F -purity deforms in Q -Gorenstein rings
aa r X i v : . [ m a t h . A C ] S e p F -PURITY DEFORMS IN Q -GORENSTEIN RINGS THOMAS POLSTRA AND AUSTYN SIMPSONA
BSTRACT . We show that F -purity deforms in local Q -Gorenstein rings of prime characteristic p > . Fur-thermore, we show that F -purity is m -adically stable in local Cohen-Macaulay Q -Gorenstein rings.
1. I
NTRODUCTION
This article explores two related questions pertaining to a property P of local rings: when can P be liftedfrom Cartier divisors, and when can P be spread out in the m -adic topology? Keeping consistent with thecommutative algebra literature, we say that a property P of local rings deforms (resp. is m -adically stable )if given a local ring ( R, m , k ) and a non-zero-divisor f ∈ m such that R/ ( f ) is P , R is P too (resp. thereexists N > so that R/ ( f + ǫ ) is P for all ǫ ∈ m N ). Provided mild assumptions on the property P (whichmany classes of good singularities satisfy), it’s known that m -adic stability of P implies deformation of P [DS20, Theorem 2.4].In the present article, we examine deformation and stability of F -purity. Recall that a ring R of primecharacteristic p > is F -pure if F : R → R is a pure morphism. The notion of F -purity dates backto work of Hochster and Roberts [HR76] and has since become a prominent figure in the vast web of F -singularities , so-called for being defined in terms of the Frobenius endomorphism r r p . One of the firstquestions considered regarding this class of singularities is whether it deforms [Fed83].Deformation of F -singularities has been an active avenue of research since [Fed83] wherein Fedderinitiated the still-open exploration of whether F -injectivity deforms. Stability of F -singularities is a com-paratively much younger program originating in [DS20] (drawing upon earlier work of Polstra and Smirnov[PS18]). The reader will appreciate the following summary of the (somewhat simplified) state of affairs inthis direction, current up to the present article. P P deforms? P is m -adically stable? F -injective Yes [Fed83, Thm. 3.4(1)] C-M
Yes [DS20, Cor. 4.9]
C-M F -pure No in general [Fed83]; Yes [Fed83] Gor ,[Sch09, Prop. 7.2] Q -Gor p No in general [DS20, Thm. 5.3] F -rational Yes [HH94, Thm. 4.2(h)] Yes [DS20, Cor. 3.9] F Strongly F -regular No in general [Sin99, Thm. 1.1]; Yes [Abe02, 2.2.4]+[AKM98] Q -Gor No in general [DS20, 5.3]; Yes [DS20, Thm. 5.11] Q -Gor ◦ , F F -nilpotent No [ST17, Ex. 2.7(2)] No [ST17, Ex. 2.7(2)]+[DS20, Thm. 2.4]+[KMPS19, Thm. 5.5] F -anti-nilpotent Yes [MQ18, Thm. 4.2(i)] F -full Yes [MQ18, Thm. 4.2(ii)] See also [HMS14, Theorem 3.7 & Corollary 4.7], [MQ18, Theorem 5.11] and [DM20] Q -Gor R is Q -Gorenstein Q -Gor ◦ R is Q -Gorenstein on the punctured spectrum Q -Gor p R is Q -Gorenstein of index relatively prime to p C-M R is Cohen-Macaulay Gor R is Gorenstein F R is F -finite In the above table, note that Fedder shows that F -purity deforms in Gorenstein rings, but fails to deformin general. More interestingly, Singh shows that F -regularity (a more exclusive class of singularities) neednot even deform to something F -pure. Nevertheless, it is natural to make the following conjecture, a solutionto which is the primary focus of this article. Conjecture 1.1.
Let ( R, m , k ) be a local F -finite Q -Gorenstein ring of prime characteristic p > . Supposethat f ∈ R is a non-zero-divisor such that R/ ( f ) is normal and F -pure, then R is F -pure. Simpson was supported by NSF RTG grant DMS . A positive solution to Conjecture 1.1 is remarkable for several reasons, perhaps most notable of which isthe relation that F -purity bears to log canonicity . By work of Hara and Watanabe [HW02], it is known thatnormal Q -Gorenstein singularities of dense F -pure type are log canonical. There is strong suspicion amongexperts that the converse is true, although this conjecture remains open. As noted in [Ish18, Theorem 9.1.17],log canonical singularities deform under certain Q -Gorenstein hypotheses using an inversion of adjunctionresult of Kawakita [Kaw06]. Thus, in light of the relationship between log canonicity and F -purity, one isled to hope that a version of Conjecture 1.1 is true.As indicated by the above table, Schwede was the first to record a proof of Conjecture 1.1 in rings whichare Q -Gorenstein of index relatively prime to p , but this was something well-understood by experts beforethe publication of [Sch09], see [HW02, Remark 4.10(1)]. As indicated by the abstract, we completelyresolve the conjecture and show further that the property of being F -pure is m -adically stable under theadditional hypothesis that R is Cohen-Macaulay. Theorem A. (Theorems 3.4 & 3.6) Let ( R, m , k ) be a local F -finite Q -Gorenstein ring of prime charac-teristic p > . Suppose that f ∈ R is a non-zero-divisor such that R/ ( f ) is Gorenstein in codimension , ( S ) , and F -pure. Then R is F -pure. Moreover, if R is Cohen-Macaulay then there exists N > so that R/ ( f + ǫ ) is F -pure for all ǫ ∈ m N . We caution the reader that we do not require Q -Gorenstein rings to be Cohen-Macaulay or normal; seeDefinition 2.1 for a precise definition.1.1. Notation, Conventions, and organization of the paper.
