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Forcing algebras, syzygy bundles, and tight closure

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Abstract

We give a survey about some recent work on tight closure and Hilbert-Kunz theory from the viewpoint of vector bundles. This work is based in understanding tight closure in terms of forcing algebras and the cohomological dimension of torsors of syzygy bundles. These geometric methods allowed to answer some fundamental questions of tight closure, in particular the equality between tight closure and plus closure in graded dimension two over a finite field and the rationality of the Hilbert-Kunz multiplicity in graded dimension two. Moreover, this approach showed that tight closure may behave weirdly under arithmetic and geometric deformations, and provided a negative answer to the localization problem.

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Authors and Affiliations

  1. Fachbereich für Mathematik und Informatik, Universität Osnabrück, Osnabrück, Germany

    Holger Brenner

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  1. Holger Brenner
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Correspondence to Holger Brenner .

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Editors and Affiliations

  1. , Dipartimento di Matematica, Università degli Studi Roma Tre, Largo San Leonardo Murialdo 1, Roma, 00146, Italy

    Marco Fontana

  2. , Department of Mathematics and Statistics, King Fahd University of Petroleum & Mine, Dhahran, 31261, Saudi Arabia

    Salah-Eddine Kabbaj

  3. , Department of Mathematical Sciences, New Mexico State University, Las Cruces, 88003, New Mexico, USA

    Bruce Olberding

  4. , Department of Mathematics, Reed College, Southeast Woodstock Blvd. 3203, Portland, 97202-8199, Oregon, USA

    Irena Swanson

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Brenner, H. (2011). Forcing algebras, syzygy bundles, and tight closure. In: Fontana, M., Kabbaj, SE., Olberding, B., Swanson, I. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6990-3_4

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