Li proved that the tensor product of a finitely generated non-free $R$-module $M$ with the Frobenius endomorphism ${}^{\varphi^n}\!R$ is not maximal Cohen-Macaulay provided that $M$ has rank and $n\gg 0$. We replace the rank hypothesis with the weaker assumption that $M$ is locally free on the minimal prime ideals of $R$. As a consequence, we obtain, if $R$ is a one-dimensional non-regular complete reduced local ring that has a perfect residue field and prime characteristic, then ${}^{\varphi^n}\!R \otimes_{R}{}^{\varphi^n}\!R$ has torsion for all $n\gg0$. This property of the Frobenius endomorphism came as a surprise to us since, over such rings $R$, there exist non-free modules $M$ such that $M\otimes_{R}M$ is torsion-free.
- Subjects :
- Mathematics - Commutative Algebra
Subjects
Details
- Database :
- arXiv
- Journal :
- Homology, Homotopy and Applications, vol. 20(1), 2018, pp.251-257
- Publication Type :
- Report
- Accession number :
- edsarx.1706.00238
- Document Type :
- Working Paper