aa r X i v : . [ m a t h . A C ] J u l COMPATIBLE IDEALS IN GORENSTEIN RINGS
THOMAS POLSTRA AND KARL SCHWEDE
Abstract.
Suppose R is a Q -Gorenstein F -finite and F -pure ring of prime characteristic p >
0. We show that if I ⊆ R is a compatible ideal (with all p − e -linear maps) then thereexists a module finite extension R → S such that the ideal I is the sum of images of all R -linear maps S → R . Introduction
Compatibly Frobenius split ideals and subvarieties have played an important role in thestudy of rings and varieties in characteristic p >
0. They first formally appeared in [MR85]in their study of Schubert varieties, although they also implicitly played a central role in[Fed83], at the dawn of the theory of characteristic p > I are always “compatible” in that for every φ : F e ∗ R → R , wehave φ ( F e ∗ I ) ⊆ I. The test ideal [Vas98, Sch10] is the smallest nonzero compatible ideal while the splittingprime is the largest compatible proper ideal, [AE05]. Being compatibly split is also a centralpart of the theory of Frobenius split varieties [MR85, BK05]. On the other hand, it turns outthat the compatibly split ideals are also closely related to the theory of log canonical centers from birational complex geometry, [Sch10]. Thus, as we begin to move into the world ofmixed characteristic singularities, it behooves us to look for other characterizations of theseimportant special ideals.One other characterization of the test ideal, at least in a Q -Gorenstein domain, is that itis the smallest possible nonzero imageHom R ( S, R ) eval @1 −−−−→ R where S ⊇ R is a finite extension, [BST11, Smi94]. But what about the other compatibleideals? Are they also images of finite extensions? We answer this in the case R is Q -Gorenstein with index not divisible by p (for example if ω ( n ) ∼ = R for some n not divisibleby p ).To do this, we prove a more finely tuned version of the celebrated Equational Lemma ,killing certain cohomology classes while leaving others nonzero (by keeping the extension´etale over certain primes). Let R be a Noetherian F -finite ring of prime characteristic p > Polstra was supported in part by NSF Postdoctoral Research Fellowship DMS at any minimal prime relations on parameters of R inside a finite extension of R . Consequently, the absoluteintegral closure of R is a big Cohen-Macaulay algebra. In fact, Hueneke and Lyubeznik[HL07] showed that one can even kill all lower local cohomology in a single finite extensioninstead of going all the way to R + , see [SS12, Bha12a, Bha12b] for generalizations.Instead of killing intermediate local cohomology however, we are interested in studyingthe top local cohomology and killing cohomology classes that belong to the tight closure ofzero. This translates to constructing the parameter test module, see [Smi94], and via finitecovers correspond to the test ideal, as done in [BST11]. Main Theorem (Corollary 3.3) . Let R be a Noetherian F -finite and F -pure Q -Gorensteinring of prime characteristic p > such that the index of ω R is not divisible by p > .Suppose I ⊆ R is a compatible ideal of R . Then there exists a finite extension R → S sothat I = Im(Hom R ( S, R ) → R ) . If R is a domain then S can be chosen to be a domain. Acknowledgements.
The authors would like to thank Bhargav Bhatt and Javier Carvajal-Rojas for valuable discussions. 2.
Background
Suppose R is an F -finite ring of prime characteristic p >
0. An ideal of I ⊆ R is saidto be compatible if for all e ∈ N every ϕ ∈ Hom R ( F e ∗ R, R ) naturally restricts to a map inHom
R/I ( F e ∗ R/I, R/I ), i.e. ϕ ( F e ∗ I ) ⊆ I . Every compatible ideal in an F -pure ring is radical.We sketch the argument now. If x n ∈ I then x p e ∈ I for all e ≫
0. Because R is F -pure forevery e ∈ N there exits ϕ ∈ Hom R ( F e ∗ R, R ) such that ϕ ( F e ∗
1) = 1. Therefore, since we areassuming the ideal I is compatible we have that ϕ ( F e ∗ x p e ) = xϕ ( F e ∗
1) = x ∈ I . Moreover,compatible ideals are easily seen to be closed under finite sums and finite intersections. Infact, it follows from this that there are finitely many compatible ideals in an F -pure ring by[EH08, Theorem 3.1] and [Sha07] in the local case, [Sch09, KM09] more generally. In fact,there are even bounds on how many such ideals there can be [ST10b, HW15]. Definition 2.1 (The trace ideal ) . Suppose R is a ring with canonical module ω R and S isan R -algebra. The trace ideal of S , denoted τ S/R is the image of the evaluation-at-1 map:Hom R ( S, R ) → R. This ideal is also called the order ideal in [EG82]. Note, in the case that S is a finite R -module, then Hom R ( S, R ) = ω S/R .The following lemma, which is well known to experts, but for which we could not find areference, states that the trace ideal of a finite extension of R defines a compatible ideal of R . Lemma 2.2.
