Poincaré series of multiplier and test ideals
aa r X i v : . [ m a t h . A C ] F e b POINCAR ´E SERIES OF MULTIPLIER AND TEST IDEALS
JOSEP `ALVAREZ MONTANER AND LUIS N ´U ˜NEZ-BETANCOURT Abstract.
We prove the rationality of the Poincar´e series of multiplier ideals in any di-mension and thus extending the main results for surfaces of Galindo and Monserrat andAlberich-Carrami˜nana et al. Our results also hold for Poincar´e series of test ideals. In orderto do so, we introduce a theory of Hilbert functions indexed over R which gives an unifiedtreatment of both cases. Introduction
Let A be a commutative Noetherian ring containing a field K . Assume that A is eitherlocal or graded with maximal ideal m and let a be an m -primary ideal. Depending on thecharacteristic of the base field we may find two parallel sets of invariants associated to thepair ( A, a c ) where c is a real parameter. In characteristic zero we have the theory of multiplierideals which play a prominent role in birational geometry and are defined using resolution ofsingularities (see [Laz04] for more insight). In positive characteristic we may find the so-called test ideals which originated from the theory of tight closure [HH90, HY03] and are definedusing the Frobenius endomorphism [BMS08, Sch11b, Bli13]. Despite its different origins,it is known that under some conditions on A , the reduction mod p of a multiplier ideal isthe corresponding test ideal [Smi00, Har01, HY03, Tak04, MS11, dFDTT15, CEMS18] (seealso [ST12, BFS13]). Moreover, both theories share a lot of common properties which wesummarize as saying that they form a filtration of m -primary ideals J : A ! J α ! J α ! . . . ! J α i ! . . . and the indices where there is an strict inequality is, under some assumptions on A , a discreteset of rational numbers [Laz04, CEMS18, BMS08, TT08, KLZ09, BSTZ10, Sch11a, ST14].The multiplicity of c ∈ R > is defined as m ( c ) = dim K ( J c − ε / J c ) , for ε > Poincar´e series of J P J ( T ) = X c ∈ R > dim K ( J c − ε / J c ) T c . The natural question is whether this is a rational function, in the sense that it belongs tothe field of fractional functions Q ( z ) where the indeterminate z corresponds to a fractionalpower T /e for a suitable e ∈ N > .Galindo and Monserrat [GM10] proved that this rationality property holds for multiplierideals associated to simple m -primary ideals in a complex smooth surface and providedan explicit formula. These results were extended later on by Alberich-Carrami˜nana et al. Partially supported by grants MTM2015-69135-P (MINECO/FEDER), 2017SGR-932 (AGAUR) andPID2019-103849GB-I00 (AEI/10.13039/501100011033). Partially supported by CONACYT Grant 284598 and C´atedras Marcos Moshinsky. AADG17] (see also [AADG20]) to the case of multiplier ideals associated to any m -primaryideal in a complex surface with rational singularities. The techniques used in both casesrely on the theory of singularities in dimension two and, in particular, the fact that thedata coming from the log-resolution of any ideal can be encoded in a combinatorial objectsuch as the dual graph . In the case of simple ideals, the divisors corresponding to the starvertices of the graph measure the difference between a multiplier ideal and its preceding.In general one needs the notion of maximal jumping divisor [AADG17] to account for thisdifference. The formula obtained for the Poincar´e series is then described in terms of the excesses of these maximal jumping divisors. During the preparation of this manuscript, welearned that Pande [Pan21] has extended these results to the case of smooth varieties inarbitrary dimension.In this work, we show the rationality of the Poincar´e series of multiplier ideals of m -primary ideals in any normal variety in arbitrary dimension (see Theorem 3.2 and Corollary4.8 for the Cohen-Macaulay case). Furthermore, we also prove the rationality of the Poincar´eseries for test ideals of m -primary ideals in F -finite rings that are strongly F -regular in thepunctured spectrum (see Theorem 3.8 and Corollary 4.9 for the Cohen-Macaulay case). Asa particular case, we obtain the rationality of P J ( T ) for ideals in normal surfaces in primecharacteristic.Our approach is completely algebraic, and it provides an unified proof of the rationality ofthe Poincar´e series for both the multiplier and the test ideals in any dimension as long as wehave discreteness of the jumping numbers and Skoda’s theorem. We point out that our mainresults does not require the rationality of the jumping numbers. Examples of non-rationaljumping numbers of multiplier ideals exist by work of Urbinati [Urb12]. The rationalityof the Poincar´e series in this case means that it belongs to the field of fractional functions Q ( T α , . . . , T α s ), where α , . . . , α s ∈ R is a finite set of jumping numbers.To such purpose we