# Dagger closure in regular rings containing a field

###### Abstract.

We prove that dagger closure is trivial in regular domains containing a field and that graded dagger closure is trivial in polynomial rings over a field. We also prove that Heitmann’s full rank one closure coincides with tight closure in positive characteristic under some mild finiteness conditions. Furthermore, we prove that dagger closure is always contained in solid closure and that the forcing algebra for an element contained in dagger closure is parasolid.

###### 2010 Mathematics Subject Classification:

Primary 13A35, 13A18; Secondary 13D45## Introduction

Dagger closure is an attempt to provide a tight closure like closure operation that is valid in any characteristic without reducing to positive characteristic. More precisely, an element belongs to the dagger closure of an ideal in a domain if it is multiplied into the extended ideal in the absolute integral closure (i. e. the integral closure of in an algebraic closure of its field of fractions) by “arbitrarily small” elements. In order to make sense of the notion “arbitrarily small” valuations are used.

Dagger closure was first introduced by Hochster and Huneke in [hochsterhunekedagger] for a complete local domain of positive characteristic. There they also proved that dagger closure coincides with tight closure in this setting [ibid.,Theorem 3.1]. Also Heitmann’s *full rank one closure* which he used to prove the direct summand conjecture in mixed characteristic in dimension (cf. [heitmanndirectsummand]) is a variant of dagger closure tailored to mixed characteristics.

One important feature of such a tight closure like operation should be that it is trivial on ideals in regular rings, since this property opens relations to singularity theory. It is also a crucial feature with respect to the direct summand conjecture in mixed characteristic. In this paper we prove this property for dagger closure for regular rings containing a field (see Theorem 2.5). The argument we provide uses the existence of big Cohen Macaulay algebras.

In [brennerstaeblerdaggersolid] we prove that a graded variant of dagger closure agrees with solid closure in graded dimension . And in [staeblerdaggerabelian] the second author proves an inclusion result for certain section rings of abelian varieties. This implies in particular that dagger closure is non-trivial in all dimensions.

This article is based on parts of the Ph.D. thesis of the second author ([staeblerthesis]). Related ideas concerning dagger closure and “almost zero” have also been studied in [robertssinghannihilators] and [asgbhattaremakrsonbcm].

## 1. Preliminaries

In this section we recall the original definition of dagger closure by Hochster and Huneke and the graded version of dagger closure as defined in [brennerstaeblerdaggersolid]. Then we introduce a definition of dagger closure for arbitrary domains and relate it and the graded version to the original definition of Hochster and Huneke. We will also prove that they both coincide with tight closure in positive characteristic. Our definition is in principle applicable if the ring does not contain a field but we will not prove any results in this case.

We recall that for a noetherian domain the absolute integral closure of is the integral closure of in an algebraic closure of its field of fractions . By a result of Hochster and Huneke ([hochsterhunekeinfinitebig, Lemma 4.1]) there is for an -graded domain a maximal -graded subring of which extends the grading of – we will denote this ring by .

We can now state the definition of Hochster and Huneke.

###### 1.1 Definition.

Let be a complete local domain. Fix a valuation on with values in such that and . Extend to so that it takes values in . Then an element belongs to the *dagger closure * of an ideal if for all positive there exists an element with such that .

###### 1.2 Proposition.

Let be a -graded domain. The map sending to , where is the minimal homogeneous component of induces a valuation on with values in . This valuation will be referred to as the *valuation induced by the grading*.

###### Proof.

Standard. ∎

###### 1.3 Definition.

Let denote an -graded domain and let be an ideal of . Let be the valuation on induced by the grading on . Then an element belongs to the *graded dagger closure* of an ideal if for all positive there exists an element with such that .
If is not a domain we say that if for all minimal primes of .^{1}^{1}1Note that this is well-defined since the minimal primes are homogeneous (see [brunsherzog, Lemma 1.5.6 (a)]).