Let R be an F -finite ring of prime characteristic p > . Suppose R → S is afinite extension of R and τ S/R is the trace ideal. Then I is a compatible ideal of R . OMPATIBLE IDEALS IN GORENSTEIN RINGS 3
Proof.
Fix some φ ∈ Hom R ( F e ∗ R, R ). The result follows immediately from the commutativityof the following diagram:Hom F e ∗ R ( F e ∗ S, F e ∗ R ) ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ F e ∗ Hom R ( S, R ) (cid:15) (cid:15) e φ / / Hom R ( S, R ) (cid:15) (cid:15) F e ∗ R φ / / R Here the vertical maps are obtained by evaluation at 1, and e φ is obtained as the followingcomposition: e φ : Hom F e ∗ R ( F e ∗ S, F e ∗ R ) ֒ → Hom R ( F e ∗ S, F e ∗ R ) restrictsource −−−−→ Hom R ( S, F e ∗ R ) Hom R ( S,φ ) −−−−−−→ Hom R ( S, R ) . (cid:3) Since the trace ideal is compatible, in an F -pure ring, all trace ideals are radical and R modulo a trace ideal is automatically F -pure as well.We recall the notion of a quasi-Gorenstein and Q -Gorenstein ring. Definition 2.3.
An S2 ring is called quasi-Gorenstein (or 1-Gorenstein) if ω R is locally free.A G1 and S2 local ring is called Q -Gorenstein if some symbolic (equivalently reflexive orS2-ified) power of ω R is locally free. The index of a Q -Gorenstein local ring is the smallestsuch power that makes it locally free, in other words it is the smallest n such that ω ( n ) R islocally free. Lemma 2.4.
Let R be an F -finite and quasi-Gorenstein ring of prime characteristic p > and let Q ∈ Spec( R ) . Suppose R ֒ → S is a finite extension of R . Then the following areequivalent:(1) The trace ideal τ S/R is contained in Q .(2) The map of local cohomology modules H ht( Q ) QR Q ( R Q ) → H ht( Q ) QR Q ( S Q ) is not injective.Proof. Without loss of generality we may assume R = ( R, m ) is a complete local ring ofKrull dimension d with maximal ideal Q = m . The Matlis dual of H d m ( R ) → H d m ( S ) is thetrace map Tr S/R : ω S/R = Hom R ( S, R ) → R . The local cohomology map is injective if andonly if the dual map is injective, in other words if the image is not contained in m . (cid:3) Suppose ( R, m , k ) is a local quasi-Gorenstein ring of prime characteristic p > d . Then Matlis duality provides to us a one-to-one correspondence between com-patible ideals of R and Frobenius stable submodules of H d m ( R ), see [BB11, Proposition 5.2]for details. We record this correspondence as a lemma for future reference. Lemma 2.5.
Let ( R, m , k ) be an F -finite and complete local quasi-Gorenstein ring of primecharacteristic p > . Then there is a one-to-one correspondence between compatible idealsof R and Frobenius stable submodules of H d m ( R ) . If ( − ) ∨ denotes Malis duality then thecorrespondence is given by Gorenstein in codimension 1
THOMAS POLSTRA AND KARL SCHWEDE (1) If I is a compatible ideal of R then ( R/I ) ∨ is a Frobenius stable submodule of H d m ( R ) .(2) If N ⊆ H d m ( R ) is Frobenius stable submodule of H d m ( R ) then Ann R ( N ∨ ) is a compatibleideal of R . Proof of Main Theorem
Theorem 3.1.