develop a theory of Hilbert functions indexed over R that shouldbe of independent interest. More precisely, in Section 2 we develop the notion of R -good a -filtrations associated to a finitely generated A -module which is an extension of the well-known theory of good a -filtrations. In this general framework we can define the multiplicityof any module in the filtration and the corresponding Poincar´e series. The main result isTheorem 2.5 where we prove the rationality of such a series. In Section 3 we specialize ourmain result to the case of multiplier ideals and test ideals. We also extend to arbitrarydimension the notion of maximal jumping divisor (see Definition 3.3) and give a formulafor the multiplicity (see Proposition 3.5). In Section 4 we provide a different approach tothe theory of R -good a -filtrations in the case of Cohen-Macaulay rings that gives a simplerformula for the Poincar´e series (see Theorem 4.5). By comparing our results with the onespreviously obtaining by geometric methods, we yield an algebraic formula for the excessassociated to the maximal jumping divisor (see Proposition 4.12). Acknowledgements:
Part of this work was done during a research stay of the first authorat CIMAT, Guanajuato supported by a Salvador de Maradiaga grant (ref. PRX 19/00405).He wants to thank the people at CIMAT for the warm welcome. We are grateful to SwarajPande for sharing a preliminary version of his work. We also acknowledge helpful discussionswith V´ıctor Gonz´alez-Alonso and Mart´ı Lahoz. . R -good filtrations Let A be a commutative Noetherian ring. Assume that A is either local or graded withmaximal ideal m and let a be an m -primary ideal. The theory of good a -filtrations givesan approach to the study of Hilbert functions that covers most of the classical results inan unified way. We start recalling briefly this notion but we refer to Rossi and Valla’smonograph [RV10] and the references therein for more insight.Let M be a finitely generated A -module such that λ ( M/ a M ) < ∞ , where λ ( · ) denotesthe length as A -module. A good a -filtration on M is a decreasing filtration M : M = M ⊇ M ⊇ · · · by A -submodules of M such that M j +1 = a M j for j ≫ Hilbert and the
Hilbert-Samuel function of M with respect tothe filtration M defined as H M ( j ) := λ ( M j /M j +1 ) and H M ( j ) := λ ( M/M j )respectively. Moreover, we consider the Hilbert and the
Hilbert-Samuel series HS M ( T ) := X j ≥ λ ( M j /M j +1 ) T j and HS M ( T ) := X j ≥ λ ( M/M j ) T j . Notice that we have HS M ( T ) = (1 − T ) HS M ( T ). As a consequence of the Hilbert-SerreTheorem, we can express them as rational functions HS M ( T ) = (1 − T ) HS M ( T ) = (1 − T ) h M ( T )(1 − T ) d +1 , where h M ( T ) ∈ Z [ T ] satisfies h M (1) = 0 and d is the Krull dimension of M . The polynomial h M ( T ) is the h -polynomial of M .The aim of this section is to extend the notion of good a -filtrations by allowing filtrationsindexed over R and thus mimicking properties satisfied by filtrations given by multiplier andtest ideals. Definition 2.1.
Let M be a finitely generated A -module such that λ ( M/ a M ) < ∞ . An R -good a -filtration is a decreasing filtration M := { M α } α ≥ of submodules of M = M ,indexed by a discrete set of positive real numbers such that M α +1 = a M α for all α > j with j ≫ Q -good a -filtration when the set of indices is contained in Q .Indeed, we may think of M as a filtration of submodules M c indexed over R for which thereexist an increasing sequence of real numbers 0 < α < α < . . . such that M α i = M c ! M α i +1 for any c ∈ [ α i , α i +1 ). In particular we have a discrete filtration of submodules M : M ! M α ! M α ! . . . ! M α i ! . . . and we say that the α i are the jumping numbers of M . A crucial observation is that, oncewe fix an index c ∈ R , the filtration M c : M c ⊇ M c +1 ⊇ M c +2 ⊇ · · · is a good a -filtration. efinition 2.2. Let M := { M c } c ≥ be an R -good a -filtration. We define the multiplicity of c ∈ R > as m ( c ) := λ ( M c − ε /M c )for ε > c is a jumping number if andonly if m ( c ) > Definition 2.3.
Let M := { M c } c ≥ be an R -good a -filtration. We define the Poincar´e seriesof M as P M ( T ) = X c ∈ R > m ( c ) T c . The question that we want to address is whether the Poincar´e series is rational in the sensethat it belongs to the field of fractional functions Q ( T α , . . . , T α s ), where α , . . . , α s ∈ R isa finite set of jumping numbers. In the case of Q -good a -filtrations, the rationality of thePoincar´e series means that it belongs to the field of fractional functions Q ( T /e ) where e ∈ N > is the least common multiple of the denominators of all the jumping numbers. Proposition 2.4.
Let M := { M c } c ≥ be an R -good a -filtration. Given c ∈ R > we havethat X j ≥ m ( c + j ) T j is a rational function in Q ( T ). Proof.
