Tight closure, coherence, and localization at single elements

In this note, a condition (\emph{open persistence}) is presented under which a (pre)closure operation on submodules (resp. ideals) over rings of global sections over a scheme $X$ can be extended to a (pre)closure operation on sheaves of submodules of a coherent $\mathcal{O}_X$-module (resp. sheaves of ideals in $\mathcal{O}_X$). A second condition (\emph{glueability}) is given for such an operation to behave nicely. It is shown that for an operation that satisfies both conditions, the question of whether the operation commutes with localization at single elements is equivalent to the question of whether the new operation preserves quasi-coherence. It is shown that both conditions hold for tight closure and some of its important variants, thus yielding a geometric reframing of the open question of whether tight closure localizes at single elements. A new singularity type (\emph{semi F-regularity}) arises, which sits between F-regularity and weak F-regularity. The paper ends with (1) a case where semi F-regularity and weak F-regularity coincide, and (2) a case where they cannot coincide without implying a solution to a major conjecture.

Tight closure theory has long been known to be connected with algebraic geometry -e.g. in the connection between test ideals and multiplier ideals [Smi00a], theorems about line bundles [Smi00b], the connection between F-singularities and singularities arising from resolution of singularities [Smi97,Har98], etc.. Hence, it was an early goal to establish that the tight closure operation, like the integral closure operation, would commute with localization at arbitrary multiplicatively closed sets [HH89].Further pressure came from the theorem that if this were true, then tight closure would commute with arbitrary regular base change in all cases of geometric interest [HH94].The question lay open for some 20 years until it was answered in the negative [BM10].Since then, the general attitude in the commutative algebra community has been that the tight closure operation itself is ultimately nongeometric, and that the true geometry that comes from tight closure theory lies in artifacts that arise from its study (e.g.test ideals, the notion of strong F-regularity, etc.).
In this work, we take a different tack.Namely, even though tight closure does not exhibit all of the localization behavior one desires, it does behave well with respect to localization in some ways, and this can be exploited to create notions of tight closure for quasi-coherent sheaves over an open subset of a Noetherian scheme of prime characteristic.Arising from these ideas, we obtain a singularity type between weak F-regularity and F-regularity, and we recast the question of whether tight closure commutes with localization at a single element of a ring in terms of a question about the coherence of certain sheaves of ideals or submodules.
Our approach is quite general.Indeed, we isolate two characteristics of a closure operation, or even a preclosure operation, on ideals or submodules that allow for a sheaf theory based on that operation.If a (pre)closure operation is openly persistent (meaning the operation is compatible with restriction to affine open subsets of an affine scheme), then a natural extension from a (pre)closure operation ideals or submodules to sheaves of ideals or of submodules is constructed.In fact, it is shown that the resulting operation on sheaves of ideals, or on sheaves of submodules of a given quasi-coherent module, is itself a (pre)closure operation on the corresponding poset.Interestingly, though the ambient sheaf must be quasi-coherent, the subsheaf need not be quasi-coherent in order to construct the closure sheaf.This is important because we do not know whether the closure of a coherent sheaf of ideals must be quasi-coherent.
If in addition a (pre)closure operation is glueable (a technical condition), and the sheaf of ideals or submodules is quasi-coherent, then the new operation becomes easier to work with than otherwise, as one has attractive equivalent definitions for it.Quasi-coherence becomes a key question, as it is shown that for a glueable preclosure operation on ideals or submodules, commutation with localization at single elements is equivalent to quasi-coherence being maintained when the resulting operation is applied to a quasi-coherent subsheaf.We then show that any preclosure operation that commutes with localization at single elements, as well as tight closure and its variants like a t -tight closure, are openly persistent and glueable.Hence all of the above results apply, and the single-element tight closure localization question is recast as a question about coherence of tight closure sheaves.
We obtain a new F-singularity type between that of F-regularity and weak F-regularity, which we dub semi F-regularity.It is defined by saying that every sheaf of ideals is tightly closed.We show that it is equivalent to weak F-regularity in Jacobson rings, but that they cannot be equivalent in local rings without implying that F-regularity is the same as weak F-regularity.

