Expected resurgence of ideals defining Gorenstein rings
aa r X i v : . [ m a t h . A C ] J u l EXPECTED RESURGENCE OF IDEALS DEFINING GORENSTEINRINGS
ELO´ISA GRIFO, CRAIG HUNEKE, AND VIVEK MUKUNDAN
Abstract.
Building on previous work by the same authors, we show that certain idealsdefining Gorenstein rings have expected resurgence, and thus satisfy the stable HarbourneConjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ringin a regular ring, provided its symbolic powers are given by saturations with the maximalideal. While this property is not suitable for reduction to characteristic p , we show that asimilar result holds in equicharacteristic 0 under the additional hypothesis that the symbolicRees algebra of I is noetherian. Introduction
In this paper we extend recent work by the same three authors [GHM], which studiedwhat is called the stable Harbourne conjecture and its relationship to expected resurgence.Building on the sufficient conditions from [GHM], in the present paper we show that thestable Harbourne Conjecture holds for certain ideals defining Gorenstein rings. The originalconjecture of Harbourne [BDRH +
09, HH13] concerns homogeneous ideals I in k [ P n ], k afield, and their symbolic powers I ( n ) = T P ∈ Ass( I ) ( I n R P ∩ R ), and depends on the big height of I , the largest height (or codimension) of any minimal prime of I . We slightly rewrite theoriginal conjecture here for radical ideals in any regular ring: Conjecture 1.1 (Harbourne) . Let I be a self-radical ideal of big height c in a regular ring R . Then for all n > I ( cn − c +1) ⊆ I n . This conjecture is part of a larger program to find the best possible a for each b suchthat I ( a ) ⊆ I b , a program known as the Containment Problem. On the one hand, thecontainment I ( cn ) ⊆ I n holds for all n > I of big height2 that is the first counterexample to I (3) ⊆ I for certain configurations of points in projectivespace P . This example has been extended to entire classes of counterexamples coming fromvery special configurations in P n [HS15, BDRH +
18, MS18b, CGM +
16, Dra17, DS20].However, Harbourne’s Conjecture is satisfied by various classes of ideals: those defininggeneral points in P [BH10a] and P [Dum15], squarefree monomial ideals, or more generallyideals defining equicharacteristic rings with mild singularities [GH19], such as Veronese ordeterminantal rings. More precisely, I satisfies Harbourne’s Conjecture whenever R/I isF-pure in characteristic p , or of dense F-pure type in equicharacteristic 0 [GH19]. There areeven classes of ideals satisfying Harbourne’s Conjecture over certain singular rings [GMS19]. Mathematics Subject Classification.
Primary: 13A15. Secondary: 13H05.
Key words and phrases. symbolic powers, containment problem, Harbourne’s Conjecture, resurgence,Gorenstein ideals.
There are, however, no known counterexamples to the following stable version of theconjecture:
Conjecture 1.2 (Stable Harbourne) . Let I be a self-radical ideal of big height c in a regularring R . Then for all n ≫ I ( cn − c +1) ⊆ I n . This stable version of Harbourne’s Conjecture holds for very general and generic pointconfigurations in P n [TX20, Theorem 2.2 and Remark 2.3], for sufficiently large sets of generalpoints in P n [BGHN20, Corollary 4.3], for ideals defining space monomial curves ( t a , t b , t c )over fields of characteristic other than 3 [GHM, Theorem 4.3], and for homogeneous idealsgenerated in small degree in equicharacteristic 0 [GHM, Theorem 3.1]. Moreover, varioussufficient conditions for the stable Harbourne Conjecture are given in [Gri20, GHM]. Manyof these results rely on the resurgence of I , as defined by Bocci and Harbourne [BH10a]: Definition 1.3 (Bocci-Harbourne [BH10a]) . The resurgence of an ideal I is given by ρ ( I ) = sup n ms | I ( m ) I s o . Notice that in particular, if m > ρ ( I ) · r , then one is guaranteed that I ( m ) ⊆ I r . Re-lated invariants have also been studied, such as the asymptotic resurgence [GHVT13], whichcan be computed via integral closures [DFMS19]. The resurgence can often be bounded byother invariants [BH10a, Theorem 1.2.1], and sometimes even computed explicitly withoutcompletely solving the Containment Problem [BH10b, DHN +