Recall that the Hilbert series HS M c − ε ( T ) and HS M c ( T ) associated to the good a -filtrations M c − ε and M c are rational functions. From the short exact sequence0 / / M c /M c + j / / M c − ε /M c + j / / M c − ε /M c / / X j ≥ λ ( M c − ε /M c + j ) T j = HS M c ( T ) + m ( c ) 11 − T .
Analogously, from the short exact sequence0 / / M c − ε + j /M c + j / / M c − ε /M c + j / / M c − ε /M c − ε + j / / X j ≥ m ( c + j ) T j = X j ≥ λ ( M c − ε /M c + j ) T j − HS M c − ε ( T )= m ( c ) 11 − T + HS M c ( T ) − HS M c − ε ( T )= m ( c )1 − T + h M c ( T ) − h M c − ε ( T )(1 − T ) d +1 and thus it is a rational function. Here, h M c ( T ) and h M c − ε ( T ) are the h -polynomials of thegood a -filtrations M c − ε and M c respectively. (cid:3) heorem 2.5. Let M := { M c } c ≥ be an R -good a -filtration. Then, the Poincar´e series P M ( T ) is rational. Moreover we have P M ( T ) = X c ∈ (0 , (cid:18) m ( c )1 − T + h M c ( T ) − h M c − ε ( T )(1 − T ) d +1 (cid:19) T c , where h M c ( T ) and h M c − ε ( T ) are the h -polynomials of the good a -filtrations M c − ε and M c respectively. Proof.
We have P M ( T ) = X c ∈ R > m ( c ) T c = . X c ∈ (0 , X j ∈ Z ≥ m ( c + j ) T j T c and thus the result follows from Proposition 2.4. (cid:3) Poincar´e series of multiplier and test ideals
In this section we turn our attention to the case where A contains a field K and the R -good a -filtration that we consider is given by a filtration of m -primary ideals J : A ! J α ! J α ! . . . ! J α i ! . . . In this setting, the multiplicity of c ∈ R > is m ( c ) = dim K ( J c − ε / J c ) , for ε > J is P J ( T ) = X c ∈ R > dim K ( J c − ε / J c ) T c . The aim of this section is to specialize the results we obtained in the previous section tothe case of multiplier ideals and test ideals.3.1.
Multiplier ideals.
Let ( A, m ) be a normal local ring containing an algebraically closedfield K of characteristic zero and a ⊆ A an ideal. Under these general assumptions we ensurethe existence of canonical divisors K X on X = Spec A which are not necessarily Q -Cartier.Then we may find some effective boundary divisor ∆ such that K X + ∆ is Q -Cartier withindex m large enough. Now, given a log-resolution π : X ′ → X of the triple ( X, ∆ , a ) wepick a canonical divisor K X ′ in X ′ such that π ∗ K X ′ = K X and let F be an effective divisorsuch that a · O X ′ = O X ′ ( − F ).The multiplier ideal associated to the triple ( X, ∆ , a c ) for some real number c ∈ R > isdefined as J ( X, ∆ , a c ) = π ∗ O X ′ (cid:18)(cid:24) K X ′ − m π ∗ ( m ( K X + ∆)) − cF (cid:25)(cid:19) . This construction allowed de Fernex and Hacon [dFH09] to define the multiplier ideal J ( a c )associated to a and c as the unique maximal element of the set of multiplier ideals J ( X, ∆ , a c )where ∆ varies among all the effective divisors such that K X + ∆ is Q -Cartier. The keypoint in their proof is the existence of such a divisor ∆ that realizes the multiplier ideal as ( a c ) = J ( X, ∆ , a c ). In this general framework we have that the local vanishing theorem stillhold [dFEM14, Theorem 4.1.19]. Namely, for any c ∈ R > we have R π ∗ O X ′ (cid:18)(cid:24) K X ′ − m π ∗ ( m ( K X + ∆)) − cF (cid:25)(cid:19) = 0 . Remark 3.1. If A is Q -Gorenstein, the canonical module K X is Q -Cartier so no boundary∆ is required in the definition of multiplier ideal. Namely we have J ( a c ) = π ∗ O X ′ (cid:18)(cid:24) K X ′ − m π ∗ ( mK X ) − cF (cid:25)(cid:19) . From its construction we have that the multiplier ideals form a filtration A ! J ( a α ) ! J ( a α ) ! ... ! J ( a α i ) ! ... and the α i where we have a strict inclusion of ideals are the jumping numbers of the ideal a .Assume in addition that a is an m -primary ideal and thus F is a divisor with exceptionalsupport. Then any multiplier ideal J ( a c ) is m -primary as well. To ensure that J = { J ( a c ) } c ≥ is an R -good a -filtration we notice the following: · Discreteness : If a is m -primary, the number of multiplier ideals in any interval [ c , c ]is smaller or equal than dim K J ( a c ) / J ( a c ). · Skoda’s theorem [dFH09, Corollary 5.7]: For any c > dim A we have J ( a c ) = a · J ( a c − ) . There are cases where the jumping numbers are not rational as shown by Urbinati [Urb12].Known cases where the jumping numbers form a discrete set of rational numbers and thusthe filtration J = { J ( a c ) } c ≥ is a Q -good a -filtration are: · X is Q -Gorenstein. · The symbolic Rees algebra R ( − ( K X + ∆)) := L n ≥ O X ( − n ( K X + ∆)) is finitelygenerated [CEMS18, Remark 2.26]. Theorem 3.2.