Persistence and closure operations on subsheaves
The purpose of this section is to give conditions under which a (pre)closure operation on ideals or submodules over rings arising from a scheme lifts to a (pre)closure operation on sheaves of ideals or of submodules over the scheme.For unexplained terminology in algebraic geometry, see [Har77].For background on closure operations, see the survey [Eps12] or the book [Ell19].
Definition 1.1.Let (S, ≤) be a partially ordered set.A preclosure operation on S is a function cl ∶ S → S (written s ↦ s cl ) that is: • extensive (i.e., s ≤ s cl for all s ∈ S) and • order-preserving (i.e., whenever s, t ∈ S with s ≤ t, we have s cl ≤ t cl ).A preclosure operation cl on S is a closure operation on S if it is also • idempotent (i.e.(s cl ) cl = s cl for all s ∈ S).Let R be a ring and M an R-module.Set (Id(R), ⊆) to be the poset of ideals of R. Then a (pre)closure operation on (the ideals of ) R is a (pre)closure operation on the poset Id(R).Let (Sub(M ), ⊆) be the poset of R-submodules of M .Then a (pre)closure operation on (the submodules of ) M is a (pre)closure operation on Sub(M ).
If R is a class of rings and M is a class of modules over various members of R, and we have a definition of a (pre)closure operation cl M on each M ∈ M , we say cl is defined on (submodules in) Definition 1.2.Let R be a ring.Let cl be a preclosure operation, such that for any open affine subset U = Spec S ⊆ Spec R, cl is defined on the ideals of S (resp.the submodules of any (finite) S-module).Then we say cl is openly persistent on ideals (resp.(finite) modules) over Spec R if for any ideal I of R and x ∈ I cl (resp.any R-submodule inclusion L ⊆ M (with M finite) and x ∈ L cl M ), and any affine open subset U ⊆ Spec R with corresponding ring map ϕ ∶ R → S, we have ϕ(x) ∈ (IS) cl (resp.x ⊗ 1 ∈ (LS) cl M ⊗ R S ).More generally, for a scheme X, we say cl is openly persistent on ideals (resp.(finite) modules) over X if for any open affine U ⊆ X, cl is openly persistent on ideals (resp.(finite) modules) over U .
Remark 1.3.Suppose cl is a closure operation that is persistent over flat ring maps.Then it is openly persistent.This is because the ring homomorphisms R → S arising from open immersions are always flat [Har77, Proposition III.9.2(a)].
Remark 1.4.If cl is openly persistent over Spec R, then for any f ∈ R, we always have Example 1.5.Open persistence does not always imply commuting with localization, even at multiplicative sets generated by a single element.To see this, first recall the v-and t-operations.Namely, if R is an integral domain with fraction field F , then for any fractional ideal a of R, one defines a −1 ∶= {x ∈ F xa ⊆ R. Then we set (0) v = (0), and for a nonzero ideal I, we set I v ∶= (I −1 ) −1 .If I = I v , we say I is divisorial (e.g., any principal ideal).We let I t ∶= ⋃{J v J ⊆ I, J is a finitely generated ideal}.Then both v and t are closure operations, with t ≤ v (see [Ell19,p. 15]).Clearly if R is Noetherian, then t = v.Note also that whenever I is an ideal of a ring R and R → S is flat, then I t S ⊆ (IS) t [Ell19, Proposition 1.2.3 (1) ⇒ (5)].Therefore, t is openly persistent by Remark 1.3.Now let T = Q[x, y] (x,y) , let m be the maximal ideal of T , let p be a positive prime number, and consider the subring m is divisorial as an ideal of R. Hence it is t-closed.On the other hand, by [Mat86, Exercise 12.4], since T is a Krull domain, any proper nonzero divisorial ideal of T must have height 1, and ht m = 2, so m is not divisorial in T (hence also not t-closed since T is Noetherian).Thus, (mR This example is due to Evan Houston.
Lemma 1.6.Let cl be an openly persistent preclosure operation on modules over a scheme X.Let M be a quasi-coherent O X -module, and L a sheaf of By open persistence of cl, we have Note that ϕ is an isomorphism because M is quasi-coherent [Liu02, Chapter 5, Exercise 1.4], and i is injective since B is flat over A. The fact that j is injective is part of what "sheaf of submodules" means.It follows that ψ is injective, and that the image of ψ can be identified with under these identifications.Theorem/Definition 1.7.Let cl be an openly persistent preclosure operation on modules over a scheme X.Then we can extend it to a preclosure operation on subsheaves of quasi-coherent sheaves of modules.
