Characteristic p methods in characteristic zero via ultraproducts

  • Hans Schoutens


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.


Local Ring Characteristic Zero Noetherian Ring Regular Ring Schubert Variety 
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Authors and Affiliations

  1. 1.Department of MathematicsCity University of New YorkNew YorkUSA

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