Let ( A, m ) be a normal local ring of dimension d containing an algebraicallyclosed field K of characteristic zero, a ⊆ A an m -primary ideal and let J := { J ( a c ) } c ≥ bethe filtration given by multiplier ideals. Then, the Poincar´e series P J ( T ) is rational. Indeed,we have P J ( T ) = X c ∈ (0 , (cid:18) m ( c )1 − T + h J ( a c ) ( T ) − h J ( a c − ε ) ( T )(1 − T ) d +1 (cid:19) T c , where h J ( a c ) ( T ) is the h -polynomial associated to the multiplier ideal J ( a c ). Proof.
The result follows from Theorem 2.5. (cid:3)
When A is the local ring at a rational singularity of a surface, Alberich-Carrami˜nana etal. [AADG17, Theorem 4.1] gave a precise formula for the multiplicity m ( c ) of any given c ∈ R > , and consequently an explicit description of the Poincar´e series. We may follow thesame approach to get a partial extension of their formula. efinition 3.3. Let ( X, ∆ , a c ) be a triple. The maximal jumping divisor associated to c ∈ R > is H c = (cid:24) K X ′ − m π ∗ ( m ( K X + ∆)) − ( c − ε ) F (cid:25) − (cid:24) K X ′ − m π ∗ ( m ( K X + ∆)) − cF (cid:25) where ε is small enough. Remark 3.4.
Denote K X ′ − m π ∗ ( m ( K X + ∆)) = P i k i E i and F = P i e i E i , where the E i ’sare the exceptional components of π . Then H c can be defined as the reduced divisor whosecomponents are the E i such that k i − ce i ∈ Z . In particular we have H c = H c +1 for all c ∈ R > . Proposition 3.5.
Let ( X, ∆ , a c ) be a triple. Then, the multiplicity of c ∈ R > is m ( c ) = h (cid:18) H c , O H c (cid:18)(cid:24) K X ′ − m π ∗ ( m ( K X + ∆)) − cF (cid:25) + H c (cid:19)(cid:19) Proof.
To avoid heavy notation, let K π := K X ′ − m π ∗ ( m ( K X + ∆)). Consider the shortexact sequence0 −→ O X ′ ( ⌈ K π − cF ⌉ ) −→ O X ′ ( ⌈ K π − cF ⌉ + H c ) −→ O H c ( ⌈ K π − cF ⌉ + H c ) −→ X and applying local vanishing for multiplier ideals we get the shortexact sequence0 −→ π ∗ O X ′ ( ⌈ K π − cF ⌉ ) −→ π ∗ O X ′ ( ⌈ K π − cF ⌉ + H c ) −→−→ H ( H c , O H c ( ⌈ K π − cF ⌉ + H c )) ⊗ K O −→ −→ J ( a c ) −→ J ( a ( c − ε ) ) −→ H ( H c , O H c ( ⌈ K π − cF ⌉ + H c )) ⊗ K O −→ c is just m ( c ) = h ( H c , O H c ( ⌈ K π − cF ⌉ + H c )). (cid:3) Question 3.6.
The key ingredient for the explicit formula of the Poincar´e series of mul-tiplier ideals in dimension 2 given by Alberich-Carrami˜nana et al. [AADG17] is that themultiplicities satisfy m ( c + k ) − m ( c ) = kρ c , where ρ c := − F · H c are the excesses associatedto the maximal jumping divisor H c . Pande [Pan21] proved that m ( c + j ) is a polynomialfunction in j of degree less than d in the case of smooth varieties in arbitrary dimension d .These results motivate the following question regarding multiplicities for m -primary idealsin normal rings. Is there a polynomial expression in terms of j for m ( c + j ) − m ( c ) = h ( H c , O H c ( ⌈ K π − cF ⌉ + H c + jF )) − h ( H c , O H c ( ⌈ K π − cF ⌉ + H c ))?3.2. Test ideals.