Let R be an F -finite and F -pure quasi-Gorenstein ring of prime characteristic p > and suppose that I ⊆ R is a compatible ideal. Then there exists a finite extension R → S such that I = Im(Hom R ( S, R ) → R ) . If R is a domain then S can be chosen to be adomain. Our proof is closely related to the method of [HL07]. However because of our assumptionswe are able to use prime avoidance to control the form of the equations that the elementswe are adjoining satisfy.
Proof.
Recall that compatible ideals in an F -pure ring are always radical ideals. Suppose I = P ∩ · · · ∩ P t and each P i ∈ Spec( R ). Let Q = { Q , . . . , Q m } be the finitely manycompatible prime ideals of R for which I is not contained in. We will show that there existsa finite map R → S so that I = Im(Hom R ( S, R ) → R ) by constructing a finite R -algebra S so that(1) Im(Hom R ( S, R ) → R ) ⊆ I (2) and R → S is ´etale at all primes Q ∈ Q .This will show I = Im(Hom R ( S, R ) → R ) = τ S/R . Indeed, if R → S is ´etale at each Q ∈ Q then R → S splits at each Q ∈ Q and the trace ideal of R → S cannot be contained in suchprimes. By Lemma 2.2 the ideal τ S/R is radical and therefore must agree with I since itsprime components are all compatible by Lemma 2.2.Consider the local ring R P j . Suppose that R P j is d j -dimensional. The maximal ideal of R P j is compatible and therefore the socle of H d j P j ( R P j ) is a 1-dimensional Frobenius stablesubmodule by Lemma 2.5. If η j ∈ H d j P j ( R P j ) generates the socle then there exists a u j ∈ R P j such that η pj = u j η j . Because R is F -pure, in particular F -injective, the element u j is a unitof R P j . By clearing denominators and replacing η j by a suitable multiple of itself by a unitof R P j , we may assume that u j ∈ R .We claim that we may alter the element u j so that η pj = u j η j and u j is a unit of R Q i for each 1 ≤ i ≤ m . Suppose { q , q , . . . , q ℓ } is the collection of maximal elements of Q with respect to inclusion and has been written so that u j avoids q ∪ · · · ∪ q i but u j is anelement of q i +1 ∩ · · · ∩ q m . The prime ideals q , . . . , q ℓ are mutually incomparable and no P i is contained in some q n . So we can choose an element a ∈ P ∩ · · · ∩ P t ∩ q ∩ · · · ∩ q i whichavoids q i +1 ∪ · · · ∪ q m . Observe that aη j = 0 and therefore we may replace u j by u j + a andstill have that η pj = u j η j . The element u j now avoids every element of Q , i.e. u j is a unit ofthe localizations R Q i for each 1 ≤ i ≤ m .Prime avoidance allows us to choose parameters x , . . . , x N of R with the following prop-erties:(1) if R P i is d i -dimensional then x , . . . , x d i . is a system of parameters of R P i ;(2) each x i avoids every element of Q . OMPATIBLE IDEALS IN GORENSTEIN RINGS 5
Suppose that R P j is d j -dimensional and ˇ C • ( x , . . . , x d j ; R P j ) is the ˇCech complex on x , . . . , x d j . We realize the local cohomology module H dP j ( R P j ) as the ˇCech cohomologymodule H d ( ˇ C • ( x , . . . , x d j ; R P j )) and choose α j ∈ ˇ C d ( x , . . . , x d j ; R P j ) a representative of η j .Let g j ( T ) = T p − u j T , a monic polynomial over R . Then g j ( η j ) = 0 and so there exists β j ∈ ˇ C d j − ( x , . . . , x d j ; R P j ) so that g j ( α j ) = ∂ d j − ( β j ). Suppose that β j = r i,j x j · · · b x i · · · x j d d ! di =1 . Let { T i,j } ≤ j ≤ t ≤ i ≤ dj be variables and consider the single variable polynomials f i,j := g T i,j x j · · · c x i · · · x j d d ! − r i,j x j · · · c x i · · · x j d d . Multiplying f i,j by ( x j · · · c x i · · · x j d d ) p produces a monic polynomial ˜ f i,j in R [ T i,j ] so that d ˜ f i,j dT i,j = u j ( x j · · · c x i · · · x j d d ) p − . Observe that each of the derivatives d ˜ f i,j dT i,j are units in the localized rings R Q for each Q ∈ Q .Therefore the R -algebra R ′ = R [ T i,j ] ≤ j ≤ t ≤ i ≤ dj / ( ˜ f i,j ) ≤ j ≤ t ≤ i ≤ dj is ´etale when localized at