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p −1-Linear Maps in Algebra and Geometry

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Commutative Algebra

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.

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Notes

  1. 1.

    A coherent Cartier module \(\mathcal{F}\) if nilpotent is some power of the structural map is zero.

  2. 2.

    An abstract scheme with a finite Frobenius is called F-finite.

  3. 3.

    Here \(\frac{1} {{p}^{e}} \cdot\mathbb{Z}\) is the subgroup of ℚ generated by \(\frac{1} {{p}^{e}}\).

  4. 4.

    Formal sums of codimension 1 subvarieties with rational coefficients.

  5. 5.

    On a projective variety X, you can take the definition of big to be a divisor which has a multiple which is linearly equivalent to an ample divisor plus an effective divisor [55, Corollary 2.2.7].

  6. 6.

    Or even semilocal.

  7. 7.

    This means that f i is not a zero divisor in \(S/\langle f_{1},\ldots,f_{i-1}\rangle\) for all i > 0.

  8. 8.

    Essentially étale means essentially of finite type and formally étale, i.e., a morphism that can be factored as a localization followed by a finite type étale morphism.

  9. 9.

    That is, a full Abelian subcategory which is closed under extensions; see [5].

  10. 10.

    It is important to note that while we call it an algebra, it is not generally an R-algebra because R is not central.

  11. 11.

    This definition differs slightly from the original one given in [4] where one requires equality for every minimal prime of σ(M) instead of R. Though this yields different results in general, in light of the Kashiwara equivalence Proposition 8.1.5, the respective theories imply each other.

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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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  1. Institut für Mathematik, Johannes Gutenberg-Universität Mainz, 55099, Mainz, Germany

    Manuel Blickle

  2. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Karl Schwede

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Blickle, M., Schwede, K. (2013). p −1-Linear Maps in Algebra and Geometry. In: Peeva, I. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5292-8_5

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