In particular, let M be a quasi-coherent O X -module, and let L ⊆ M be a sheaf of submodules.Define a sheaf L cl M of submodules of M as follows: Moreover, this is a preclosure operation on subsheaves of M, in that and cl is a closure operation (i.e.idempotent), then we obtain a closure operation on subsheaves on M, in that we also have idempotence.That is: Proof.First we prove that for any open U ⊆ X,

The locality property for L cl
M follows from that of M. To see the equivalence in the two definitions, let U be an open set and Finally, let L be a sheaf of O X -submodules of M, where X is Noetherian, M is coherent, and cl is idempotent on submodules.Let U be an open subset of X, and let Then by open persistence of cl, we have , where both the containment and the last equality follow from the fact that cl is a closure operation on O X (W )-submodules of M(W ).Since x ∈ U was arbitrary and M .But we already know the subsheaf relation holds in the other direction.Hence, Remark 1.8.Suppose one only had a (pre)closure operation on ideals, and not submodules, then the above holds for sheaves of ideals -with the same proof, restricting our attention to M = O X .That is, we have the following: Theorem/Definition 1.9.Let cl be an openly persistent preclosure operation on ideals over a scheme X.Then we can extend it to a preclosure operation on sheaves of ideals.
In particular, let I ⊆ O X be a sheaf of ideals.Define a sheaf I cl of ideals as follows: For U ⊆ X open and s ∈ O X (U ), we say Given s ∈ O X (U ), we have s ∈ I cl (U ) if and only if there is some collection The above definition makes I cl a sheaf of ideals of O X .Moreover, this is a preclosure operation on sheaves of ideals, in that • For any ideal sheaf I of O X , I ⊆ I cl , and • Given an ideal sheaf inclusion J ⊆ I, J cl is a subsheaf of I cl .If, moreover, X is Noetherian and cl is a closure operation (i.e.idempotent), then we obtain a closure operation on ideal sheaves, in that we have idempotence.That is: • For any ideal sheaf I, we have (I cl ) cl = I cl .

Glueable (pre)closure operations
In this section, we provide a second condition (glueability) under which (pre)closures of sheaves are particularly well behaved.
Definition 2.1.Let R be a ring and cl be an openly persistent preclosure operation on submodules (resp.ideals) over Spec R. Let I be an ideal, and let (f α ) α∈A be a generating set for I. Let g ∈ √ I. Let L ⊆ M be R-modules (resp.let J be an ideal) and z ∈ M (resp.z ∈ R) such that for all α ∈ A, we have If cl is an openly persistent preclosure operation on submodules (resp.ideals) over a scheme X, we say it is glueable over X if for all affine open U = Spec R ⊆ X, cl is glueable over R.
Proposition 2.2.Let R be a ring.Let cl be an openly persistent preclosure operation on submodules (or ideals) over Spec R that commutes with localizations at single elements.Then cl is glueable over R.
Proof.We prove the module case; the proof of the ideal case is identical.
Let z, f α , I, g, L, M be as in the definition of glueability.Then for all α, we have z 1 ∈ (L fα ) cl M fα = (L cl M ) fα .Thus, for each α, there is some positive integer n α such that f nα α z ∈ L cl M .Since radicals of ideals are insensitive to powers of generators, we have g ∈ ({f nα α } α∈A ), whence for some positive integer N we have g N ∈ ({f nα α } α∈A ).Thus, there exist α 1 , . . ., α s ∈ A and r 1 , . . ., r s ∈ R with It is important at this point to establish some properties of glueable preclosure operations.
Proposition 2.3.Let cl be a glueable preclosure operation on submodules over R. Let I be an ideal of R, and let (f α ) α∈A , (g β ) β∈B be two generating sets of ideals that have the same radical as I. Let M be an R-module, L ⊆ M a submodule, and z ∈ M .Then z 1 ∈ (L fα ) cl M fα over the ring R fα for all α ∈ A, if and only if z 1 ∈ (L g β ) cl Mg β over the ring R g β for all β ∈ B.