Let A be a commutative Noetherian ring containing a field K of charac-teristic p >
0. The theory of test ideals has its origins in the work of Hochster and Hunekeon tight closure [HH90]. In the case of A being a regular ring, Hara and Yoshida [HY03]extended the notion of test ideals to pairs ( A, a c ) where a ⊆ A is an ideal. Their constructionhas been generalized in subsequent works [BMS08, BMS09, TT08, BSTZ10, Sch11b, Bli13]using the theory of Cartier operators .Assume that A is F -finite. Then, the test ideal τ ( a c ) associated to a and some realnumber c ∈ R ≥ is the smallest nonzero ideal which is compatible with any Cartier operator ∈ L e ≥ Hom A ( F e ∗ A, A ) · F e ∗ a ⌈ cp e ⌉ , where F e ∗ is the Frobenius functor. In this situation wealso have a filtration A ! τ ( a α ) ! τ ( a α ) ! ... ! τ ( a α i ) ! ... and the α i where we have a strict inclusion of ideals are called the F -jumping numbers ofthe ideal a .We now give a sufficient condition to have that τ ( a c ) is m -primary. Lemma 3.7.
Let ( A, m ) be a local F -finite Noetherian ring containing a field K of charac-teristic p > a ⊆ A be an m -primary ideal. Assume that A p is a strongly F -regularring for all prime ideals p = m . Then, the test ideals τ ( a c ) are m -primary or A . Proof.
Since test ideals localize [Bli13, Proposition 3.2], we have that τ ( a c ) p = τ ( a c p ) = τ ( A c p ) = τ ( A p ) = A p for all prime ideals p = m , because A p is strongly F -regular. Thereforerad( τ ( a c )) ⊇ m . (cid:3) Under these extra assumptions we have that τ = { τ ( a c ) } c ≥ is an R -good a -filtration: · Discreteness : If a is m -primary and A is strongly F -regular in the punctured spec-trum, the number of test ideals in any interval [ c , c ] is smaller or equal thandim K τ ( a c ) /τ ( a c ). · Skoda’s theorem [Bli13, HT04, ST14]: For any c > dim A we have τ ( a c ) = a · τ ( a c − ) . Known cases where the F -jumping numbers form a discrete set of rational numbers andthus the filtration τ = { τ ( a c ) } c ≥ is a Q -good a -filtration are: · ( A, m ) is an F -finite, normal Q -Gorenstein local domain [BMS08, TT08, KLZ09,BSTZ10, Sch11a, ST14]. · A is an F -finite ring which is a direct summand of a regular ring [AHN17]. Theorem 3.8.
Let ( A, m ) be an F -finite local ring of dimension d containing a field K ofcharacteristic p > a be an m -primary ideal. Assume that A p is a strongly F -regularring for all prime ideals p = m . Let τ = { τ ( a c ) } c ≥ be the filtration given by test ideals.Then, the Poincar´e series P τ ( T ) is rational. Indeed, we have P τ ( T ) = X c ∈ (0 , (cid:18) m ( c )1 − T + h τ ( a c ) ( T ) − h τ ( a c − ε ) ( T )(1 − T ) d +1 (cid:19) T c where h τ ( a c ) ( T ) is the h -polynomial associated to the test ideal τ ( a c ). Proof.
The result follows from Theorem 2.5. (cid:3)
Motivated by the case of multiplier ideals [GM10, AADG17, Pan21], we would like to havea precise description of the multiplicities of F -jumping numbers since it would yield a moreexplicit formula for the Poincar´e series. More precisely we ask the following Question 3.9.
Is the multiplicity of test ideals of m -primary ideals in a strongly F -regularring, m ( c + j ), a polynomial function in j of degree less than d ? . Poincar´e series in Cohen-Macaulay rings
Let ( A, m , K ) be a Cohen-Macaulay local ring of dimension d . Let a be an m -primaryideal generated by a regular sequence f , . . . , f d . Let J = { J c } c ≥ be an R -good a -filtrationof m -primary ideals satisfying Skoda’s theorem so J c = a J c − for all c > d . The Poincar´eseries of J is P J ( T ) = X c ∈ R > m ( c ) T c = X c ∈ (0 , X j ≥ m ( c + j ) T j ! T c and zooming in the summands we have X j ≥ m ( c + j ) T j = m ( c )+ m ( c +1) T + · · · + m ( c + d − T d − + T d − X j ≥ λ ( a j J c + d − − ε / a j J c + d − ) T j The aim of this section is to work towards finding a more explicit formula for the Poincar´eseries in Cohen-Macaulay rings, especially in the case that J is a filtration of multiplier or testideals where we require that K is an infinite field. Namely, let ( A, m ) be a local Noetherianring containing an infinite field K and let a be any m -primary ideal. Every minimal reductionof a can be generated by a superficial sequence of length equal to the analytical spread of a [HS06, Theorem 8.6.3]. Since a is m -primary, ℓ ( a ) = dim( A ). If A is Cohen-Macaulay thissuperficial sequence is indeed a regular sequence. Therefore we have a = ( f , . . . , f d ), where( · ) denotes the integral closure. Multiplier ideals and test ideal are invariant up to integralclosure so we may assume that a is generated by a regular sequence. Setup 4.1.