15, BDRH + I is strictly less than its big height. Inthis case, we say that I has expected resurgence . In this paper, we expand on the sufficientconditions for expected resurgence from [GHM] to prove the following main results: Theorem A (Theorem 2.3 and Theorem 4.3) . Let I be a homogeneous ideal of big height c > R with homogeneous maximal ideal m , or in a regularlocal ring ( R, m ), and assume that R contains a field k . Suppose that: • I ( n ) = ( I n : m ∞ ) for all n >
1, and • R/I is Gorenstein.Then I has expected resurgence whenever k has prime characteristic or the symbolic Reesalgebra of I is noetherian.In the prime characteristic p setting, the key step is to show that there exists q = p e such that I ( cq − c +1) ⊆ m I q ; in fact, I ( cq − c +1) ⊆ m I [ q ] . Since this statement depends heavilyon the characteristic, it is not suitable for reduction to characteristic p techniques, whichis why an additional assumption is needed in equicharacteristic 0. However, the followingstrengthening of [ELS01, HH02] holds independently of the characteristic: Theorem B (Theorem 2.3 and Theorem 4.1) . Let I be a homogeneous ideal of height c > R with homogeneous maximal ideal m , or in a regular localring ( R, m ), and assume that R contains a field k . If R/I is Gorenstein, then I ( cn ) ⊆ m I n for all n > XPECTED RESURGENCE OF IDEALS DEFINING GORENSTEIN RINGS 3
While this might appear similar to [TY08, Theorem 3.1], there is a crucial differencebetween the two, since [TY08, Theorem 3.1] only guarantees I ( ct − c +1) ⊆ m I t − , and thatone power difference is what will allow us to apply [GHM, Theorem 3.3] and conclude that I satisfies expected resurgence. Our methods are also necessarily different from [TY08,Theorem 3.1], since that one extra power of I is a priori difficult to obtain and directly usesthe assumption that R/I is Gorenstein.In Section 2, we establish the main result in the case of prime characteristic. In Sec-tion 3, we extend the methods of [GHM] to a more general setting that will allow for ourequicharacteristic 0 results. Finally, we study the equicharacteristic 0 case in Section 4.2.
Prime characteristic
Discussion 2.1.
Let I be a homogeneous ideal of height c > R over a field k of prime characteristic p and with homogeneous maximal ideal m , or anideal in a regular local ring R of characteristic p with maximal ideal m . Assume that R/I isGorenstein. We consider the minimal free resolution of
R/I over R :0 / / R φ c / / R β c − φ c − / / R β c − / / · · · / / R β φ / / R / / R/I / / . Since R is regular of prime characteristic p , [PS73, Theorem 1.7] shows that the complexof free modules0 / / R φ [ p ] c / / R β c − φ [ p ] c − / / R β c − / / · · · / / R β φ [ p ]1 / / R / / R/I [ p ] / / R/I [ p ] . After choosing bases of the freemodules, the maps in this resolution are simply the entrywise pth powers of the entries ofthe corresponding matrices in the resolution of R/I . The exactness of this resolution provesthat
R/I [ p ] is also Gorenstein.The natural quotient map R/I [ p ] −→ R/I induces a map between the resolutions of
R/I and
R/I [ p ] , as follows:0 / / R φ c / / R β c − φ c − / / R β c − / / · · · / / R β φ / / R / / R/I / / / / R φ [ p ] c / / α c O O R β c − φ [ p ] c − / / α c − O O R β c − / / α c − O O · · · / / R β φ [ p ]1 / / α O O R / / R/I [ p ] / / O O α c = µ ∆ , the map defined by multiplication by some element ∆, which is uniqueup to homotopy. Since the entries of φ c and φ [ p ] c for a minimal generating set for I and I [ p ] ,respectively, it follows that ∆ is unique modulo I [ p ] . By standard linkage theory, ∆ is agenerator for the colon ideal ( I [ p ] : I ) modulo I [ p ] . Duality shows that ( I [ p ] : ∆) = I . See[KM84, Theorem 1.2] for details.A critical observation that we need for our main results is the following proposition: Proposition 2.2.
Let the notation be as in Discussion 2.1. Then ∆ · I ⊆ m · I [ p ] . Proof.
