# Characteristic-free test ideals

## Abstract

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

## 1. Introduction

The test ideal originated in the study of tight closure Reference HH90. Since then, it has been used to define a classification of singularities in rings of characteristic Reference HH90Reference HH94Reference HH89, which aligns well with the classification of singularities in equal characteristic 0 Reference Smi00Reference Har01. The general idea is that the larger the test ideal, the closer the ring is to being regular, and the smaller the test ideal, the singular the ring is. The gap in the literature on test ideals is the mixed characteristic case. Recent work of Ma and Schwede Reference MS18aReference MS18b has begun to fill in this gap, from the perspective of test ideals of pairs. However, most existing results are heavily dependent on the characteristic of the ring, and it is not always known whether corresponding definitions actually agree. In this paper, we study a generalization of the test ideal in a characteristic-free setting. We study test ideals from the perspective of closure operations, mimicking the approach of Hochster and Huneke Reference Hoc07 with regard to the tight closure test ideal but broadening our definition to include test ideals coming from arbitrary closure operations.

We are motivated by work of the second named author on the connections between closure operations given by big Cohen-Macaulay modules and algebras, and the singularities of the ring Reference R.G16bReference RG18, and encouraged by the fact that these connections hold in all characteristics. More precisely, in Reference R.G16b, the second named author proved that a ring is regular if and only if all closure operations satisfying certain axioms (Dietz closures) act trivially on modules over the ring. Since big Cohen-Macaulay modules give Dietz closures, we expect further connections to hold between the singularities of the ring and the big Cohen-Macaulay module closures over the ring, and we give some of those connections in this paper. In order to do this, we define and study the test ideals given by closures coming from big Cohen-Macaulay modules and algebras. See Section 3 for details.

We prove that the test ideal of a module closure has multiple equivalent definitions, which we use to get our main results connecting singularities to big Cohen-Macaulay module test ideals.

In particular, the second result is similar to the result that the tight closure test ideal

for particular elements Reference HT04. This perspective on the tight closure test ideal is one of the major tools used to study it, as described in Reference ST12. Our second definition also coincides with the trace ideal of the module as studied in ,Reference Lam99Reference Lin17. By drawing this connection, we open the door for future results on test ideals using the theory of trace ideals, and vice versa. In an upcoming paper with Neil Epstein Reference ERG21, the second named author has generalized this to a duality between closure operations and interior operations on finitely-generated and Artinian modules over complete local rings.

One important consequence of these results is that when the ring is complete and cl is a big Cohen-Macaulay module closure, is nonzero (Corollary 3.16).

We also define a finitistic test ideal of an arbitrary closure operation and discuss cases where it is equal to the (big) test ideal of the same closure operation. In the Gorenstein case, the test ideal of an algebra closure is the whole ring if and only if the corresponding finitistic test ideal is also the whole ring (Proposition 3.10).

One advantage to working with test ideals of module closures is that, as a consequence of Theorem 3.12, when the module is finitely-generated, we can compute its test ideal in Macaulay2. This is in contrast to the tight closure test ideal, which is difficult to compute in general. In Section 5, we compute examples of test ideals of finitely-generated Cohen-Macaulay modules, and in some cases are able to compute or approximate the “smallest” Cohen-Macaulay test ideal.

In summary, our results on the classification of singularities via test ideals are:

We apply our techniques to the case of mixed characteristic rings in Section 6. We propose a mixed characteristic closure operation that satisfies Dietz’s axioms (these guarantee that it acts like a big Cohen-Macaulay module closure–see Reference Die10Reference R.G16b for details), and prove that its test ideal can be viewed in three different ways similar to those we gave for module closures earlier. In addition to demonstrating how our results can be used in mixed characteristic, this section shows how our proof techniques can be applied to a broader group of closures than module closures.

Throughout the paper, will denote a commutative Noetherian ring, though some of the under consideration will not be Noetherian. -algebras

## 2. Preliminaries

In this section we recall the concepts of closure operations and trace ideals. We record their basic properties for later use and give the appropriate references for their proofs.

### 2.1. Closure operations

Given a submodule of a module we would like to find a submodule of , containing that also satisfies some desired properties. This idea is encoded in the following familiar definition.

A particularly important family of closures are Dietz closures, originally defined in Reference Die10Reference Die18. A local domain has a Dietz closure if and only if it has a big Cohen-Macaulay module Reference Die10.

Associated to any -module we define a closure operation as follows.

When is an the previous definition can be simplified to -algebra, if and only if

The following examples show that familiar ideals and closure operations are particular examples of module closures.

For reference, we list some properties of closure operations and refer the reader to Reference R.G16b, Lemma 3.1, Reference Die10, Lemma 1.2, and Reference Die18, Lemma 1.3 for the proofs.

When the closure operation satisfies the functoriality and semi-residuality axioms, the elements of the ring multiplying the closure inside the original module can be seen as an annihilator. More precisely:

Proposition 2.16 gives information about the behavior of module closures under ring extension.

The generation property enables us to compare the closures given by and Before we give the precise result we need a lemma. .

Proposition 2.20 is the result of a conversation with Yongwei Yao, and gives one case where we have containment of module closures.

Theorem 2.21 characterizes regular rings in terms of the behaviour of Dietz closures. This result describes an important connection between the behavior of big Cohen-Macaulay module closure operations and the singularities of the ring.

Note that this result holds regardless of the characteristic of as by ,Reference HH92Reference And18, we know that big Cohen-Macaulay algebras (and in particular big Cohen-Macaulay modules) exist over complete local domains of any characteristic.

In fact, the proof of this statement in Reference R.G16b uses the fact that big Cohen-Macaulay modules over regular rings are faithfully flat Reference HH92, and we get the following corollary to Theorem 2.21 and its proof in Reference R.G16b:

We also have the following: