MULTIPLIER IDEALS OF SUMS VIA CELLULAR RESOLUTIONS

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1 MULTIPLIER IDEALS OF SUMS VIA CELLULAR RESOLUTIONS SHIN-YAO JOW AND EZRA MILLER Abstract. Fix nonzero ideal sheaves a 1,..., a r and b on a noral Q-Gorenstein colex variety X. For any ositive real nubers α and β, we construct a resolution of the ultilier ideal J ((a a r ) α b β ) by sheaves that are direct sus of ultilier ideals J (a λ1 1 aλr r The resolution is cellular, in the sense that its boundary as are encoded by the algebraic chain colex of a regular CW-colex. The CW-colex is naturally exressed as a triangulation of the silex of nonnegative real vectors λ R r bβ ) for various λ R r 0 satisfying r λ i = α. with r λ i = α. The acyclicity of our resolution reduces to that of a cellular free resolution, suorted on, of a related onoial ideal. Our resolution ilies the ultilier ideal su forula J (X, (a a r ) α b β ) = λ 1+ +λ r=α J (X, a λ1 1 aλr r bβ ), generalizing Takagi s forula for two suands [Tak05], and recovering Howald s ultilier ideal forula for onoial ideals [How01] as a secial case. Our resolution also yields a new exactness roof for the Skoda colex [Laz04, Section 9.6.C]. Introduction Let X be a sooth colex algebraic variety and let a O X be an ideal sheaf. Alications in algebraic geoetry of the ultilier ideal sheaves J (a α ) = J (X, a α ) O X for real nubers α > 0 have led to investigations of their behavior with resect to natural algebraic oerations. For exale, Deailly, Ein, and Lazarsfeld [DEL00] roved that given two ideal sheaves a 1 and a 2, one has J ((a 1 a 2 ) α ) J (a α 1 )J (aα 2 ). For the subtler case of sus, on the other hand, Mustaţǎ [Mus02] showed that J ((a 1 + a 2 ) α ) J (a1 α t )J (a t 2 ), 0 t α and Takagi [Tak05] later refined this to (1) J ((a 1 + a 2 ) α ) = 0 t α J (a α t 1 a t 2), Date: 4 Seteber Key words and hrases. Multilier ideal, suation forula, cellular resolution, onoial ideal, hoology-anifold-with-boundary. 1

2 2 SHIN-YAO JOW AND EZRA MILLER where he roved it ore generally when X is noral and Q-Gorenstein. Takagi used characteristic ethods to deduce (1), which akes his work distinctly different fro [DEL00] and [Mus02], where the arguents roceed by geoetric techniques such as log resolutions and sheaf cohoology. Our urose is to show that such geoetric techniques, cobined with cobinatorial ethods fro toology and coutative algebra, can recover Takagi s equality (1) and generalize it. As a consequence, we derive a new roof (Corollary 3) of Howald s forula for ultilier ideals of onoial ideals [How01], and deonstrate how it can be reforulated to hold for all ideals. In addition, we obtain a new cellular exactness roof (Corollary 4.4) for the Skoda colex [Laz04, Section 9.6.C] via the contractibility of silices. Our ain results center around the following, which constitutes art of Theore 3.6. Theore 1. Fix nonzero ideal sheaves a 1,..., a r, b on a noral Q-Gorenstein colex variety X and α, β > 0. There is a resolution 0 J r 1 J 0 0 of the ultilier ideal J ((a a r ) α b β ) by sheaves J i that are finite direct sus of ultilier ideals of the for J (a λ 1 r bβ ) for various nonnegative λ R r with r λ i = α. Every distinct ideal sheaf of that for aears as a suand of J 0. Part of the final clai of Theore 1 is that there are only finitely any distinct ultilier ideals of the for J (X, a λ 1 r bβ ) for λ λ r = α. (This fact alone is not very hard to see fro the definition of ultilier ideals.) In articular, the surjection J 0 J (X, (a a r ) α b β ) in our resolution ilies the following (finite) suation forula; see also Section 4 for refineents. Corollary 2. J (X, (a a r ) α b β ) = λ 1 + +λ r=α J (X, a λ 1 r b β ). Corollary 2 reduces the calculation of the ultilier ideals of arbitrary olynoial ideals to those of rincial ideals. In the secial case of a onoial ideal a = x γ 1,..., x γr, generated by the onoials in the olynoial ring C[x 1,..., x d ] with exonent vectors γ 1,..., γ r N d, the suation forula becoes articularly exlicit. For a subset Γ R d, let conv Γ denote its convex hull. By the integer art of a vector ν = (ν 1,..., ν d ) R d, we ean the vector ( ν 1,..., ν d ) Z d whose entries are the greatest integers less than or equal to the coordinates of ν. Corollary 3. If a = x γ 1,..., x γr is a onoial ideal in C[x 1,..., x d ], then J (a α ) is generated by the onoials in C[x 1,..., x d ] whose exonent vectors are the integer arts of the vectors in conv{α γ 1,..., α γ r } R d. Proof. Using Corollary 2 with a j = x γ j, it suffices to note that the divisor of a single onoial has sile noral crossings, so no log resolution is necessary. It is easy to check that for α = 1, the vectors in the conclusion of Corollary 3 are recisely those fro Howald s result [How01], naely the vectors γ N d such that γ + (1,..., 1) lies in the interior of the convex hull of all exonents of onoials in a.