From the commutative diagram in Discussion 2.1, we see that φ c ◦ α c = α c − ◦ φ [ p ] c .Thus I (∆) ⊆ I [ p ] · I ( α c − ). Thus it is enough to show I ( α c − ) ⊆ m . Suppose I ( α c − ) = R . GRIFO, HUNEKE, AND MUKUNDAN
Possibly after a suitable change of basis, we may assume that α c − = id ⊕ α ′ , and if e denotesthe first standard basis element of R β c − , α c − ( e ) = e . Then φ c − ◦ α c − ( e ) = φ c − ( e ). If φ c − ( e ) = ( v , . . . , v βc − ), notice that all v i ∈ m , since I ( φ c − ) ⊆ m (note that c > φ c − ◦ α c − = α c − ◦ φ [ p ] c − so φ c − ( e ) = α c − ◦ φ [ p ] c − ( e ) . But then v i ∈ ( v , . . . , v β c − ) [ q ] , which is a contradiction. (cid:3) Theorem 2.3.
Let I be a homogeneous ideal of height c > R containing a field k of characteristic p and with homogeneous maximal ideal m , or anideal in a regular local ring R of characteristic p with maximal ideal m . Assume that R/I is Gorenstein. Then I ( cq − c +1) ⊆ m I [ q ] for every q = p e > pc . Moreover, for every n > I ( cn ) ⊆ m I n . Proof.
We show the first statement. Equivalently, we will show that I ( cqp − c +1) ⊆ m I [ qp ] forall q = p e > c . In this case, I ( cqp − c +1) ⊆ I ( cqp − q +1) , and by [TY08, Theorem 3.1] using n = 1 , k = cqp − q − c , we have I ( cqp − q +1) ⊆ m I ( cqp − q − c +1) . On the other hand, I ( cqp − q − c +1) I [ q ] ⊆ I ( cqp − c +1) ⊆ I [ qp ] . The second containment is well-known; see for example [GH19, Lemma 2.6]. Thus we have I ( cqp − c +1) ⊆ m I ( cqp − q − c +1) ⊆ m ( I [ qp ] : I [ q ] ) = m ( I [ p ] : I ) [ q ] . Since
R/I is Gorenstein, Discussion 2.1 shows that (cid:0) I [ p ] : I (cid:1) = I [ p ] + (∆) for some ∆ ∈ R .Since I ( cqp − c +1) ⊆ I [ qp ] , we also have I ( cqp − c +1) ⊆ m ( I [ p ] : I ) [ q ] ∩ I [ qp ] ⊆ m ( I [ qp ] + (∆ q )) ∩ I [ qp ] ⊆ m I [ qp ] + m (∆ q ) ∩ I [ qp ] . Thus it is enough to show that (∆ q ) ∩ I [ qp ] ⊆ m I [ qp ] . (2.1)If r ∆ q ∈ I [ qp ] , then r ∈ I [ qp ] : ∆ q = ( I [ p ] : ∆) [ q ] = I [ q ] . The last equality follows as in Discussion 2.1. This shows that (∆ q ) ∩ I [ qp ] ⊆ ( I ∆) [ q ] . ByProposition 2.2, I (∆) ⊆ m I [ p ] . The first statement follows.To prove the second statement, assume not, and choose an element f ∈ I ( cn ) , with f / ∈ m I n .By the first part of the theorem, there exists q > n such that I ( cq − c +1) ⊆ m I [ q ] . Observe that I ( cnq ) = I ( cqn ) = I ( cn +( cq − c ) n )) ⊆ (cid:0) I ( cq − c +1) (cid:1) n , where the last containment follows by [HH02, Theorem 1.1 (a)]. Since I ( cq − c +1) ⊆ m I [ q ] , I ( cnq ) ⊆ (cid:0) I ( cq − c +1) (cid:1) n ⊆ (cid:0) m I [ q ] (cid:1) n = m n (cid:0) I [ q ] (cid:1) n . Take any z ∈ (cid:0) m [ q ] : m n (cid:1) . Then f q ∈ (cid:0) I ( cn ) (cid:1) q ⊆ I ( cnq ) ⊆ m n (cid:0) I [ q ] (cid:1) n , XPECTED RESURGENCE OF IDEALS DEFINING GORENSTEIN RINGS 5 so zf q ∈ ( m I n ) [ q ] . Then z ∈ (cid:16) ( m I n ) [ q ] : f q (cid:17) = ( m I n : f ) [ q ] ⊆ m [ q ] , since we assumed that f / ∈ m I n . But this implies that (cid:0) m [ q ] : m n (cid:1) = m [ q ] , which is impossible by choice q > n . (cid:3) Remark 2.4.