###### 1.4 Definition.

Let be a domain and an ideal. Then an element belongs to the *dagger closure* of if for every valuation of rank at most one on and every positive there exists with and .

We recall that a valuation of rank one is a non-trivial valuation whose value group is contained in . A valuation of rank zero is precisely the trivial valuation (see [zariskisamuel, vol. 2, VI §10 Theorem 15]) and we include this only to force that if is a field of positive characteristic. Indeed, if for instance then there is no non-trivial valuation on . See also [heitmannplusextended, Remark 1.4]. Henceforth, the term “dagger closure” refers to Definition 1.4 unless explicitly stated otherwise.

###### 1.5 Remark.

In the definition of dagger closure it is enough to only consider valuations such that , where is the valuation ring associated to . Indeed, means that all elements of have non-negative valuation. If there is an element such that and is an ideal then with respect to . To see this let . Then and for sufficiently large .

If then in any case.

###### 1.6 Lemma.

Let be a complete local noetherian domain containing a field. Then there exists a -valued valuation on which is non-negative on and positive on .

###### Proof.

As is a noetherian integral domain it is dominated by a discrete valuation ring – cf. [swansonhunekeintegralclosure, Theorem 6.3.3], the case a field is trivial. ∎

###### 1.7 Lemma.

Let be an extension of integral domains and an ideal. Then .

###### Proof.

Follows similarly to [heitmannplusextended, Lemma 1.6]. ∎

###### 1.8 Proposition.

Let be a domain of characteristic essentially of finite type over an excellent local ring. Then dagger closure as in Definition 1.4 coincides with tight closure.

###### Proof.

Dagger closure always contains tight closure (see the argument in the proof of Theorem [hochsterhunekedagger, Theorem 3.1] on p. 235) so we only have to show the other inclusion. Passing to the normalisation we may assume that is normal. By virtue of [hochsterhuneketightzero, Theorem 1.4.11] has a completely stable test element. Hence, by [hochsterhuneketightzero, Theorem 1.4.7 (g)] an element is contained in the tight closure of an ideal if and only if is contained in the tight closure of the extended ideal for the completion of at every maximal ideal . Furthermore, Lemma 1.7 yields that . Therefore, we may assume that is a complete local domain (since is normal and excellent, its completions at maximal ideals are again domains by [EGAIV, 7.8.3 (vii)]).

Let now be an ideal, and assume that . In particular, this implies that is multiplied into by elements of small order with respect to an extension of a valuation as in Lemma 1.6 above. But by [hochsterhunekedagger, Theorem 3.1] this yields that . ∎

The following result compares our definition of dagger closure with that of Hochster and Huneke.

###### 1.9 Corollary.

Let be a complete local noetherian domain. Then dagger closure is contained in the dagger closure in the sense of Definition 1.1. If is of positive characteristic or if the ideal is primary to then the two closures coincide.

###### Proof.

If an element is contained in dagger closure then a fortiori in the dagger closure of Hochster and Huneke. In positive characteristic the result is immediate from Proposition 1.8 and [hochsterhunekedagger, Theorem 3.1].

Assume now that the ideal is primary. We have already seen in Remark 1.5 that we only need to look at valuations that are non-negative on . If they are in addition positive on then they are equivalent by a theorem of Izumi ([izumimeasure]). So the only remaining case is that and there is an such that . Since is primary for so with respect to this fixed valuation the dagger closure of is . ∎

Of course, it is interesting to ask how Definition 1.4 compares to the graded version of dagger closure (Definition 1.3). We do have a result in positive characteristic. First of all, we need the following

###### 1.10 Lemma.

Let be an -graded domain finitely generated over a field . Then the valuation by grading extends to a -valued valuation on the completion of which is non-negative on and positive on . Moreover, is again a domain.

###### Proof.

This is provided by the “Untergrad”-function in [schejastorch2, proof of Satz 62.12] which also implies the last assertion. Alternatively, there is an ad-hoc argument in [staeblerthesis, proof of Lemma 9.10]. ∎

###### 1.11 Proposition.

Let be an -graded domain finitely generated over a field of positive characteristic. Then graded dagger closure coincides with dagger closure as in Definition 1.4.