each elementof Q . Denote by t i,j the images of T i,j in R ′ .In what follows, elements of the total ring of fractions of R map to the total ring offractions of R ′ , and we identify them with their images. Consider the following elements ofˇ C d j − ( x , . . . , x d j ; R ′ P j ) and ˇ C d j ( x , . . . , x d j ; R ′ P j ) respectively:(1) β j = (cid:18) t i,j x n ··· b x i ··· x ndd (cid:19) di =1 ;(2) α j = α j − ∂ d j − ( β j ).Then g j ( β j ) = β j and therefore g j ( α j ) = g j ( α j ) − g j ( ∂ d j − ( β j )) = ∂ d j − ( β j ) − ∂ d j − ( g j ( β j )) = 0 . Therefore α j is an element of the total ring of fractions of R ′ satisfying the monic poly-nomial g j ( T ). In particular, the R -algebra obtained by adjoining this element to R ′ isisomorphic to R ′ [ T ] / ( g ( T )) modulo an intersection of a subset of minimal primes. Let S = R ′ [ T j ] ≤ j ≤ t / ( g j ( T j )) ≤ j ≤ t . Then each α j ∈ ˇ C d j ( x , . . . , x d j ; S P j ) is an element of S P j and therefore represents the 0-element of H d j P j ( S P j ), i.e. the image of η j in H d j P j ( S P j ) is 0.Furthermore, just as above, we see that S Q is ´etale over R Q for each Q ∈ Q .The R -algebra S is obtained by the variables T i,j and then modding out by monic poly-nomials in each of the variables. Hence S is a finite extension of R . By Lemma 2.4 andLemma 2.5 we find that the trace ideal τ S/R is contained in I . By Lemma 2.2 we know that τ S/R is compatible. In particular, the trace ideal is a radical ideal and it remains to observethat I is not contained in any element of Q . But this is indeed the case since if Q ∈ Q then R → S is ´etale at Q and hence the trace ideal of R → S at Q agrees with the unit ideal. (cid:3) THOMAS POLSTRA AND KARL SCHWEDE
Our goal for the rest of the section is to generalize Theorem 3.1 to the Q -Gorenstein case,at least when the index is not divisible by p . First we present a lemma very closely relatedto work of Speyer [Spe20], also c.f. [ST10a]. Lemma 3.2.
Suppose that R is F -finite and that R ⊆ T is a finite split extension of S2reduced rings that is ´etale in codimension over R and that every minimal prime of T dominates a minimal prime of R . Let Tr := Tr K ( T ) /K ( R ) denote the trace map on the totalrings of fractions and suppose that Tr( T ) ⊆ R . Fix a surjective map φ : F e ∗ R → R extending(uniquely) to a map φ T : F e ∗ T → T . Then I ⊆ R is compatible with φ if and only if √ IT ⊆ T is compatible with φ T .Proof. Since R ⊆ T is ´etale in codimension 1 and the extension is split, we have that Trgenerates the S -module, Hom( S, R ), hence Tr is surjective. Additionally, since φ is surjective, I is radical and hence without loss of generality, we may assume that ( R, m ) is complete andlocal, I = m and T is semi-local and hence a finite product of complete local reduced finiteextensions T i of R . In this setting Tr simply sums over the Tr of the individual terms in theproduct. By restricting φ T to each T i , it then suffices to handle each T i separately and sowe may assume that T is itself local with maximal ideal n = √ m T .By hypothesis and [ST10a, Spe20], we have the following commutative diagram: F e ∗ T Tr (cid:15) (cid:15) φ T / / T Tr (cid:15) (cid:15) F e ∗ R (cid:127) _ (cid:15) (cid:15) φ / / R (cid:127) _ (cid:15) (cid:15) F e ∗ T φ T / / T. Indeed, the diagram exists for the normalizations, and hence restricts back to T and R sinceTr( T ) ⊆ R . We also observe that Tr( √ m T ) ⊆ m by [Spe20, Lemma 9].Note that Speyerworks only with normal domains, but we may reduce to the normal context since we assumethat Tr( T ) ⊆ R .Suppose m is compatible with φ . If φ T ( F e ∗ n ) n then φ T ( F e ∗ n ) = T which implies that R = Tr( T ) = Tr φ T ( F e ∗ n ) = φ ( F e ∗ Tr( n )) = φ ( F e ∗ m ) ⊆ m , a contradiction.Conversely if n is compatible with φ T , then the diagram above immediately implies thatso is Tr( n ) = m = n ∩ R . (cid:3) Corollary 3.3.
