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p−1-Linear Maps in Algebra and Geometry

  • Manuel Blickle
  • Karl Schwede
Chapter

Abstract

At least since Habousch’s proof of Kempf’s vanishing theorem, Frobenius splitting techniques have played a crucial role in geometric representation theory and algebraic geometry over a field of positive characteristic. In this article we survey some recent developments which grew out of the confluence of Frobenius splitting techniques and tight closure theory and which provide a framework for higher dimension geometry in positive characteristic. We focus on local properties, i.e. singularities, test ideals, and local cohomology on the one hand and global geometric applicatioms to vanishing theorems and lifting of sections on the other.

Keywords

Line Bundle Finite Type Coherent Sheaf Local Cohomology Projection Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The authors are deeply indebted to Alberto Fernandez Boix, Lance Miller, Claudiu Raicu, Kevin Tucker, Wenliang Zhang, and the referee for innumerable valuable comments on previous drafts of this chapter.

The first author was partially supported by a Heisenberg Fellowship and the SFB/TRR45. The second author was partially supported by the NSF grant DMS #1064485 and a Sloan Research Fellowship.

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Authors and Affiliations

  1. 1.Institut für MathematikJohannes Gutenberg-Universität MainzMainzGermany
  2. 2.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

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