All rings in this article are commutative withunit and Noetherian. Typically, our interests lie in the development of the theory of prime characteristicrings, especially in Section 3. If R is of prime characteristic p > , then we denote by F : R → R theFrobenius endomorphism. If I ⊆ R is an ideal, for each e ∈ N we denote by I [ p e ] the ideal h r p e | r ∈ I i .For each e ∈ N and R -module M we let F e ∗ M denote the R -module obtained from M via restriction ofscalars along F e . That is, F e ∗ M agrees with M as an abelian group, and given r ∈ R and m ∈ M we have r · F e ∗ ( m ) = F e ∗ ( r p e m ) where F e ∗ ( m ) is the element of F e ∗ M corresponding to m . Often times we assumethat R is F -finite , i.e. if M is a finitely generated R -module, so too is F e ∗ M for each e ∈ N .We occasionally assume that a local ring is the homomorphic image of a regular local ring, an assumptionthat is satisfied by every F -finite local ring, [Gab04, Remark 13.6]. If R ∼ = S/I where S is a regular localring then Ext ht( I ) S ( R, S ) is a choice of canonical module of R .In the study of m -adic stability, we occasionally use the shorthand “ R/ ( f + ǫ ) satisfies property P for all ǫ ∈ m N ≫ ” to mean that there exists an integer N > so that R/ ( f + ǫ ) is P for all ǫ ∈ m N .Section 2 is devoted to preliminary results concerning divisors, Q -Gorenstein rings, cyclic covers, canon-ical modules, and Frobenius splitting ideals. Section 3 is where we prove Theorem A.2. P RELIMINARY R ESULTS
Generalized divisors, divisorial ideals, and cyclic covers.
A ring R is ( G ) if R is Gorenstein incodimension . Suppose that ( R, m , k ) is a local ( G ) ring satisfying Serre’s condition ( S ) , and let L denote the total ring of fractions of R . Following [Har94] there is a well-defined notion of linear equivalenceon the collection of divisors which are Cartier in codimension , and hence there is a well-defined additivestructure on such divisors up to linear equivalence. If D is a Weil divisor which is Cartier in codimension then we let R ( D ) denote the corresponding divisorial ideal of R , i.e. R ( D ) = { x ∈ L \ { } | div ( x ) + D ≥ } ∪ { } . In other words, R ( D ) denotes the global sections of O Spec R ( D ) . We assume that R admits a dualizingcomplex and thus admits a canonical module ω R . Under our assumptions we have that ω R ∼ = R ( K X ) for achoice of canonical divisor K X on X = Spec( R ) which is Cartier in codimension . -PURITY DEFORMS IN Q -GORENSTEIN RINGS 3 Definition 2.1.
A ring R is said to be Q -Gorenstein if(1) R is ( G ) and ( S ) with choice of canonical divisor K X ;(2) there exists an integer n so that nK X is Cartier.The least integer n so that nK X is Cartier is referred to as the Q -Gorenstein index of R . The index of acanonical divisor is independent of choice of canonical divisor as K X is Cartier in codimension , i.e. R is ( G ) .If D ≤ is an anti-effective divisor, Cartier in codimension , then I = R ( D ) ⊆ R is an ideal of pureheight . Conversely, if I ⊆ R is an ideal of pure height , principal in codimension , then I = R ( D ) for some anti-effective divisor D which is Cartier in codimension . For every natural number N ≥ thedivisorial ideal R ( N D ) agrees with I ( N ) , the N th symbolic power of I .Suppose that D is a Q -Cartier (i.e. torsion) divisor of index n which is Cartier in codimension . Supposethat R ( nD ) = R · u . The cyclic cover of R corresponding to D is the finite R -algebra R → R D := n − M i =0 R ( − iD ) t − i ∼ = ∞ M i =0 R ( − iD ) t − i / ( u − t − n − . The map R → R D is a finite R -module homomorphism and the ring R D decomposes into a direct sum of ( S ) R -modules, hence R D is ( S ) . Furthermore, we can explicitly describe the canonical module of R D as ω R D ∼ = Hom R ( R D , R ( K X )) ∼ = n − M i =0 R ( K X + iD ) t i . The above computation commutes with localization and hence R D is Gorenstein in codimension . Indeed,if P is a height prime ideal of R D then p = P ∩ R is a height prime of R and R D ⊗ R R p ∼ = ω R D ⊗ R R p .Localizing further we find that ( ω R D ) P ∼ = ( R D ) P . Lemma 2.2.
Let ( R, m , k ) be a local Q -Gorenstein ring with canonical divisor K X on X = Spec( R ) .Suppose that R is of Q -Gorenstein composite index m · n and D := mK X . Then the cyclic cover R D corresponding to the Q -Cartier divisor D is Q -Gorenstein of index m .Proof. We identify ω R D as L n − i =0 R ( K X + iD ) t i as above. Then ω ( m ) R D ∼ = L n − i =0 R ( mK X + imD ) t im .Observe that L n − i =0 R ( mK X + imD ) t im is isomorphic to a shift of S and hence S is Q -Gorenstein of indexdividing m . Furthermore, if m ′ < m and m ′ divides m then it is easy to check that ω ( m ′ ) R D = S , else thecanonical divisor K X of X = Spec( R ) will have index strictly less than m · n . (cid:3) In Section 3 we will need to understand the behavior of divisorial ideals under base change. In particular,we will need to know that the base change of a divisorial ideal remains a divisorial ideal under suitablehypotheses. The following lemma will be a key tool in accomplishing this.
Lemma 2.3.