Proof.The forward direction follows from the definition because each g β ∈ ({f α ∶ α ∈ A}).The backward direction follows from reversing the roles of the f s and the gs.
Corollary 2.4.Let cl be a glueable preclosure operation on submodules over R. Let (f α ) α∈A be elements of R that generate the unit ideal.Let M be an R-module, L a submodule of M , and z ∈ M .Then z ∈ L cl M if and only if for each α, we have z 1 ∈ (L fα ) cl M fα .Proof.In the above proposition, let B be a singleton and g = g β = 1.
Theorem 2.5.Let cl be a glueable preclosure operation on submodules over a scheme X.Let M a quasi-coherent O X -module, and L a quasi-coherent O X -submodule of M. Let U be an open set in X and s ∈ M(U ).Then , and let V be an affine open subset of U .Since V is affine, we have V = Spec R for some R. Let {U α } α∈A be an affine open cover of U such that s Uα ∈ L(U α ) cl M(Uα) for all α ∈ A. For each α ∈ A, since U α ∩ V is an open subset of V , one can find a positive integer n α and elements g α,1 , . . ., g α,nα ∈ R such that U α ∩ V = ⋃ nα i=1 D R (g α,i ).For any maximal ideal m of R, we have m ∈ U α ∩ V for some α, whence there is some g α,i , 1 ≤ i ≤ n α , such that m ∈ D(g α,i ), which means that g α,i ∉ m.It follows that the set {g α,i α ∈ A, 1 ≤ i ≤ n α } generates the unit ideal R.
. But by quasi-coherence of the sheaves L and M, we have . Since this holds for all such pairs, and since the g α,i generate R, then by Corollary 2.4, we have s Since F is a subsheaf of M, it follows that s ∈ F(U ).Thus, F = L cl M .One can make a similar statement for glueable preclosure operations on ideals over a scheme, with identical proof: Theorem 2.6.Let cl be a glueable preclosure operation on ideals over a scheme X.
Moreover, for any affine open subset V of X, we have Indeed, I cl is unique with respect to this property.That is, suppose F is a sheaf of O X -ideals such that F(V ) = I(V ) cl for all open affine subsets V of X.Then F = I cl .

Quasi-coherence and localization at elements
This section examines conditions for quasi-coherence of (pre)closures of subsheaves, as well as conditions for subsheaves being closed.
Theorem 3.1.Let R be a ring, M an R-module, L ⊆ M , and cl a glueable preclosure operation on submodules over Spec R. The following are equivalent: (a) For any f ∈ R, we have On the other hand, by quasi-coherence of

Application to tight closure and its variants
We start with the following definition.
Definition 4.1.Let R be an F p -algebra.A p-system of ideals is a sequence of ideals b • = {b p e } ∞ e=1 such that for all pairs q, q ′ of powers of p, we have q ⊆ b pq for all powers q of p. • a F -graded system of ideals [Bli13, Definition 3.20] if it is descending (i.e.b q ⊇ b pq for all q).Hence, the notion of p-system is a common generalization of p-families and F -graded systems of ideals.
With this setup, we can introduce the following definition.Definition 4.3.Let R be a Noetherian F p -algebra, let b • be a p-system of ideals, and let L ⊆ M be R-modules.Then for z ∈ M , we say z is in the b • -tight closure of L in M if there is some c ∈ R ○ and some power q 0 of p such that for all q ≥ q 0 , we have M .We write z ∈ L * b• M .This generalizes several notions of tight closure in the literature, e.g.: • Setting b q ∶= R for all powers q of p, this just gives the tight closure operation of Hochster and Huneke [HH90].• If a 1 , . . ., a n are ideals of R such that a j ∩ R ○ ≠ ∅ for each j, and t 1 , . . ., t n are fixed positive real numbers, then setting b q ∶= a ⌈t 1 q⌉ 1 ⋯a ⌈tnq⌉ n , we recover the a t 1 1 ⋯a tn n -tight closure of Hara and Yoshida [HY03, Remark 6.2(2)].