Suppose that R is an F -finite Q -Gorenstein F -pure ring with a dualizingcomplex of index not divisible by p . Then for every compatible ideal I ⊆ R , there exists afinite extension R ⊆ S such that I is the trace ideal, I = τ S/R . Simply sum over the trace maps over the individual field extensions.
OMPATIBLE IDEALS IN GORENSTEIN RINGS 7
Proof.
By hypothesis, there exists an n such that ω ( n ) R is locally free. Since there are onlyfinitely many compatible ideals, we may choose f ∈ R not in any compatible ideal of R , suchthat ω ( n ) R [1 /f ] is actually free (indeed, it is true after localizing to obtain a semi-local ringcentered at maximal compatible ideals). A finite extension of R [1 /f ] with the trace idealequal to I [1 /f ] will then yield a finite extension S ⊇ R with trace ideal equal to I . Thuswe may assume that ω ( n ) R ∼ = R . Since n is not divisible by p , there exists an e > R ( F e ∗ R, R ) ∼ = F e ∗ ω (1 − p e ) R ∼ = F e ∗ R and thus there exists φ generating that Hom-set suchthat I is comaptible if and only if I is φ -compatible, see [Sch09, Proposition 4.1].Take a canonical cover R ⊆ T of index n . By construction ω T/R = Hom R ( T, R ) ∼ = T and let Tr T/R denote the trace map which generates this set as a T -module since R ⊆ T is´etale in codimension 1 away from V ( f ), see [Kol13, Section 2.44], note that they require ademi-normality (see [Kol13, Definition 5.1]) assumption which is satisfied for rings that areG1, S2 and F -pure. Thus for every R ⊆ T ⊆ S , we have Hom R ( S, R ) → Hom R ( T, R ) → R where the second map may be identified with Tr : T → R . But now T is quasi-Gorenstein,and so we are done by combining Lemma 3.2 and Theorem 3.1. (cid:3) Further questions and examples
There are at least two ways which one could try generalize these results, we could weakenthe Q -Gorenstein with index not divisible by p hypothesis, or we could try to weaken the F -purity hypothesis. Question 1 (Removing the Q -Gorenstein index hypothesis) . Suppose that R is an F -finite F -pure ring. Is every compatible ideal always the trace ideal of some finite ring extension? Our proof doesn’t seem to generalize to this setting. While we can take canonical covers R ⊆ T , and the canonical cover is still F -pure by [CR17], we do not see that the associatedtrace map satisfies Tr( n ) ⊆ m for all prime ideals n ∈ T lying over m ∈ R . This only seemsto be apparent to us at the minimal primes of the non- Q -Gorenstein locus.Instead, as we try to weaken the F -purity hypothesis (say while keeping the quasi-Gorenstein hypothesis which implies that locally there there exists Φ ∈ Hom R ( F ∗ R, R )generating the Hom-set), we quickly notice that there are potentially infinitely many com-patible ideals. However, there is a distinguished finite set of compatible ideals, the fixedideals . In this setting, an ideal is (Φ-)fixed ifΦ( F ∗ I ) = I, instead of merely ⊆ . See [BB11] for the fact that there are only finitely many such idealsand the test ideal is the smallest not contained in any minimal prime. It is also easy to seethat for a surjective Φ the compatible ideals are always fixed. Question 2 (Weakening the F -purity hypothesis) . Suppose R is an F -finite quasi-Gorensteinring. Is every fixed ideal I always the trace ideal of some finite ring extension? If one studies this question in the local case where √ I = m and k is perfect, then there isan associated finitely generated Frobenius fixed k -subvector space of H d m ( R ) correspondingto I . However, we were not able to gain enough control over the relations induced by theequational lemma to mimic our approach in Theorem 3.1. THOMAS POLSTRA AND KARL SCHWEDE
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