Let ( A, m , K ) be a Cohen-Macaulay local ring of dimension d . Let J ⊆ A be an m -primary ideal and a = ( f , · · · , f d ) a parameter ideal. Consider a free resolution(1) · · · / / A β / / A β / / A / / A/ a j / / , where β = (cid:0) j +( d − d − (cid:1) is the number of generators of a j . After tensoring with A/J , we get(2) · · · / / ( A/J ) β φ Jj / / ( A/J ) β ϕ Jj / / ( A/J ) / / A/ ( a j + J ) / / , The morphisms ϕ Jj and φ Jj plays a role in what follows. If the ideal J is clear from thecontext we simply denote ϕ j and φ j . Notice also that φ j = 0 for j = 0. Lemma 4.1.
Let ( A, m , K ) be a Cohen-Macaulay local ring of dimension d . Let J ⊆ A bean m -primary ideal and a = ( f , · · · , f d ) a parameter ideal. Then, for every j ∈ Z > we have λ ( J/ a j J ) = λ ( A/ a j ) − λ (Im φ j ) + ( β − λ ( A/J )where β = (cid:0) j +( d − d − (cid:1) . Proof.
From the short exact sequence, 0 → J → A → A/J → , we have the induced longexact sequence 0 → Tor A ( A/ a j , A/J ) → J/ a j J → A/ a j → A/ ( a j + J ) → . ollowing Notation 4.1 we have Tor A ( A/ a j , A/J ) = ker ϕ j / Im φ j and A/ ( a j + J ) =( A/J ) / Im ϕ j . Then, λ ( J/ a j J ) = λ ( A/ a j ) + λ (Tor A ( A/ a j , A/J )) − λ ( A/ ( a j + J ))= λ ( A/ a j ) + [ λ (ker ϕ j ) − λ (Im φ j )] − [ λ ( A/J ) − λ (Im ϕ j )]= λ ( A/ a j ) − λ (Im φ j ) − λ ( A/J ) + [ λ (ker ϕ j ) + λ (Im ϕ j )]= λ ( A/ a j ) − λ (Im φ j ) − λ ( A/J ) + λ (( A/J ) β )= λ ( A/ a j ) − λ (Im φ j ) − λ ( A/J ) + β λ ( A/J )= λ ( A/ a j ) − λ (Im φ j ) + ( β − λ ( A/J ) (cid:3) Lemma 4.2.
Let ( A, m , K ) be a Cohen-Macaulay local ring of dimension d . Let J ⊆ K ⊆ A be m -primary ideals and a = ( f , · · · , f d ) a parameter ideal. Then, X j ≥ λ ( a j K/ a j J ) T j = λ ( K/J )(1 − T ) d + X j ≥ [ λ (Im φ Kj ) − λ (Im φ Jj )] T j . Proof.
From the short exact sequences0 → a j K/ a j J → K/ a j J → K/ a j K → , → J/ a j J → K/ a j J → K/J → λ ( a j K/ a j J ) = λ ( K/J ) + λ ( J/ a j J ) − λ ( K/ a j K ). Thus, applying Lemma 4.1 to theideals J and K , we get λ ( a j K/ a j J ) = λ ( K/J ) + [ λ ( A/ a j ) − λ (Im φ Jj ) + ( β − λ ( A/J )] − [ λ ( A/ a j ) − λ (Im φ Kj ) + ( β − λ ( A/K )]= λ ( K/J ) + ( β − λ ( A/J ) − λ ( A/K )) + [ λ (Im φ Kj ) − λ (Im φ Jj )]= β λ ( K/J ) + [ λ (Im φ Kj ) − λ (Im φ Jj )] , where β = (cid:0) j +( d − d − (cid:1) . Then the result follows since P j ≥ (cid:0) j +( d − d − (cid:1) T j = − T ) d . (cid:3) In order to get some control on λ (Im φ j ) we use the following result of Kodiyalam [Kod93,Theorem 2] in the form that we need in the present work. Proposition 4.3.
Let ( A, m , K ) be a local ring of dimension d and let a , J be m -primaryideals. Then, for all i ≥
0, the function λ (Tor Ai ( A/ a j , A/J )) is a polynomial of degree d − j ≫ λ and the fact that Tor modules are the homologymodules of the complex (2), we get Corollary 4.4.
Under Setup 4.1, the function λ (Im φ Jj ) is a polynomial of degree d − j ≫ heorem 4.5. Let ( A, m , K ) be a Cohen-Macaulay local ring of dimension d . Let a =( f , · · · , f d ) be a parameter ideal and J = { J c } c ≥ an R -good a -filtration of m -primary idealssatisfying J c = a J c − for all c > d . Then, there exists α , . . . , α d ∈ Z and p ( T ) ∈ Z [ T ] suchthat P J ( T ) = X c ∈ (0 , (cid:18) m ( c ) + · · · + m ( c + d − T d − + m ( c + d − T d − (1 − T ) d + T d (cid:18) α d (1 − T ) d + · · · + α (1 − T ) + p ( T ) (cid:19)(cid:19) T c . Proof.
We have X j ≥ m ( c + j ) T j = m ( c )+ m ( c +1) T + · · · + m ( c + d − T d − + T d − X j ≥ λ ( a j J c + d − − ε / a j J c + d − ) T j so applying Lemma 4.2 with K = J c + d − − ε and J = J c + d − we get P J ( T ) = X c ∈ (0 , (cid:18) m ( c ) + · · · + m ( c + d − T d − + m ( c + d − T d − (1 − T ) d + T d − X j ≥ [ λ (Im φ J c + d − − ε j ) − λ (Im φ J c + d − j )] T j ! T c . Using Corollary 4.4 we have that for j ≫ λ (Im φ J c + d − − ε j ) − λ (Im φ J c + d − j ) is apolynomial of degree d − α d (cid:18) ( j −
1) + d − d − (cid:19) + · · · + α (cid:18) ( j −
1) + 22 (cid:19) + α j + α Therefore, there exists k ∈ Z > such that T d − X j ≥ [ λ (Im φ J c + d − − ε j ) − λ (Im φ J c + d − j )] T j == T d q ( T ) + X j ≥ k (cid:20) α d (cid:18) ( j −
1) + d − d − (cid:19) + · · · + α (cid:18) ( j −
1) + 22 (cid:19) + α j + α (cid:21) T j − ! = T d (cid:18) q ( T ) + (cid:18) α d (1 − T ) d − q d ( T ) (cid:19) + · · · + (cid:18) α (1 − T ) − q ( T ) (cid:19)(cid:19) where q ( T ) , q d ( T ) , . . . , q ( T ) ∈ Z ( T ) have degree ≤ k − p ( T ) = q ( T ) − q d ( T ) − · · · − q ( T ). (cid:3) The following result is a direct consequence of Theorem 4.5. orollary 4.6. Let ( A, m , K ) be a Cohen-Macaulay local ring of dimension d . Let a =( f , · · · , f d ) be a parameter ideal and J = { J c } c ≥ an R -good a -filtration of m -primary idealssatisfying J c = a J c − for all c > d . Then, the function m ( c + j ) is a polynomial function on j of degree less than d for j ≫ Remark 4.7.
In the case of multiplier ideals in a smooth variety, Pande proved that thisresult holds for all j [Pan21, Theorem 3.2].Now we also specialize Theorem 4.5 to the case of multiplier and test ideals. Corollary 4.8.
Suppose ( A, m , K ) is a normal Cohen-Macaulay local ring of dimension d over an algebraically closed field of characteristic zero, a ⊆ A is any m -primary ideal and J := { J ( a c ) } c ≥ is the filtration given by multiplier ideals. Then, P J ( T ) = X c ∈ (0 , (cid:18) m ( c ) + · · · + m ( c + d − T d − + m ( c + d − T d − (1 − T ) d + T d (cid:18) α d (1 − T ) d + · · · + α (1 − T ) + p ( T ) (cid:19)(cid:19) T c . Proof.
For every m -primary ideal a there exist a parameter ideal with the same integralclosure. Since the multiplier ideals are the same for an ideal and its integral closure [Laz04,Variation 9.6.39] (see also [dFH09, Corollary 5.7]), the result follow from Theorem 4.5. (cid:3) Corollary 4.9.
Suppose that ( A, m , K ) is an F -finite Cohen-Macaulay local domain of di-mension d over an infinite field of characteristic p > A p is a strongly F -regular ring for allprime ideals p = m , a is any m -primary ideal and τ = { τ ( a c ) } c ≥ is the filtration given bytest ideals. Then, P τ ( T ) = X c ∈ (0 , (cid:18) m ( c ) + · · · + m ( c + d − T d − + m ( c + d − T d − (1 − T ) d + T d (cid:18) α d (1 − T ) d + · · · + α (1 − T ) + p ( T ) (cid:19)(cid:19) T c . Proof.
For every m -primary ideal a there exist a parameter ideal with the same integralclosure, because K is infinite. Since the test ideals are the same for an ideal and its integralclosure [HT04, Proof of Theorem 4.1] (see also [BMS08, Lemma 2.27]), the result follow fromTheorem 4.5. (cid:3) Remark 4.10.
Let ( A, m , K ) be an F -finite normal local ring of dimension 2 over an infinitefield of characteristic p >
0. Then the condition of being strongly F -regular in the puncturedspectrum and being Cohen-Macaulay is automatically satisfied4.1. The case of multiplier ideals in dimension two revisited.
Let ( A, m , K ) be aCohen-Macaulay local ring of dimension 2. Let a = ( f , f ) be a parameter ideal and J = { J c } c ≥ an R -good a -filtration of m -primary ideals satisfying J c = a J c − for all c > P J ( T ) = X c ∈ (0 , (cid:18) m ( c ) + m ( c + 1) T (1 − T ) + T (cid:18) α (1 − T ) + α (1 − T ) + p ( T ) (cid:19)(cid:19) T c . We see that, at least for the case of multiplier ideals in a complex surface with a rationalsingularity, this formula is much simpler. To do so we compare our formula with the oneobtained in that case.