Let ( R, m , k ) be a local ring which is ( S ) and ( G ) . Suppose that J ( R is a canonical idealof R . Let f ∈ R be a non-zero-divisor so that f is regular on R/J . If for each p ∈ V ( f ) we have that depth( J ( n ) R p ) ≥ min { ht( p ) , } , e.g. J ( n ) is Cohen-Macaulay, then J ( n ) /f J ( n ) ∼ = (( J, f ) / ( f )) ( n ) .Proof. Consider the short exact sequence → J ( n ) → R → R/J ( n ) → . Since f is regular on R and R/J ( n ) we have that Tor ( R/ ( f ) , R/J ( n ) ) ∼ = H ( f ; R/J ( n ) ) = 0 , hence thereis a short exact sequence → J ( n ) f J ( n ) → R ( f ) → R ( J ( n ) , f ) → , THOMAS POLSTRA AND AUSTYN SIMPSON and therefore J ( n ) /f J ( n ) ∼ = ( J ( n ) , f ) / ( f ) .We notice that the ideals ( J ( n ) , f ) / ( f ) and (( J, f ) / ( f )) ( n ) agree at codimension points of Spec( R/ ( f )) by the assumption that R/ ( f ) is ( G ) . Moreover, our assumptions imply that ( J ( n ) , f ) / ( f ) is an ( S ) R/ ( f ) -module, which together with the previous sentence tells us that ( J ( n ) , f ) / ( f ) ∼ = (( J, f ) / ( f )) ( n ) asclaimed. (cid:3) Q -Gorenstein rings and the ( G ) property. The following result shows that being Gorenstein incodimension deforms and is m -adically stable. The proof uses a standard trick involving the Krull inter-section theorem which we use elsewhere in this article with details omitted. Proposition 2.4.
Let ( R, m , k ) be a local equidimensional and catenary ring which admits a canonicalmodule. If f ∈ R is a non-zero-divisor and R/ ( f ) is ( G ) , then R is ( G ) . If R is further assumed to beexcellent, then the rings R/ ( f + ǫ ) are ( G ) for all ǫ ∈ m N ≫ .Proof. Suppose that p is a height prime. Then the ideal ( p , f ) has height no more than since we areassuming R is equidimensional and catenary and hence we can choose a height prime q ∈ V (( p , f )) .Then R q / ( f ) R q is Gorenstein by assumption. The property of being Gorenstein deforms [Sta18, Tag 0BJJ]and therefore R q is Gorenstein. Localizing further we find that R p is Gorenstein as well, hence R is ( G ) .The non-Gorenstein locus of an excellent local ring is a closed subset (e.g. by [Mat86, Theorem 24.6]).Therefore since the ring R is Gorenstein in codimension , the height primes in the non-Gorenstein locusof R form a finite set, say { p , . . . , p t } . Furthermore, f p i for all i by the assumption that R/ ( f ) is ( G ) . Moreover, T N ( m N + p i ) = p i for all i by Krull’s intersection theorem, so there exists N i > sothat f m N i + p i . Choosing N ≥ max i { N i } we obtain that f + ǫ p i for all i and all ǫ ∈ m N . Hence, R/ ( f + ǫ ) is ( G ) for all ǫ ∈ m N . (cid:3) Lemma 2.5.
Let ( R, m , k ) be a local ring and f ∈ R a non-zero-divisor of R such that R/ ( f ) is ( G ) and ( S ) . Further suppose that R is the homomorphic image of a regular local ring. Let J ( R be a canonicalideal of R such that f is regular on R/J . If (( J, f ) / ( f )) un denotes the intersection of the minimal primarycomponents of the ideal ( J, f ) / ( f ) of R/ ( f ) then (( J, f ) / ( f )) un ∼ = ω R/ ( f ) .Proof. Suppose S is a regular local ring, R ∼ = S/I , and ht( I ) = h . Consider the short exact sequence → R · f −→ R → R/ ( f ) → . Then there is an exact sequence → J · f −→ J → ω R/ ( f ) → Ext h +1 S ( R, S ) and hence there is a left exact sequence → Jf J ∼ = ( J, f )( f ) → ω R/ ( f ) → Ext h +1 S ( R, S ) . By the assumption that R/ ( f ) is ( S ) , we know that Ext h +1 S ( R, S ) is not supported at any height com-ponent of R/ ( f ) . Hence ( J, f ) / ( f ) → ω R/ ( f ) is an isomorphism at codimension points of R/ ( f ) . By[Har94, Proposition 1.11] we have that if ( − ) ∗ denotes Hom R ( − , R/ ( f )) then (cid:18) ( J, f )( f ) (cid:19) ∗∗ ∼ = ω R/ ( f ) . Therefore the lemma is proven as (( J, f ) / ( f )) ∗∗ ∼ = (( J, f ) / ( f )) un . (cid:3) -PURITY DEFORMS IN Q -GORENSTEIN RINGS 5 Proposition 2.6.
Let ( R, m , k ) be an excellent equidimensional local Q -Gorenstein ring of index n satis-fying Serre’s condition ( S ) . Further suppose that R is the homomorphic image of a regular local ring.Suppose that f ∈ m is a non-zero-divisor and R/ ( f ) is ( G ) and ( S ) . Then R/ ( f ) is Q -Gorenstein ofindex dividing n . Moreover, we may choose canonical ideal J ( R and element a ∈ J so that(1) J ( n ) = ( a ) and J/f J ∼ = ( J, f ) / ( f ) ;(2) (( J, f ) / ( f )) un ∼ = ω R/ ( f ) and ((( J, f ) / ( f )) un ) ( n ) = ( a, f ) / ( f ) .Proof. By Proposition 2.4 the ring R is ( G ) . We first show the existence of a canonical ideal J ⊆ R sothat (( J, f ) / ( f )) un ⊆ R/ ( f ) is a canonical ideal of R/ ( f ) . We will then show that ((( J, f ) / ( f )) un ) ( n ) isa principal ideal of R/ ( f ) , i.e. the Q -Gorenstein index of R/ ( f ) divides the Q -Gorenstein index of R asclaimed.Suppose that ω R is a canonical module of R . Let W be the complement of the union of the height components of f . Then ( ω R ) W ∼ = R W . Thus there exists u ∈ ω R and ideal J ⊆ R not contained in anyheight component of ( f ) , so that ω R ∼ = J · u . Then J ⊆ R is a canonical ideal of R . Moreover, since thecomponents of J are disjoint from the components of ( f ) we may assume that J is an ideal of pure height and f is a regular element of R/J . As in the proof of Lemma 2.3 we have that
J/f J ∼ = ( J, f ) / ( f ) and wehave by Lemma 2.5 that (( J, f ) / ( f )) un ∼ = ω R/ ( f ) .Suppose that J ( n ) = ( a ) . We claim that ((( J, f ) / ( f )) un ) ( n ) = ( a, f ) / ( f ) . Equivalently, we need toshow that if p is a height of prime of R containing f then (( J n , f ) / ( f )) R p = (( a, f ) / ( f )) R p . We areassuming R/ ( f ) is ( G ) and hence R p / ( f ) R p is Gorenstein. The property of being Gorenstein deforms andtherefore R p is Gorenstein. In particular, J N R p = J ( N ) R p for every N ∈ N and so (( J n , f ) / ( f )) R p = (cid:16) ( J ( n ) , f ) / ( f ) (cid:17) R p = (( a, f ) / ( f )) R p as claimed. Therefore the index of R/ ( f ) divides the index of R . (cid:3) Splitting ideals.
Our study of m -adic stability of F -purity requires us to study the Frobenius splittingideals of R . To this end we suppose that ( R, m , k ) is a local F -finite ring of prime characteristic p > . The e th Frobenius splitting ideal of R is the ideal I e ( R ) := { c ∈ R | R F e ∗ c −−−−→ F e ∗ R is not pure } . Frobenius splitting ideals were introduced by Enescu and Yao in [EY11] and play a prominent role in primecharacteristic commutative algebra, especially in the study of F -regularity and F -purity. It is important forus to notice that a ring R is F -pure if and only if I e ( R ) is a proper ideal for some (equivalently every) naturalnumber e ∈ N .If R is Cohen-Macaulay (e.g if R is F -regular) the Frobenius splitting ideals I e ( R ) may be realized ascertain colon ideals, and this viewpoint has been central to the study of the “weak implies strong” conjectureand F -signature theory (see [AP19, HL02, PT18, PS18, WY04] for example). Proposition 2.7 ([PT18, Lemma 6.2]) . Let ( R, m , k ) be a local Cohen-Macaulay ring of prime character-istic p > . Suppose that R admits a canonical ideal J ( R , = x ∈ J a non-zero-divisor, x , . . . , x d parameters on R/ ( x ) , and suppose that u generates the socle of the -dimensional Gorenstein quotient R/ ( J, x , . . . , x d ) . Then for each e ∈ N there exists t e ∈ N so that for all t ≥ t e I e ( R ) = ( x t − J, x t , . . . , x td ) [ p e ] : R ( x · · · x d ) ( t − p e u p e . The study of Frobenius splittings often requires carefully manipulating the colon ideals described above.We first discuss a powerful technique at our disposal in this vein when the parameter element x is chosenso that x multiplies the canonical ideal J into a principal ideal contained in J . Such elements can be foundprovided R is ( G ) . Choosing such a parameter element conveniently allows us to switch freely between THOMAS POLSTRA AND AUSTYN SIMPSON bracket powers of the canonical ideal with symbolic powers thereof, which we record in more detail in thefollowing lemma.
Lemma 2.8.
Let ( R, m , k ) be a local F -finite Cohen-Macaulay ( G ) ring of prime characteristic p > and of Krull dimension d at least . Suppose that J ( R is a choice of canonical ideal of R and x ∈ J isa non-zero-divisor. Then there exists a parameter element x on R/ ( x ) and element = a ∈ J such that x J ⊆ ( a ) . Moreover, for any choice of parameters y , . . . , y d on R/ ( x , x ) and e, N ∈ N we have that ( J [ p e ] , x Np e , y , . . . , y d ) : R x ( N − p e = ( J ( p e ) , x Np e , y , . . . , y d ) : R x ( N − p e = ( J ( p e ) , x p e , y , . . . , y d ) : R x p e = ( J [ p e ] , x p e , y , . . . , y d ) : R x p e . Proof.
We refer the reader to [PT18, Lemma 6.7(i)] where the first named author and Tucker record a proofof this lemma under the additional assumption that R is a normal domain. We observe that the normalityassumption is not necessary and one only needs that J is principal in codimension for the methodology of[PT18] to be applicable. (cid:3) Another tactic that we employ in the study of the colon ideals appearing in Proposition 2.7 allows us toremove the x term. The following lemma is well-known to experts, but we record a detailed proof for thereader’s convenience. Lemma 2.9.
Let ( R, m , k ) be a local F -finite Cohen-Macaulay ring of prime characteristic p > and ofKrull dimension d . Suppose that J ( R is a choice of canonical ideal of R and x ∈ J is a non-zero-divisor.Then for any choice of parameters y , . . . , y d of R/ ( x ) we have that (( x t − J ) [ p e ] , y , . . . , y d ) : R x ( t − p e = ( J [ p e ] , y , . . . , y d ) . Proof.