In the proof above, we used [TY08, Theorem 3.1], which says that if ( R, m )is an excellent regular local ring of characteristic p and I is a radical ideal of big height I in R , then I (( h + k ) n +1) ⊆ m (cid:0) I ( k +1) (cid:1) n for all n > k >
0. We note, however, that this result also holds in the case where R is instead a standard graded regular ring containing a field k of characteristic p and withhomogeneous maximal ideal m . Indeed, the graded case follows directly from the local case,once we notice that it is sufficient to check that (cid:0) I (( h + k ) n +1) (cid:1) P ⊆ (cid:16) m (cid:0) I ( k +1) (cid:1) n (cid:17) P at each prime ideal P of R ; when P = m , the statement is precisely [TY08, Theorem 3.1],while for P = m the statement becomes I (( h + k ) n +1) ⊆ (cid:0) I ( k +1) (cid:1) n , which is [HH02, Theorem 1.1 (a)].We recall [GHM, Theorem 3.3]: Theorem 2.5 (Theorem 3.3 in [GHM]) . Let I be a radical ideal of big height c > R, m ) containing a field, or a quasi-homogeneous radical ideal of bigheight c > m . If I ( ct − c +1) ⊆ m I t for some fixed t, and if I p has the property that I ( n ) P = I nP for all P = m and for all n , then ρ ( I ) < c .As a consequence, we can now prove the following: Corollary 2.6.
Let I be a radical ideal of height c > R, m ) ofcharacteristic p . Suppose that R/I is Gorenstein. Further assume that for all primes P notequal to the maximal ideal m , I nP = I ( n ) P . Then the resurgence of I is expected, i.e., ρ ( I ) < c .In particular, I satisfies the stable Harbourne conjecture. Proof.
Using Theorem 2.3, we obtain that I ( cq − c +1) ⊆ m I [ q ] for every q = p e > pc . Thensimply put t = q for large enough q in the statement of Theorem 2.5. (cid:3) Remark 2.7.
The condition that I nP = I ( n ) P for all primes P not equal to the maximal idealcomes up often in the statements of our theorems. It is not an easy question in general todecide whether or not the symbolic powers of an ideal agree with the usual powers, althoughthis equality holds whenever I is generated by a regular sequence. In particular, if I isself-radical in a regular ring R and the dimension of R/I is one, then I P is generated by a GRIFO, HUNEKE, AND MUKUNDAN regular sequence for all primes P not equal to the maximal ideal. More generally if R/I hasan isolated singularity, or in the graded case if the associated projective variety is smooth,then I nP = I ( n ) P for all n and for all primes P with P = m . This condition is also equivalentto I ( n ) = ( I n : m ∞ ) for all n > Remark 2.8.
One of the important evidences in the proof of Theorem 2.3 is the containment(2.1). When R is graded, then the containment (2.1) can also be verified by comparing thedegree of the generators on both sides of the relation. Consequently, it would be enough tocheck if deg ∆ is more than the maximal degree of a minimal generating set of the homoge-neous ideal I . When translated into the language of Betti numbers, the previous statementis true due the existence of extremal Betti numbers of I . Roughly speaking, an extremalBetti number β i,j is the non-zero top left corner in a block of zeroes in the betti diagram for I (we refer to [BCP99] for a formal definition). The existence of extremal Betti numbers hasbeen proved in [BCP99, Theorem 1.2].3. Expected Resurgence
In this section, we extend some the sufficient conditions for expected resurgence from[GHM] in such a way that will allow us to prove expected resurgence for suitable idealsdefining Gorenstein rings in equicharacteristic 0. We also search for other characteristic-freeconditions which imply expected resurgence, focusing on the condition that the symbolic Reesalgebra of I is noetherian. Considering such a condition is particularly important since evena strong result in characteristic p will not necessarily give any conclusion in equicharacteristic0, despite of the usual technique of reduction to characteristic p . This is due to the fact thatresurgence is an asymptotic invariant, and one can only generally conclude something for alllarge values. In characteristic p , these large values often depend on the characteristic itself,for example as in the statement of Theorem 2.3. As the characteristic grows, so too do thesebounds, making it impossible to apply standard reduction techniques.The symbolic Rees algebra of an ideal I in a ring R is the graded