Theorem 4.11 ([AADG17, Theorem 6.1]) . Let ( A, m ) be the local ring of a complex surfacewith a rational singularity, a ⊆ A an m -primary ideal and let J := { J ( a c ) } c ≥ be the filtrationgiven by multiplier ideals. Then P J ( T ) = X c ∈ (0 , (cid:18) m ( c )1 − T + ρ c T (1 − T ) (cid:19) T c where ρ c := − F · H c is the excess associated to the maximal jumping divisor H c .If we compare both formulas we observe P J ( T ) = X c ∈ (0 , (cid:18) m ( c ) + ( m ( c + 1) − m ( c )) T + m ( c ) T (1 − T ) + T (cid:18) α (1 − T ) + α (1 − T ) + p ( T ) (cid:19)(cid:19) T c . = X c ∈ (0 , (cid:18) m ( c )1 − T + ρ c T (1 − T ) + T (1 − T ) (cid:0) m ( c ) + α + α (1 − T ) + p ( T )(1 − T ) (cid:1)(cid:19) T c and we conclude that m ( c ) = − α , α = 0 and p ( T ) = 0. If we take a closer look to theseconditions we obtain a reformulation of [AADG17, Proposition 4.5] which, in particular,gives an algebraic formula for the excesses. Proposition 4.12.
Let ( A, m ) be the local ring of a complex surface with a rational singu-larity, a ⊆ A an m -primary ideal and let J := { J ( a c ) } c ≥ be the filtration given by multiplierideals. Then, ρ c = 1 j (cid:16) λ (Tor A ( A/ a j , A/ J ( a c +1 ))) − λ (Tor A ( A/ a j , A/ J ( a c +1 − ε ))) (cid:17) for every j ≥
1, where is the excess associated to the maximal jumping divisor H c . Inparticular, m ( c + j ) − m ( c ) = λ (Tor A ( A/ a j , A/ J ( a c +1 ))) − λ (Tor A ( A/ a j , A/ J ( a c +1 − ε )))for every j ≥ Proof.
First recall that the morphisms φ Jj in Setup 4.1 for an m -primary ideal J ⊆ A are0 / / ( A/J ) j φ Jj / / ( A/J ) j +1 ϕ Jj / / ( A/J ) / / A/ ( a j + J ) / / , and thus λ (Im φ Jj ) = λ (( A/J ) j ) − λ (ker φ Jj ) = jλ ( A/J ) − λ (Tor A ( A/ a j , A/J )) . For simplicity we denote λ c +1 j and λ c +1 − εj when we refer to λ (Im φ Jj ) with J being themultiplier ideals J ( a c +1 ) and J ( a c +1 − ε ) respectively. Then, as in the proof of Theorem 4.5, e have X j ≥ [ λ c +1 − εj − λ c +1 j ] T j − = q ( T ) + (cid:18) α (1 − T ) − q ( T ) (cid:19) + (cid:18) α (1 − T ) − q ( T ) (cid:19) where, for some k ≫ q ( T ) = ( λ c +1 − ε − λ c +11 ) + ( λ c +1 − ε − λ c +12 ) T + · · · + ( λ c +1 − εk − − λ c +1 k − ) T k − . q ( T ) = α (1 + 2 T + · · · + ( k − T k − ). q ( T ) = α (1 + T + · · · + T k − ).Since α = 0, α = − m ( c ) and0 = p ( T ) = ( λ c +1 − ε − λ c +11 + m ( c ))+( λ c +1 − ε − λ c +12 +2 m ( c )) T + · · · +( λ c +1 − εk − − λ c +1 k − +( k − m ( c )) T k − we get for j = 1 , . . . , k − jm ( c ) = λ c +1 j − λ c +1 − ε = jλ ( A/ J ( a c +1 )) − λ (Tor A ( A/ a j , A/ J ( a c +1 ))) − jλ ( A/ J ( a c +1 − ε )) + λ (Tor A ( A/ a j , A/ J ( a c +1 − ε )))= jm ( c + 1) + λ (Tor A ( A/ a j , A/ J ( a c +1 − ε ))) − λ (Tor A ( A/ a j , A/ J ( a c +1 )))Therefore jρ c = λ (Tor A ( A/ a j , A/ J ( a c +1 ))) − λ (Tor A ( A/ a j , A/ J ( a c +1 − ε )))The same formula also holds for j ≥ k since we have λ c +1 − εj − λ c +1 j = α j = − m ( c ) j. (cid:3) References [AADG17] Maria Alberich-Carrami˜nana, Josep `Alvarez Montaner, Ferran Dachs-Cadefau, andV´ıctor Gonz´alez-Alonso. Poincar´e series of multiplier ideals in two-dimensional localrings with rational singularities.
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