###### Proof.

We prove that graded dagger closure coincides with tight closure in this setting. Then the result follows by Proposition 1.8.

As before one inclusion is clear. Denote the completion of at by and note that it is an integral domain by the previous lemma. Now, if an element is contained in for some ideal then is multiplied into by elements of arbitrarily small order with respect to the valuation in Lemma 1.10. Therefore, by [hochsterhunekedagger, Theorem 3.1]. Since has a completely stable test element (fields are excellent local rings – thus [hochsterhuneketightzero, Theorem 1.4.11] applies) this implies . ∎

###### 1.12 Corollary.

Let be an -graded ring finitely generated over a field of positive characteristic. Then graded dagger closure coincides with tight closure.

###### Proof.

Since both definitions reduce to the domain case by killing minimal primes the result follows from the proof of Proposition 1.11. ∎

###### 1.13 Proposition.

Let be a noetherian domain. Then dagger closure coincides with integral closure for principal ideals and dagger closure is always contained in integral closure.

###### Proof.

Let be a principal ideal in and let be the normalisation of . If then (this follows from the first part of the proof of [brunsherzog, Proposition 10.2.3] which does not require the noetherian hypothesis) and hence immediately .

For the other inclusion and the second assertion one can argue as in [heitmannplusextended, Proposition 2.6 (a)]). ∎

## 2. Dagger closure in regular domains

We now want to prove our main result, namely that for every ideal in a regular noetherian domain containing a field we have . The key ingredient will be the existence of big Cohen-Macaulay algebras.

###### 2.1 Definition.

Let be a noetherian local ring. A *big balanced Cohen-Macaulay algebra* is an -algebra such that the images of every system of parameters in form a regular system in .

The existence of such algebras has been proven by Hochster and Huneke if is an excellent local domain containing a field (see [hochsterhunekeinfinitebig]). In fact, they also proved that if, in addition, is of positive characteristic then is a big balanced Cohen-Macaulay algebra. Moreover, Hochster and Huneke proved in [hochsterhunekeapplicationsBCMA] that this assignment can be made weakly functorial in a certain sense.

###### 2.2 Lemma.

Let be a complete local noetherian domain containing a field and let be a rank one valuation on which is non-negative on . Fix elements in . Then there is a big balanced Cohen-Macaulay algebra for such that if then .

###### Proof.

Note that is excellent. By [hochstertightsolid, Theorem 5.6 (a)] (or [hochsterhunekeapplicationsBCMA, Theorem 5.12]), is then contained in the big equational tight closure of . Hence, it is a fortiori contained in the integral closure of . Now the result follows by the characterisation of integral closure in terms of valuations (cf. [brunsherzog, Proposition 10.2.4 (a)]). ∎

###### 2.3 Lemma.

Let be a finite extension of local domains. Let be a valuation on which is non-negative on . Then is non-negative on .

###### Proof.

Let , we find an equation . Applying yields . Hence, . ∎

###### 2.4 Definition.

Let be a domain and an -module. We say that is *almost zero* if for every valuation on of rank at most one and for every the element is annihilated by an element with .

###### 2.5 Theorem.

Let be a regular noetherian domain containing a field and an ideal. Then .

###### Proof.

Localising at a maximal ideal and completing we may assume that is a noetherian complete regular local domain by Lemma 1.7 and since the completion of a regular noetherian local ring is again a regular noetherian local domain.

Since implies and since is the intersection of -primary ideals (see e. g. [hunekeapplication, Ex. 1.1]) we may assume that is -primary.