Clearly the ideal on the right-hand side of the claimed equality is contained in the left-hand side.Suppose that r ∈ (( x t − J ) [ p e ] , y , . . . , y d ) : R x ( t − p e , i.e. rx ( t − p e ∈ (( x t − J ) [ p e ] , y , . . . , y d ) . Equivalently, there exists an element j ∈ J [ p e ] such that ( r − j ) x ( t − p e ∈ ( y , . . . , y d ) . The element x ( t − p e is a non-zero-divisor on R/ ( y , . . . , y d ) and therefore r − j ∈ ( y , . . . , y d ) and therefore r ∈ ( J [ p e ] , y , . . . , y d ) as claimed. (cid:3)
3. D
EFORMATION AND STABILITY OF F - PURITY F -purity deforms in Q -Gorenstein rings. Let us discuss our strategy to resolving Conjecture 1.1.Suppose that ( R, m , k ) is a Q -Gorenstein F -finite local ring of prime characteristic p > and f ∈ R isa non-zero-divisor such that R/ ( f ) is ( G ) , ( S ) , and F -pure. Suppose that K X is a choice of canonicaldivisor on X = Spec( R ) , np e K X ∼ , n is relatively prime to p , and R ( np e K X ) = R · u . We let D be thedivisor nK X and S = L ∞ i =0 R ( − iD ) t − i / ( u − t − ( p e − − be the cyclic cover of R corresponding to thedivisor D . Then the ring R is a direct summand of S and therefore if S is F -pure then R is F -pure. The ring S is Q -Gorenstein of index relatively prime to p , see Lemma 2.2. Thus, if we are able to show S/f S is F -pure, then we have reduced solving Conjecture 1.1 to the scenario that the Q -Gorenstein index is relativelyprime to the characteristic. As discussed in the introduction, deformation of F -purity in this scenario was -PURITY DEFORMS IN Q -GORENSTEIN RINGS 7 well-understood by experts before being recorded in the literature by Schwede, [Sch09, Proposition 7.2].Our strategy to show that S/f S is F -pure is to show that S/f S is a cyclic cover of R/ ( f ) and then utilize[Car17, Proposition 4.21] to conclude that S/f S is F -pure. For the sake of convenience, we record a proof that F -purity deforms in Q -Gorenstein rings whoseindex is relatively prime to the characteristic. We do this since the reader might wish to avoid the extratechnicalities of [Sch09] where the deformation of F -purity along a Weil divisor D of a pair ( R, ∆) isconsidered. Proposition 3.1.
Let ( R, m , k ) be a local F -finite ring of prime characteristic p > . Suppose that R is Q -Gorenstein of index relatively prime to p and f ∈ R is a non-zero-divisor such that R/ ( f ) is ( G ) , ( S ) ,and F -pure. Then R is F -pure.Proof. Because the index of K X is relatively prime to p there exists an e so that (1 − p e ) K X ∼ . Toshow that R is F -pure we will show that every R -linear map F e ∗ R/ ( f ) ϕ −→ R/ ( f ) can be lifted to a map Φ : F e ∗ R → R . That is there exists Φ : F e ∗ R → R so that the following diagram is commutative: F e ∗ R RF e ∗ R/ ( f ) R/ ( f ) . Φ ϕ Equivalently, we wish to show that the natural map
Hom R ( F e ∗ R, R ) → Hom R/ ( f ) ( F e ∗ R/ ( f ) , R/ ( f )) ob-tained by applying Hom R ( F e ∗ R, − ) to R → R/ ( f ) is onto.First suppose that R is Gorenstein and consider the short exact sequence → R · f −→ R → R/ ( f ) → . Then
Ext R ( F e ∗ R, R ) = 0 since F e ∗ R is Cohen-Macaulay and R ∼ = R ( K X ) is the canonical module of R ,[BH93, Theorem 3.3.10]. Therefore the natural map Hom R ( F e ∗ R, R ) → Hom R/ ( f ) ( F e ∗ R/ ( f ) , R/ ( f )) isindeed onto under the Gorenstein hypothesis.Now we show Hom R ( F e ∗ R, R ) → Hom R/ ( f ) ( F e ∗ R/ ( f ) , R/ ( f )) is onto under the milder hypothesis that R is Q -Gorenstein of index relatively prime to p . Recall that we choose e large enough so that (1 − p e ) K X ∼ . The module Hom R ( F e ∗ R, R ) can be identified with F e ∗ R , Hom R ( F e ∗ R, R ) ∼ = Hom R ( F eR ⊗ R ( K X ) , R ( K X )) ∼ = Hom R ( F e ∗ ( p e K X ) , R ( K X )) ∼ = F e ∗ Hom R ( R ( p e K X ) , R ( K X )) ∼ = F e ∗ R ((1 − p e ) K X ) ∼ = F e ∗ R. The ring R/ ( f ) is Q -Gorenstein of index which divides the index of R by Lemma 2.6. Therefore it is alsothe case that Hom R/ ( f ) ( F e ∗ R/ ( f ) , R/ ( f )) ∼ = F e ∗ R/ ( f ) . In particular, when viewed as an F e ∗ R/ ( f ) -module,the image of Hom R ( F e ∗ R, R ) → Hom R/ ( f ) ( F e ∗ R/ ( f ) , R/ ( f )) is cyclic and therefore ( S ) . We can thencheck that the image agrees with with entire module at the codimension points of Spec( R/ ( f )) by [Har94,Theorem 1.12]. We are assuming R/ ( f ) is ( G ) and so the surjectivity of the desired map follows by thesurjectivity of the map in the Gorenstein scenario. (cid:3) Suppose that ( R, m , k ) is a ( G ) and ( S ) local ring. If D is a divisor and f ∈ R a non-zero-divisor of R such that R/ ( f ) is ( G ) and ( S ) and R ( D ) /f R ( D ) is principal codimension R/ ( f ) -module, then welet D | V ( f ) denote a choice of divisor of R/ ( f ) so that R/ ( f )( D | V ( f ) ) is isomorphic to the reflexification Carvajal-Rojas makes the running assumption in [Car17] that all rings are essentially of finite type over an algebraically closedfield. Furthermore, it is assumed in [Car17, Proposition 4.21] that R is normal. However, the proof of [Car17, Proposition 4.21]works verbatim under the milder assumptions that ( R, m , k ) is any F -finite local ring which is ( G ) , ( S ) , and D is a divisor whichis Cartier in codimension . THOMAS POLSTRA AND AUSTYN SIMPSON ( R ( D ) /f R ( D )) un := Hom R/ ( f ) (Hom R/ ( f ) ( R ( D ) /f R ( D ) , R/ ( f )) , R/ ( f )) of R ( D ) /f R ( D ) . The fol-lowing lemma provides a criterion for when the cyclic cover of a torsion divisor base changes to a cycliccover of R/ ( f ) . Lemma 3.2.