algebra R s ( I ) = M n > I ( n ) t n ⊆ R [ t ] . In general, the symbolic Rees algebra can fail to be noetherian, even if I is a prime ideal ofheight d − d , with the first such example with non-noetheriansymbolic Rees algebra found by Roberts in [Rob85]. Even for the special case of the definingideal of a space monomial curve ( t a , t b , t c ), the symbolic Rees algebras are not noetherian[GNW94] for certain choices of a , b , and c . The condition that the symbolic Rees algebrais noetherian does have nice consequences, such as the fact that the resurgence is rational[DD20].While determining whether or not the symbolic Rees algebra of a given radical ideal isnoetherian is a difficult question in general, this does indeed happen for various interestingclasses of ideals. For example, the symbolic Rees algebra of a monomial ideal is alwaysnoetherian [Lyu88, Proposition 1]. Morales gave a criterion for when primes defining curveshave noetherian symbolic Rees algebras [Mor91]; see also [Hun87]. In [Cut91], Cutkoskyshowed that the symbolic Rees algebra of the defining ideal of k [ t a , t b , t c ] is noetherian aslong as ( a + b + c ) > abc . XPECTED RESURGENCE OF IDEALS DEFINING GORENSTEIN RINGS 7
We first need a result similar to [GHM, Lemma 2.5], but stronger:
Lemma 3.1.
Let ( R, m ) be a regular local ring of dimension d , or R a standard graded ringcontaining a field k with homogeneous maximal ideal m . Let I be an ideal of big height c such that I ( n ) = ( I n : m ∞ ) for all n >
1. If I ( cn ) ⊆ m ⌊ nα ⌋ I n for all n ≫
0, then ρ ( I ) < c . Proof.
By [GHM, Lemma 2.5], it is enough to find positive t < r such that I ( tcn ) ⊆ I rn forall n ≫
0. Since I rn + d ⊆ I rn , it is enough to show that I ( ctn ) ⊆ I rn + d for all n ≫ t , and let v ∈ R be nonzero such that v (cid:16) m ⌊ tα ⌋ I t (cid:17) n ⊆ m ⌊ tα ⌋ n I nt for all n . For why such a v exists, see for example [SH06, Corollary 6.8.12]. Fix integers l and k with the following properties:(1) m ln I ( n ) ⊆ I n for all n > vJ ⊆ I n for some ideal J , then J ⊆ I n − k .Such l exists by [Swa97, Main Theorem], given our assumption that m is the only possibleassociated prime for I n . The integer k exists since the integral closures of powers of I are determined by the (finitely many) Rees valuations v , ..., v m of I ; see [SH06, Theorem10.2.2]. If k is chosen so that v i ( v ) v i ( I k ) for every such v i , then vJ ⊆ I n implies that v i ( J ) > v i ( I n − k ) for 1 i m , which then implies that J ⊆ I n − k .Given any integer b , and n ≫ vI ( ctn ) ) b +1 ⊆ (cid:16) v m ⌊ tα ⌋ n I tn (cid:17) b I ( ctn ) ⊆ m b ⌊ tα ⌋ n I bnt I ( ctn ) . (3.1)Now we claim that we can chose b (independent of t and r ) such that b (cid:22) tα (cid:23) > lct. To do that, take b > lcα . b (cid:22) tα (cid:23) = bu > lcαu = lcαu + lcαu = lc ( t − v ) + lcαu = lct − lcv + lcαu = lct + lc ( αu − v ) > lctn. With our choice of b , we have m b ⌊ tα ⌋ n I ( cnt ) ⊆ m lctn I ( cnt ) ⊆ I ctn , and thus ( vI ( ctn ) ) b +1 ⊆ I ctn + btn . So far, everything we have shown holds for any positive integer t . We will now show thatgiven a fixed b > lcα and k as above, we can find positive integers t < r such that ctn + btn > ( rn + d + k )( b + 1) . GRIFO, HUNEKE, AND MUKUNDAN
With such values, we will have ( vI ( ctn ) ) b +1 ⊆ (cid:0) I rn + d + k (cid:1) b +1 , which shows that vI ( ctn ) ⊆ I rn + d + k . As a consequence, we have I ( ctn ) ⊆ I rn + d ⊆ I rn . So all that remains to show that given b > lcα , we can chose positive t < r such that ctn + btn > ( rn + d + k )( b + 1) . To do that, choose an integer r > r > γ ( d + l )1 − γ , where γ = b +1 b + c . Then r − rγ > γ ( d + k ) so r − > ( r + d + k ) γ Pick any integer t such that r > t > ( r + d + k ) γ. For all n >
1, we have t > ( r + d + k ) γ > (cid:18) r + d + kn (cid:19) γ = (cid:18) r + d + kn (cid:19) (cid:18) b + 1 b + c (cid:19) . (3.2)Multiplying both sides by n , we get tn > ( rn + d + k ) (cid:18) b + 1 b + c (cid:19) , so tn ( b + c ) > ( rn + d + k )( b + 1) . And this concludes our proof that I ( ctn ) ⊆ I rn . By [GHM, Lemma 2.5], this implies that ρ ( I ) < c . (cid:3) Our main result in this section assumes the symbolic Rees algebra of I is noetherian.Under mild circumstances such as excellence, this is equivalent to the statement that thereexists l such that I ( ln ) = (cid:0) I ( l ) (cid:1) n for all n >