• Given a collection {a n } ∞ n=1 of ideals such that a n ∩ R ○ ≠ ∅ for all n, and such that a n a m ⊆ a n+m (i.e. a graded family of ideals in the terminology of [ELS01]), if we set b q = a q for each power q of p, then the b • -tight closure coincides with the a • -tight closure of Hara [Har05, Definition 2.7] Lemma 4.4.Let R be a Noetherian F p -algebra and b • a p-system of ideals.Then b • -tight closure is a preclosure operation on R-submodules.
Accordingly, let y, z ∈ L * b• M , and r ∈ R. Then there exist powers q 0 , q 1 of p and c, d ∈ R ○ such that whenever q ≥ q 0 (resp.q ≥ q 1 ), we have cb q y q M ⊆ L [q] M (resp.db q z q M ⊆ L [q] M ).Hence for all q ≥ max{q 0 , q 1 }, we have cdb q (y + rz) q M ⊆ d ⋅ (cb q y q M ) + cr q ⋅ (db q z q M ) ⊆ L [q] M .Thus y + rz ∈ L * b• M .To extend this definition to sheaves of submodules, we need the following result.
e=1 be the family of R(U )-ideals given by b(U ) q ∶= b q R(U ).The following are equivalent: Then whenever we have D(f β ) ⊆ U α , it follows that for any β ∈ Σ and any powers q, q ′ of p that b(D(f β )) q b(D(f β )) Also, since the f β generate R, there exist β 1 , . . ., β t ∈ Σ such that R = (f β 1 , . . ., f βt ).Now let q, q ′ be powers of p and let x ∈ b q and y ∈ b q ′ , so that for each 1 ≤ j ≤ t, x 1 ∈ b(D(f β j )) q and y 1 ∈ b(D(f β j )) q ′ .Then we have Thus there is some m j ∈ N with f m j β j xy q ∈ b qq ′ .But since the f m j β j generate R, there exist a 1 , . . ., a t ∈ R with ∑ t j=1 a j f Definition/Proposition 4.6.Let X be a locally Noetherian F p -scheme, and let b • ∶= {b p e } ∞ e=1 be a sequence of quasi-coherent ideal sheaves over X. Theorem 4.8.Let X be a locally Noetherian F p -scheme.Let b • be a psystem of ideal sheaves over X.Then * b • is a glueable preclosure operation on submodules over X.
We know from Lemma 4.4 that whenever b • is a p-system of ideals in a Noetherian ring R of characteristic p, * b • is a preclosure operation.It remains to show open persistence and glueability.We begin with a lemma that is presumably (see [HH94, parenthetical comment within the statement of Theorem 6.22]) well-known.Lemma 4.9.Let ϕ ∶ R → S be a homomorphism of commutative rings that satisfies going-down.Then ϕ(R ○ ) ⊆ S ○ .In particular, this holds if ϕ is flat.
Proof.Let c ∈ R ○ .Let Q be a prime ideal of S with ϕ(c) ∈ Q.Then c ∈ q ∶= ϕ −1 (Q), so by assumption, q cannot be a minimal prime of R. Let p be a prime ideal of R properly contained in q.Then since ϕ satisfies the going-down property, there is some prime ideal P of S with P ⊆ Q and ϕ −1 (P ) = p.Since p ≠ q, it follows that P ≠ Q.Thus, Q is not a minimal prime of S. Since Q ∈ Spec S was arbitrary with ϕ(c) ∈ Q, it follows that ϕ(c) ∈ S ○ .The final statement follows from [Mat86, Theorem 9.5].

Next, we prove open persistence of the operation.
Lemma 4.10.Let X be a locally Noetherian F p -scheme.Let b • be a psystem of ideal sheaves over X.Then * b • is openly persistent on submodules over X.