Let ( R, m , k ) be a ( G ) and ( S ) local F -finite ring of prime characteristic p > . Supposethat D is a torsion divisor of index N and R ( N D ) = R · u . Suppose that f ∈ R is a non-zero-divisorsuch that R/ ( f ) is ( G ) and ( S ) . For each ≤ i ≤ N − suppose that R ( − iD ) /f R ( − iD ) is an ( S ) R/ ( f ) -module which is principal in codimension . If S = L ∞ i =0 R ( − iD ) t − i / ( u − t − N − is the cycliccover of R associated with D then S/f S is the cyclic cover of R/ ( f ) with respect to D | V ( f ) .Proof. We must verify two things:(1) for each ≤ i ≤ N − the ideal R ( − iD ) /f R ( − iD ) is isomorphic to R/ ( f )( − iD | V ( f ) ) ;(2) the index of D | V ( f ) is N .Our assumptions allow us to check (1) at the height primes of Spec( R/ ( f )) . By assumption, each R ( − iD ) /f R ( − iD ) is principal at codimension points of Spec( R/ ( f )) . Hence R ( − iD ) is principalat height points of Spec( R ) containing f and so R ( − iD ) agrees with R ( − D ) i at such points of Spec( R ) .Therefore R ( − iD ) /f R ( − iD ) is indeed isomorphic to R/ ( f )( − iD | V ( f ) ) as claimed.For (2) we first notice that R ( − N D ) ∼ = R and so R ( − N D ) /f R ( − N D ) is an ( S ) and principal incodimension module of R/ ( f ) . Therefore by the above, R ( − N D ) /f R ( − N D ) = R/ ( f )( − N D V ( f ) ) ∼ = R/ ( f ) and the index of D | V ( f ) cannot exceed the index of D . However, if there was an ≤ i ≤ N − so that − iD | V ( f ) ∼ then R ( − iD | V ( f ) ) ∼ = R ( − iD ) /f R ( − iD ) ∼ = R/ ( f ) . By Nakayama’s Lemma themodule R ( − iD ) is a principal module of R and the index of D would be strictly less than N , a contradictionto our initial assumptions. (cid:3) If ( R, m , k ) is a local strongly F -regular ring and D is a torsion divisor, then R ( D ) is a direct summand of F e ∗ R for some e ∈ N , see [Mar20, Proof of Proposition 2.6]. It will likely not be the case that every torsiondivisorial ideal in an F -pure local ring is a direct summand of F e ∗ R for some e ∈ N , but the following lemmapoints out that the divisorial ideals of index p to a power are a direct summand of F e ∗ R for some e ∈ N , c.f.Lemma 3.5 below. Lemma 3.3.
Let ( R, m , k ) be a local ( G ) and ( S ) F -finite and F -pure ring of prime characteristic p > .If D is a torsion divisor of index p e then R ( D ) is a direct summand of F e ∗ R .Proof. The e th iterate of the Frobenius map R → F e ∗ R splits as an R -linear map. If we tensor with R ( D ) and reflexify we find that R ( D ) is a direct summand of F e ∗ R ( p e D ) ∼ = F e ∗ R . (cid:3) Theorem 3.4.
Let ( R, m , k ) be a local F -finite ring of prime characteristic p > . Suppose that R is Q -Gorenstein and f ∈ R is a non-zero-divisor such that R/ ( f ) is ( G ) , ( S ) , and F -pure. Then R is F -pure.Proof. If dim( R ) ≤ then R/ ( f ) being ( G ) implies that R is Gorenstein. The property of being F -pure is equivalent to being F -injective in Gorenstein rings and F -injectivity is known to deform in Cohen-Macaulay rings, see [Fed83, Lemma 3.3 and Theorem 3.4]. Thus we may assume that dim( R ) ≥ , and byinduction we may assume that R is F -pure at every non-maximal prime of Spec( R ) containing the element f . Let K X be a choice of canonical divisor of X = Spec( R ) . Suppose that K X has index np e where n is relatively prime to p . The divisor D = nK X has index p e and we suppose that R ( p e D ) = R · u . Let S = L ∞ i =0 R ( − iD ) t − i / ( u − t − p e − be the cyclic cover of R corresponding to the divisor D . The workthat follows will allow us to utilize Lemma 3.2 and establish that S/f S is a cyclic cover of R/ ( f ) .If p ∈ Spec( R ) \ { m } and f ∈ p then R ( − iD ) p is a direct summand of F e ∗ R p by Lemma 3.3. Hence depth( R p ) ≥ min { ht( p ) , } for all primes p ∈ Spec( R ) \ { m } containing f . Therefore the quotients -PURITY DEFORMS IN Q -GORENSTEIN RINGS 9 R ( − iD ) /f R ( − iD ) are ( S ) R/ ( f ) -modules on the punctured spectrum of R/ ( f ) . In particular, if C i denotes the cokernel of R ( − iD ) /f R ( − iD ) ⊆ ( R ( − iD ) /f R ( − iD )) un ∼ = R/ ( f )( − iD | V ( f ) ) then C i is a finite length R -module and H i m ( R ( − iD ) /f R ( − iD )) ∼ = H i m (cid:16) R ( − iD | V ( f ) ) (cid:17) for all i ≥ . Weaim to show that R ( − iD ) /f R ( − iD ) is an ( S ) R/ ( f ) -module for each ≤ i ≤ p e − . The work abovereduces this problem to showing depth( R ( − iD ) /f R ( − iD )) ≥ .Consider the following