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The Frobenius action on local cohomology modules in mixed characteristic

Part of: Commutative algebra: Homological methods

Published online by Cambridge University Press:  01 November 2007

Kazuma Shimomoto
Kazuma Shimomoto*
Affiliation:
School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455, USA (email: [email protected])
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Abstract

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Heitmann’s proof of the direct summand conjecture has opened a new approach to the study of homological conjectures in mixed characteristic. Inspired by his work and by the methods of almost ring theory, we discuss a normalized length for certain torsion modules, which was introduced by Faltings. Using the normalized length and the Frobenius map, we prove some results of local cohomology for local rings in mixed characteristic, which has an immediate implication for the subject of splinters studied by Singh.

Keywords

local cohomologyFrobenius mapmixed characteristicnormalized length

MSC classification

Secondary: 13D45: Local cohomology 13D22: Homological conjectures (intersection theorems)
Type
Research Article
Information
Compositio Mathematica , Volume 143 , Issue 6 , November 2007 , pp. 1478 - 1492
Copyright
Copyright © Foundation Compositio Mathematica 2007