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Characteristic p methods in characteristic zero via ultraproducts

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Commutative Algebra

Abstract

In recent decades, by exploiting the algebraic properties of the Frobenius in positive characteristic, many so-called homological conjectures and intersection conjectures have been established, culminating into the powerful theory of tight closure and big Cohen–Macaulay algebras. In the present article, I give a survey of how these methods also can be applied directly in characteristic zero by taking ultraproducts, rather than through the cumbersome lifting/reduction techniques. This has led to some new results regarding rational and log-terminal singularities, as well as some new vanishing theorems. Even in mixed characteristic, we can get positive results, albeit only asymptotically.

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  1. Department of Mathematics, City University of New York, 365 Fifth Avenue, New York, NY, 10016, USA

    Hans Schoutens

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  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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Schoutens, H. (2011). Characteristic p methods in characteristic zero via ultraproducts. 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_15

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