commutative diagram whose horizontal arrows are R -linear and whose verticalarrows are p e -linear: R ( − iD ) f R ( − iD ) R ( f ) (cid:16) − iD | V ( f ) (cid:17) R ( − ip e D ) f R ( − ip e D ) R ( f ) (cid:16) − ip e D | V ( f ) (cid:17) R ( f ) R ( f ) F e F e ∼ = ∼ = Recall that R ( p e D ) ∼ = R , so by Lemma 3.2 the middle horizontal map of the above commutative diagramis an isomorphism. By Lemma 3.3 the right most vertical map is a split map. It follows that for all i ≥ the p e -linear maps of local cohomology modules H i m ( R ( − iD ) /f R ( − iD )) → H i m ( R ( − ip e D ) /f R ( − ip e D )) is a split map of abelian groups. In particular, the above p e -linear maps on local cohomology modules areinjective.Now we consider the following commutative diagram: R ( − iD ) R ( − iD ) R ( − iD ) f R ( − iD ) 0 R ( − ip e D ) R ( − ip e D ) f R ( − ip e D ) R R ( f ) · f F e F e ∼ = ∼ = The top row of the above diagram is a short exact sequence of R -modules and the composition of the verticalmaps are p e -linear. There is an induced commutative diagram of local cohomology modules whose top row By definition, a p e -linear map of R -modules N → M is the same as an R -linear map N → F e ∗ M . is exact: H m (cid:18) R ( − iD ) f R ( − iD ) (cid:19) H m ( R ( − iD )) H m ( R ( − iD )) H m (cid:18) R ( − iD ) f R ( − iD ) (cid:19) H m ( R ) H m (cid:18) R ( f ) (cid:19) · f πF e F e ∼ = We first remark that H m ( R ) is indeed the -module. We are assuming R/ ( f ) is ( S ) and f is a non-zero-divisor, hence R has depth at least and H m ( R ) = 0 . The right most p e -linear map is injective.Therefore the map π is the -map and H m ( R ( − iD )) = f H m ( R ( − iD )) . The module R ( − iD ) is an ( S ) R -module and therefore H m ( R ( − iD )) can be checked to be of finite length by Matlis duality. ByNakayama’s Lemma we have that H m ( R ( − iD )) = 0 , therefore H m ( R ( − iD ) /f R ( − iD )) = 0 , and themodule R ( − iD ) /f R ( − iD ) is an ( S ) R/ ( f ) -module for each ≤ i ≤ p e − .Suppose that R ( p e D ) = R · u and let S = L ∞ i =0 R ( − iD ) t − i / ( u − t − p e − be the cyclic cover of R corresponding to D . The ring R is F -pure if and only if S is F -pure by [Car17, Proposition 4.21].Moreover, the ring S is Q -Gorenstein of index n , a number relatively prime to p , see Lemma 2.2. Thus toverify S is F -pure it suffices to check S/f S is F -pure by Proposition 3.1. We are assuming that R/ ( f ) is F -pure and the work above allows us to utilize Lemma 3.2 and claim that S/f S is the cyclic cover of R/ ( f ) corresponding to the divisor D | V ( f ) . Therefore S/f S is F -pure by a second application of [Car17,Proposition 4.21] and we conclude that R is F -pure. (cid:3) F -purity is m -adically stable in Cohen-Macaulay Q -Gorenstein rings. Our strategy to show F -purity is m -adically stable in a Cohen-Macaulay Q -Gorenstein ring ( R, m , k ) is different than our strategyfor deformation. We study the Frobenius splitting ideals of R/ ( f + ǫ ) and compare them to the Frobeniussplitting ideals of the quotient R/ ( f ) . In light of Proposition 2.7 and Lemma 2.8 it is advantageous for thereto be some natural number e so that if J ( R is a canonical ideal of R then J ( p e ) is a Cohen-Macaulay R -module. Lemma 3.5.
Let ( R, m , k ) be a Q -Gorenstein F -pure local ring of prime characteristic p > . Then thereexists integers e , e ∈ N with e ≥ so that R ( p e K X ) is a direct summand of F e ∗ R ( K X ) . In particular, if R is Cohen-Macaulay then there exists an e ≥ so that R ( p e K X ) is Cohen-Macaulay.Proof. Since R is F -pure we have that for every integer e ≥ that F e ∗ R ∼ = R ⊕ M has a free summand.Applying Hom R ( − , R ( K X )) we find that Hom R ( F e ∗ R, R ( K X )) ∼ = F e ∗ R ( K X ) ∼ = R ( K X ) ⊕ M ′ has an R ( K X ) -summand for every e ≥ . Suppose that the Q -Gorenstein index of R is np e where p does notdivide n . We choose e ≥ so that n divides p e − . If we apply − ⊗ R R (( p e − K X ) and then reflexifythe direct sum decomposition F e ∗ R ( K X ) ∼ = R ( K X ) ⊕ M ′ we find that F e ∗ R ( K X + p e ( p e − K X ) ∼ = F e ∗ R ( K X ) ∼ = R ( p e K X ) ⊕ M ′′ . (cid:3) Theorem 3.6.
Let ( R, m , k ) be an equidimensional local F -finite Q -Gorenstein Cohen-Macaulay ring ofprime characteristic p > . Suppose that f ∈ m is a non-zero-divisor and R/ ( f ) is ( G ) and F -pure. Then R/ ( f + ǫ ) is F -pure for all ǫ ∈ m N ≫ . -PURITY DEFORMS IN Q -GORENSTEIN RINGS 11 Proof.