1. See for example [Ree58, Lemma 2] or [Gri18,Lemma 5.2].
Theorem 3.2.
Let I be a homogeneous ideal of big height c > R with homogeneous maximal ideal m , or an ideal in a regular local ring ( R, m ).Suppose that:a) I ( cn ) ⊆ m I n for all n ≫
0; andb) the symbolic Rees algebra of I is noetherian.Then there exists an integer l such that I ( cn ) ⊆ m ⌊ nl ⌋ I n for all n > Proof.
Assumption b) implies that there exists t such that I ( tn ) = (cid:0) I ( t ) (cid:1) n for all n >
1; seefor example [Ree58, Lemma 2]. Notice that this containment also holds if we replace t byany multiple of t , and so we may assume without loss of generality that l = tk is chosenlarge enough so that • I ( cn ) ⊆ m I n for all n > l , and XPECTED RESURGENCE OF IDEALS DEFINING GORENSTEIN RINGS 9 • I ( ln ) = (cid:0) I ( l ) (cid:1) n for all n > n is a multiple of l , say n = lk . Indeed, I ( clk ) = (cid:0) I ( cl ) (cid:1) k ⊆ (cid:16) m I l (cid:17) k ⊆ m k I lk ⊆ m ⌊ kll ⌋ I kl . Now take f ∈ I ( cn ) and n > f l ∈ (cid:0) I ( cn ) (cid:1) l ⊆ I ( cnl ) ⊆ m n I ln ⊆ (cid:16) m ⌊ nl ⌋ I n (cid:17) l . Thus f ∈ m ⌊ nl ⌋ I n . We have now shown that I ( cn ) ⊆ m ⌊ nl ⌋ I n for all n > (cid:3) In fact, the conclusion of the above theorem is stronger than what we need to applyLemma 3.1 and conclude that I has expected resurgence. Corollary 3.3.
Let I be a homogeneous ideal of big height c > R with homogeneous maximal ideal m , or in a regular local ring ( R, m ). Suppose that:a) I ( n ) = ( I n : m ∞ ) for all n > I ( cn ) ⊆ m I n for all n ≫
0; andc) the symbolic Rees algebra of I is noetherian.Then the resurgence of I is expected. Proof.
The result follows from the previous theorem and Lemma 3.1. (cid:3) Resurgence in Characteristic Zero for Ideals Defining Gorenstein Rings
We now apply the results from Section 3 to ideals defining Gorenstein rings in equichar-acteristic 0.
Theorem 4.1.
Let I be a homogeneous ideal of height c > R containing a field k of characteristic 0 and with homogeneous maximal ideal m , or anideal in a regular local ring R of characteristic p with maximal ideal m . If R/I is Gorenstein,then for all n > I ( cn ) ⊆ m I n . Proof.
