Characteristic-free test ideals

Pérez, Felipe; R. G., Rebecca

doi:10.1090/btran/55

Characteristic-free test ideals
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by Felipe Pérez and Rebecca R. G. HTML | PDF

Trans. Amer. Math. Soc. Ser. B 8 (2021), 754-787

Abstract:

Tight closure test ideals have been central to the classification of singularities in rings of characteristic $p>0$, and via reduction to characteristic $p>0$, in equal characteristic 0 as well. Their properties and applications have been described by Schwede and Tucker [Progress in commutative algebra 2, Walter de Gruyter, Berlin, 2012]. In this paper, we extend the notion of a test ideal to arbitrary closure operations, particularly those coming from big Cohen-Macaulay modules and algebras, and prove that it shares key properties of tight closure test ideals. Our main results show how these test ideals can be used to give a characteristic-free classification of singularities, including a few specific results on the mixed characteristic case. We also compute examples of these test ideals.

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