The ring R is F -pure by Theorem 3.4. By Lemma 3.5 there exists an e ∈ N so that R ( p e K X ) is aCohen-Macaulay R -module.We aim to show that the e th Frobenius splitting ideals I e ( R/ ( f + ǫ )) of R/ ( f + ǫ ) are proper ideals for all ǫ ∈ m N ≫ . If the dimension of R is no more than then R/ ( f ) being ( G ) implies that R is Gorenstein. Inthe Gorenstein setting, F -purity and F -injectivity are equivalent, and the latter is known to deform and be m -adically stable in Cohen-Macaulay rings, see [Fed83, Lemma 3.3 and Theorem 3.4] and [DS20, Corollary4.9]. Thus we may assume R is of dimension d + 1 ≥ .We may select canonical ideal J ( R so that J/f J ∼ = ( J, f ) / ( f ) is the canonical ideal of R/ ( f ) , seeLemma 2.5. By Proposition 2.4 we know that the rings R and R/ ( f + ǫ ) are ( G ) for all ǫ ∈ m N ≫ .Furthermore, the rings R/ ( f + ǫ ) are Q -Gorenstein of index dividing the Q -Gorenstein index of R for all ǫ ∈ m N ≫ , see Proposition 2.6. Also note that for all ǫ ∈ m N ≫ we have that f + ǫ avoids all componentsof J by the Krull intersection theorem (see also the proof of Proposition 2.4). Therefore J ( n ) / ( f + ǫ ) J ( n ) ∼ =( J ( n ) , f + ǫ ) / ( f + ǫ ) .Since J ( p e ) is Cohen-Macaulay, it follows that for all ǫ ∈ m N ≫ that ( J ( p e ) , f + ǫ ) / ( f + ǫ ) is a Cohen-Macaulay R/ ( f + ǫ ) -module. By Lemma 2.3 ( J ( p e ) , f + ǫ ) / ( f + ǫ ) is an unmixed ideal of height and ( J ( p e ) , f + ǫ )( f + ǫ ) = (cid:18) ( J, f + ǫ )( f + ǫ ) (cid:19) ( p e ) . To ease notation we write R ǫ to denote the quotient R/ ( f + ǫ ) . The above tells us that J ( p e ) R ǫ =( J R ǫ ) ( p e ) for all ǫ ∈ m N ≫ . Choose a non-zero-divisor x ∈ J on R and R/ ( f ) and let W denote themultiplicatively closed set given by the complement of the union of the minimal primes of the unmixedideal ( x , f ) . Since R/ ( f ) is ( G ) the localized ideal (( J, f ) / ( f )) R W is principal. Hence there exists aparameter element x on R/ ( x , f ) and a ∈ J so that x J ⊆ ( a ) ⊆ J .We extend x to a full parameter sequence x , x , . . . , x d on R/ ( x , f ) and choose socle generator u ∈ R on R/ ( J, x , . . . , x d , f ) . Proposition 2.7 then tells us that for all t ≫ • I e ( R ǫ ) = ( x t − J, x t , . . . , x td ) [ p e ] : R ǫ ( x · · · x d ) ( t − p e u p e for all ǫ ∈ m N ≫ .Lemma 2.9 allows us to simplify the above colon ideals as follows: • I e ( R ǫ ) = ( J, x t , . . . , x td ) [ p e ] : R ǫ ( x · · · x d ) ( t − p e u p e for all ǫ ∈ m N ≫ .In light in of Lemma 2.8 the above ideals can be further simplified as • I e ( R ǫ ) = (( J R ǫ ) ( p e ) , x p e , x tp e . . . , x tp e d ) : R ǫ ( x · · · x d ) ( t − p e ( x u ) p e for all ǫ ∈ m N ≫ .Since J ( p e ) is a Cohen-Macaulay R -module, and consequently ( J R ǫ ) ( p e ) are Cohen-Macaulay R ǫ -modules,the quotient R/J ( p e ) is Cohen-Macaulay of dimension d and the quotients R ǫ / ( J R ǫ ) ( p e ) are Cohen-Macaulayof dimension d − . In particular, the above colon ideals can be simplified once more: • I e ( R ǫ ) = (( J R ǫ ) ( p e ) , x p e , x p e . . . , x p e d ) : R ǫ u p e for all ǫ ∈ m N ≫ .Another application of Lemma 2.8 allows us to realize the above colon ideals as • I e ( R ǫ ) = ( J, x , x . . . , x d ) [ p e ] : R ǫ ( x u ) p e for all ǫ ∈ m N ≫ .Observe now that for all ǫ ∈ m N ≫ I e ( R ǫ ) = (( J, x , x . . . , x d ) [ p e ] , f + ǫ ) : R ( x u ) p e ( f + ǫ ) . Thus R/ ( f ) is F -pure if and only if the following colon ideal is a proper ideal of R : (( J, x , x . . . , x d ) [ p e ] , f ) : R ( x u ) p e . Observe that if ǫ ∈ ( J, x , x . . . , x d , f ) then (( J, x , x . . . , x d ) [ p e ] , f + ǫ ) : R ( x u ) p e = ( J, x , x . . . , x d , f ) : R ( x u ) p e . Therefore I e ( R ǫ ) is a proper ideal of R ǫ for all ǫ ∈ m N ≫ , i.e. R ǫ is F -pure and the property of being F -pure is indeed m -adically stable. (cid:3) AcknowledgementsWe thank Ian Aberbach, Linquan Ma, Takumi Murayama, and Karl Schwede for useful conversationsduring the preparation of this article. We also thank Pham Hung Quy for pointing out an inaccuracy in aprevious draft. The second named author is grateful to his advisor, Kevin Tucker, for his constant encour-agement. R
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