Assume now that . Since has finite length we find by [Eisenbud, Theorem 3.1] an element whose annihilator is . Similar to [brennerstaeblerdaggersolid, Proposition 2.8] an element of is in if and only if its image is almost zero in (considered as an -module). Hence, is almost zero. Therefore, for every there is an element such that and . In particular, in and this is the case if and only if for suitable and generators of . We may thus assume that all relevant data are contained in a finite ring extension . Note that is still complete local by [Eisenbud, Corollary 7.6] since is a domain.

By [hochsterhunekeapplicationsBCMA, Theorem 3.9] there is a weakly functorial big balanced Cohen-Macaulay algebra for . We have an exact sequence of -modules. Since is flat over (by [hochsterhunekeinfinitebig, 6.7] – this is precisely where we need regular) tensoring with yields that the annihilator of in considered as an -module is . In particular, .

In order to deduce a contradiction fix a rank one valuation on which is non-negative and positive on and extend it to (this is always possible and then necessarily a rank one valuation on – see [zariskisamuel, vol. 2, VI, §4, Theorem 5’] and [zariskisamuel, vol. 2, VI, §11, Lemma 2]). We consider the restriction of to . By Lemma 2.3 this valuation is non-negative on . Hence, we can apply Lemma 2.2 to see that , where the are ideal generators of . Since is actually positive on this is a contradiction for small enough. ∎

###### 2.6 Remark.

Since the assertion of the Theorem can be reduced to the complete local case the result follows in characteristic from [hochsterhunekedagger, Theorem 3.1] and from the fact that ideals in regular rings are tightly closed. Furthermore, in positive characteristic, is a big balanced Cohen-Macaulay algebra for if is an excellent local domain (see [hochsterhunekeinfinitebig, Theorem 1.1] or [hunekeapplication, Theorem 7.1]). But this is wrong in characteristic zero if .

###### 2.7 Corollary.

Let be an -graded polynomial ring over a field and an ideal. Then .

###### Proof.

Note that since is -graded any regular ring with a field is in fact a polynomial ring – see [brunsherzog, Exercise 2.2.25 (c)].

## 3. Consequences of the main theorem

In this section we draw various consequences from our main theorem 2.5, discuss the one dimensional situation and compare dagger closure to Heitmann’s full rank one closure.

###### 3.1 Corollary.

Let be the ring of invariants of a finite group acting linearly on a polynomial ring over a field of characteristic such that does not divide the group order . Then for any ideal we have .

###### Proof.

Recall that for a local noetherian ring of dimension and parameters an element in is called a *paraclass*.

###### 3.2 Corollary.

Let be an excellent local domain of dimension containing a field . Then no paraclass is almost zero.

###### Proof.

Let be parameters and the corresponding paraclass. We may assume that is normal. For if is almost zero then a fortiori is almost zero in , where is the normalisation of . Since and is excellent and normal we may moreover assume that is a complete local domain.

By virtue of complete Noether normalisation (see [brunsherzog, Theorem A.22]) the ring is a regular local ring over which is finite. In particular, and since the annihilator of is (cf. [brennerparasolid, Lemma 2.8]) it cannot be almost zero by arguments as in the proof of Theorem 2.5. ∎

Next we shall need the notion of a parasolid algebra as introduced in [brennerparasolid]. Since we are not concerned with the mixed characteristic case here we may state a simpler definition.

###### 3.3 Definition.

Let denote a local noetherian ring of dimension containing a field. An -algebra is called *parasolid* if the image of every paraclass in does not vanish.

An algebra over noetherian ring containing a field is called *parasolid* if is parasolid over for every maximal ideal of .

###### 3.4 Lemma.

Let be a finite extension of noetherian domains containing a field and let be an -algebra. If is parasolid as an -algebra then is parasolid.

###### Proof.

We may assume that is a local complete noetherian ring with maximal ideal and that is quasi-local (cf. [brennerparasolid, Proposition 1.5], [Eisenbud, Corollary 7.6] and recall that completion is flat). Fix a paraclass in . Let be a maximal ideal in containing . Localising at one has that are parameters for . In particular, the image of in is nonzero. We have a commutative diagram