Proof.Let V ⊆ U be affine open subsets of X. Write R = O X (U ), S ∶= O X (V ), and ϕ ∶ R → S the corresponding ring map.For each power q = p e of p, write b q = b q (U ).Let L ⊆ M be R-modules and x ∈ L * b• M = L * b• M .Then there is some c ∈ R ○ and some power q 0 of p such that for all q ≥ q 0 , cb q x q M ⊆ L [q] M .Applying ϕ and (−) ⊗ R S, it follows that ϕ(c)(b q S)(x ⊗ 1) [q] M ⊗S ⊆ (LS) [q] M ⊗ R S for all q ≥ q 0 .Since ϕ(R ○ ) ⊆ S ○ (by virtue of the flatness of the open immersion V ⊆ U and Lemma 4.9; see also Remark 1.3), we have ϕ(c) ∈ S ○ .Moreover, the quasi-coherence of the ideal sheaves b q implies that b q (V ) = b q S for each power q of p. Thus, letting b • S ∶= {b p e S} ∞ e=1 , we have Proof of Theorem 4.8.We may assume X is affine.Write R = O X (X).For each power q of p, b q ∶= b q (X).Let (f α ) α∈A be elements of R, with I the ideal they generate, and let g ∈ for each α ∈ A, where b • S ∶= {b p e S} ∞ e=1 for any R-algebra S. Since g ∈ √ I, there exist α 1 , . . ., α k ∈ A such that g ∈ (f α 1 , . . ., f α k ).For each 1 ≤ i ≤ k, there is some power q i of p and some M ) fα i for all q ≥ q i .Let q 0 ∶= max{q 1 , . . ., q k } and c = ∏ k i=1 c i .Then there exist positive integers N i,q such fthat for all i and q ≥ q 0 , f M .Fix some q ≥ q 0 .Since g ∈ (f Mg .As this holds for all q ≥ q 0 , we have z 1 ∈ (L g ) * b•Rg Mg . But by quasi-coherence of the ideal sheaves b q , we have b q (D(f α )) = b q R fα for all α, and b q (D(g)) = b q R g .The result follows.
It is instructive to compare the following corollary with [HH90, Theorem 8.14], where under stable test element conditions, a similar thing is proved for localizations at maximal ideals (i.e. at the stalk level).In contrast, the following corollary has no need even of weak test elements.
Corollary 4.11.Let R be a Noetherian ring of positive prime characteristic.Let (f α ) α∈A be elements of R that generate the unit ideal.Let M be an Rmodule, L a submodule of M , and z ∈ M .Then z ∈ L * M if and only if for each α, we have z 1 ∈ (L fα ) * M fα .
Metatheorem 4.12.All the results from the previous sections of the paper apply to tight closure and its variants.
M(V ) by what we have shown above.Conversely, let s ∈ L(V ) cl M(V ) .Then for any open affine subset W of V , open persistence of cl shows that s W ∈ L(W ) cl M(W ) .Hence, s ∈ L cl M (V ).For the "indeed" statement, let U be an open subset of X and s ∈ M(U ).If s ∈ F(U ), then by the restriction axiom, for any open affine subset V of U , we have s follows that C is a subsheaf of Lcl M .On the other hand, let U be an open set and let s ∈ Lcl M (U ).Then there is some open affine cover{U α α ∈ A} of U with s Uα ∈ L(U α ) cl M (Uα).We can refine the cover so that U α = D(g α ) for some elements g α ∈ R. Then by open persistence and the assumption (a), we haves D(gα) ∈ (L gα ) cl Mg α = (L cl M ) gα = C(D(g α )).Hence, s ∈ C(U ).Therefore, Lcl M = C. Proposition 3.2.Let X be a scheme, and cl a glueable preclosure operation on modules over X.Let L ⊆ M be quasi-coherent O X -modules.The following are equivalent.(a) L = L cl M .(b) There is an open affine cover {U α α ∈ A} of X such that for each α, L(U α ) = L(U α ) cl M(Uα) , and L cl M is quasi-coherent.Proposition 3.4.Let cl a glueable preclosure operation on ideals over a scheme X.The following are equivalent: (a) For any quasi-coherent sheaf I of ideals on X, we have I cl = I.(b) For any ideal sheaf I on X, we have I cl = I.(c) For any affine open subset U of X, every ideal of O X (U ) is cl-closed.

Proposition 4. 5 .
Let R be a Noetherian F p -algebra, and let b • = {b p e } ∞ e=1 a collection of ideals indexed by the powers of p.For any affine open subset U of Spec R, let R(U ) ∶= O Spec R (U ) be the corresponding ring, with R-algebra structure induced from the open immersion U ↪ Spec R, and let b ) ⇒ (b): Choose the cover consisting of all affine open sets.(b) ⇒ (a): Refine {U α } to an open cover of the form {D(f β )} β∈Σ .