Suppose that the theorem is false, and pick n and f such that f ∈ I ( cn ) but f / ∈ m I n .First assume we are in either the graded case, or the case in which R is essentially of finite typeover k . We will reach a contradiction via standard reduction to characteristic p techniques,similarly to [HH02, Theorem 4.4]. Using the standard descent theory of [HH99, Chapter 2],we replace the field k by a finitely generated Z -algebra A , so that we have a counterexamplefor this containment in an affine A -algebra R A over A with R A ⊆ R and R ∼ = k ⊗ A R A ;after localizing at a nonzero element of A , we may assume that A is smooth over Z . We alsoretain models I A of I and m A of m , such that R A /I A is Gorenstein and such that there isan element f A ∈ I ( cn ) A such that f A / ∈ m A I nA . The result now follows from the fact that foralmost all fibers, the containment holds for the map Z −→ R A after passing to fibers overclosed points of Spec( Z ). Suppose that isn’t so. Further using [HH99, Theorem 2.3.15] wenow have a regular ring R p containing a field k p of characteristic p and with (homogeneous) maximal ideal m p , an ideal I p of height c such that R p /I p is Gorenstein, and f p ∈ I ( cn ) p suchthat f p / ∈ m p I np . This contradicts the conclusion of Theorem 2.3. This shows that for ouroriginal I , in equicharacteristic 0, and for all n > I ( cn ) ⊆ m I n .In the general case we also follow the detailed proof in [HH02]. The point is that aftercompleting and using Artin-Approximation, we can descend a counterexample to the desiredcontainment. The only difference in this case from a similar result in [HH02] is the addedcondition that R/I be Gorenstein. However, as described in Chapter 2 of [HH99], we canpreserve the shape of resolutions as we descend, and in particular can preserve the Bettinumbers in a resolution of
R/I over R . Since the height does not change (even thoughthe dimension of ambient regular ring may change), we can preserve the fact that R/I isGorenstein. This reduces to the case in which R is essentially of finite type over k , and thefirst part of the proof does this case. (cid:3) Corollary 4.2.
Let I be a homogeneous ideal of height c > R containing a field k of characteristic 0 and with homogeneous maximal ideal m , or anideal in a regular local ring R of characteristic p with maximal ideal m . Assume:a) I ( n ) = ( I n : m ∞ ) for all n > I is noetherian andc) R/I is Gorenstein.Then the resurgence of I is expected. Proof.
Recall that our assumption that the symbolic Rees algebra is noetherian implies thatthere exists l such that I ( ln ) = (cid:0) I ( l ) (cid:1) n for all n >
1. By Theorem 4.1, our conditions implythat for all n > I ( cn ) ⊆ m I n . Then by Corollary 3.3 I has expected resurgence. (cid:3) As we observed earlier, if I has expected resurgence, then I satisfies the stable Harbourneconjecture, which is an apparently much weaker condition. However, in characteristic 0,the assumption that the symbolic Rees algebra is noetherian gives the converse, so that theconditions of expected resurgence and stable Harbourne are in fact equivalent: Theorem 4.3.
Let I be a homogeneous ideal of big height c > R with homogeneous maximal ideal m , or an ideal in a power series ring over afield of characteristic 0 with maximal ideal m . Suppose that:a) I ( n ) = ( I n : m ∞ ) for all n >
1, andb) the symbolic Rees algebra of I is noetherian.If I satisfies the stable Harbourne Conjecture, then the resurgence of I is expected. Proof.
We claim that for all n ≫ I ( cn ) ⊆ m I n . First, note that in characteristic 0, any element f is integral over m D ( f ), where D ( f ) isthe ideal generated by the partial derivatives of f . In particular, I ( n ) ⊆ m I ( n − for all n .Suppose that n is large enough for stable Harbourne to hold, meaning I ( cn − c +1) ⊆ I n . Then I ( cn ) ⊆ m I ( cn − as we just showed ⊆ m I ( cn − c +1) because c > ⊆ m I n by assumptionThis shows we can apply Corollary 3.3, and I must have expected resurgence. (cid:3) XPECTED RESURGENCE OF IDEALS DEFINING GORENSTEIN RINGS 11
Remark 4.4.
In fact, the proof above shows that under these assumptions, expected resur-gence is equivalent to I ( cn − ⊆ I n for n ≫
0. Essentially, this says that I has expectedresurgence if and only if I ( cn ) ⊆ I n is not the best possible containment as n −→ ∞ . Acknowledgements
This work was done while the first author was on a year long visit to the University ofUtah; she warmly thanks the University of Utah Math Department for their hospitality. Thefirst author is supported by NSF grant DMS-2001445.
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Department of Mathematics, University of California, Riverside, Riverside, CA 92521,USA
E-mail address : [email protected] Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4135,USA
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