Characteristic-free test ideals

Abstract

Tight closure test ideals have been central to the classification of singularities in rings of characteristic p > 0 p>0 , and via reduction to characteristic p > 0 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.

The trace property in preenveloping classes

“…There has been recent interest in the theory of trace ideals over commutative rings, where a trace module in X is called a trace ideal provided X = R; see [7,10,13,14,16,20,24]. This, and other progress in the more general theory of trace modules, have shown certain trace modules to have desireable properties, like nonrigidity, that is, the existence of self extensions; see [21].…”

Uniform bounds on symbolic powers in regular rings

“…We can now define the big Cohen-Macaulay test ideals from [MS21]. See also [PRG21,Def. 3.1] for a related definition, which applies when ∆ = 0 but does not assume that K R is Q-Cartier.…”

“…In this section, we develop our new version of test ideals, which combines aspects of the big Cohen-Macaulay test ideals defined in [MS21] and [PRG21] together with aspects of the perfectoid test ideals defined in [MS18]. Throughout this section, we will use the notation from Notation 2.1.…”

“…Since [MS18], various other versions of test ideals have appeared. For normal complete Noetherian local rings R, Ma and Schwede [MS21] and Pérez and R. G. [PRG21] introduced the big Cohen-Macaulay test ideals τ B (R, ∆) for divisor pairs (R, ∆) and big Cohen-Macaulay algebras B in arbitrary characteristic, the definition of which was subsequently extended to triples (R, ∆, a t ) by Sato and Takagi [ST]. The definition of τ B (R, ∆) relies on the existence of big Cohen-Macaulay algebras B.…”

Test elements in equal characteristic semianalytic algebras

“…The key property of test ideals is that they multiply the tight closure of any ideal back into that ideal has also been generalized to arbitrary closure operations in any characteristic [ERG19,PRG19]. There are a lot of closure operations [Hei01].…”