For any affine open
subset U of X, set b(U ) • ∶= {b p e (U )} ∞ e=1 .The following are equivalent: (a) There is an affine opencover {U α } of X such that b(U α ) • is a psystem of O X (U α )-ideals for each α ∈ Λ.(b) For any affine open subset U of X, b(U ) • is a p-system of O X (U )-ideals.Under these conditions we call b • a p-system of ideal sheaves over X.Proof.We need only prove (a)⇒ (b).For each α ∈ Λ, let {V β } β∈Σα be an affine open cover of U ∩ U α .By Proposition 4.5((a)⇒ (c)), b(V β ) • is a p-system of O X (V β )-ideals for all β ∈ Σ α for all α ∈ Λ.But then by Proposition 4.5((b) ⇒ (a)), b(U ) • is a p-system of O X (U )-ideals.Definition 4.7.Let X be a locally Noetherian F p -scheme.Let b • = {b p e } ∞ e=1 be a p-system of ideal sheaves over X.For any affine open subset U ⊆ X and any R-submodule inclusion L ⊆ M (where R = O X (U )), set b(U ) • ∶= {b p e (U )} ∞ e=1 and L * b• M ∶= L * b(U )• M .

Remark 4. 13 .
In particular (using ordinary tight closure for simplicity), we define the tight closure of a subsheaf L of a quasi-coherent sheaf M by saying that a section s ∈ M(U ) is in L * M (U ) if U has an affine open cover {U α } α∈A such that s Uα ∈ L(U α ) * M(Uα) as O X (U α )-modules for all α ∈ A. Then we see above that this gives a sheaf of submodules of M, and defines a closure operation on sheaves of submodules of M (or in the case of most b • -tight closures, only a preclosure operation).In the case where L is a coherent sheaf of submodules, we have that s ∈ L * M (U ) if and only if for every affine open V in U , we have s V ∈ L(V ) * M(V ) -and L * M is the unique sheaf that admits this property.Consider the question: Does tight closure commute with localization at single elements?This seems to be open, as it was when Brenner and Monsky gave the counterexample to tight closure commuting with arbitrary localization [BM10, p. 572].We now reformulate the question as follows: Theorem 4.14.Let R be a Noetherian ring containing F p , and let I be an ideal of R. The following are equivalent (a) For any f ∈ R, (I f ) * = (I * ) f .(b) The sheaf ( Ĩ) * of O Spec R -ideals is coherent.Thus, tight closure of ideals in R commutes with localization at single elements if and only if the tight closure of any coherent sheaf of O Spec R -ideals is coherent.Similarly, if M is an R-module and L a submodule, the following are equivalent:(a) For anyf ∈ R, (L f ) * M f = (L * M ) f .(b) The sheaf L * M of O X -submodules of M is quasi-coherent.Thus,tight closure of submodules of R-modules (resp.finite R-modules) commutes with localization at single elements if and only if the tight closure of any quasi-coherent sheaf of submodules of a quasi-coherent (resp.coherent) sheaf of O Spec R -modules is always quasi-coherent.Proof.This follows from Theorem 4.8 (with n = 0) and Theorem 3.1.We introduce a new (non-local!)F-singularity class in the following theorem.Theorem 4.15.Let R be a Noetherian ring of positive prime characteristic and X = Spec R. The following are equivalent: (a) For any ideal I of R, we have I * = I, where I = Ĩ.(b) For any ideal sheaf I on X, we have I * = I.(c) For any affine open subset U of X, every ideal in O X (U ) is tightly closed.(d) For any f ∈ R and any ideal I of R, we have I f = (I f ) * (i.e.R f is weakly F-regular).(e) For any R-module inclusion L ⊆ M , with M finitely generated, we have L * M = L, where L = L and M = M .(f ) For any coherent sheaf M of O X -modules and any sheaf L of submodules, we have L * M = L. (g) For any affine open subset U of X and any finite O X (U )-module M , 0 is tightly closed in M .(h) For any f ∈ R and any inclusion of finite R-modules L ⊆ M , we have (L f ) * M f = L f as R f -modules.