3 MULTIPLIER IDEALS OF SUMS VIA CELLULAR RESOLUTIONS 3 Our aroach to Theore 1 is to construct a secific resolution satisfying the hyotheses, including the art about J 0. The resolution we construct is cellular, in a sense generalizing the anner in which resolutions of onoial ideals can be cellular [BS98]; see [MS05, Chater 4] for an introduction. In general, a colex in any abelian category could be called cellular if each hoological degree is a direct su indexed by the faces of a CW-colex, and the boundary as are deterined in a natural way fro those of the CW-colex. An eleentary way to hrase this in the resent context is as follows. (Theore 3.6 is ore recise: a secific triangulation is constructed in Section 2, and the sheaves J σ are secified in Reark 3.3.) Theore 4. Resue the notation fro Theore 1. There is a triangulation of the silex {λ R r r λ i = α and λ 0} such that J i = σ i J σ can be taken to be a direct su indexed by the set i of i-diensional faces σ, and the differential of J. is induced by natural as between ideal sheaves, using the signs fro the boundary as of. If λ 0 is a vertex, then J λ = J (X, a λ 1 r b β ). Coaring the final sentences of Theores 1 and 4, a key oint is that every ossible ultilier ideal of the for J (X, a λ 1 r b β ) occurs at soe vertex λ 0. Writing down what it eans for two such ultilier ideals to coincide, this stiulation rovides strong hints as to otential choices for triangulations; see Section 2. The roof of exactness for the cellular resolution in Theore 4 roceeds by lifting the roble to an aroriate log resolution X X; see the roof of Theore 3.6. Over X, we resolve the lifted ideal sheaf by a colex (of locally rincial ideal sheaves in O X ) that is, analytically locally at each oint of X, a cellular free colex over a olynoial ring. This cellular free colex turns out to be a cellular free resolution of an aroriate onoial ideal; its construction and roof of acyclicity occuy Section 2, articularly Proosition 2.2. Having cellularly resolved the lifted sheaf over X, the desired cellular resolution over O X is obtained by ushing forward to X and using local vanishing; again, see the roof of Theore 3.6. Thus Theores 1 and 4 constitute a certain global version of cellular free resolutions of onoial ideals. The acyclicity of the cellular free resolutions in Proosition 2.2 reduce to a silicial hoology vanishing stateent, Corollary 1.7, for silicial colexes obtained by deleting boundary faces fro certain contractible anifolds-with-boundary. We deduce Corollary 1.7 fro the following ore general stateent, which is of indeendent interest. Its ried hyothesis is satisfied by the barycentric subdivision of any olyhedral hoology-anifold-with-boundary. In what follows, to delete a silex σ fro a silicial colex M eans to reove fro M every silex containing σ.

4 4 SHIN-YAO JOW AND EZRA MILLER Proosition 5. Fix a silicial colex M whose geoetric realization M is a hoology-anifold with boundary M. Assue that M is ried, eaning that (2) if σ is a face of M, then σ M is a face of M. Then deleting any collection of boundary silices fro M results in a silicial subcolex whose (reduced) hoology is canonically isoorhic to that of M. Acknowledgeents. We are grateful to Rob Lazarsfeld for valuable discussions, and to Shunsuke Takagi for the suggestion to work with singular varieties. EM was suorted by NSF CAREER grant DMS and a University of Minnesota McKnight Land-Grant Professorshi. The latter funded his visit to the University of Michigan, which he wishes to thank for its hositality while this work was coleted. 1. Cobinatorial toological reliinaries In this section we collect soe eleentary results fro silicial toology. For additional background, see [Mun84], esecially 63 for hoology-anifolds, and [Zie95, Section 5.1] for olyhedral colexes. The reader interested solely in the construction of resolutions for Theores 1 and 4, as oosed to their acyclicity roofs, can roceed to Section 2 after Exale 1.1. The goal for the reainder of this section is the roof of Proosition 5 and its iediate consequence, Corollary 1.7. Our convention is to identify any abstract silicial colex with the underlying toological sace of any geoetric realization. Thus, fixing a vertex set V, we view as a collection of silices (finite subsets of V ), called faces of, such that any subset of any face of is a face of. For exale, we can exress the link in of a face σ as the subcolex link (σ) = {τ τ σ and τ σ = }. In what follows, all links are taken inside of the abient silicial colex, unless otherwise noted. Our otivation is the following class of silicial colexes. Exale 1.1. Let P be a olyhedral colex, such as a olyhedral subdivision of a olytoe. (This exale works just as well for any regular cell colex; see [Bjö84]). The barycentric subdivision of P is the silicial colex whose vertex set is the set of faces of P, and whose silices are the chains P 1 < < P l of faces P j P. Here P < Q eans that P is a roer face of Q. The barycentric subdivision is hoeoorhic to (the sace underlying) P; one way to realize is to let each face P 1 < < P l be the convex hull of the barycenters of the faces P 1,..., P l. The ain idea in the roof of Proosition 5 is to delete the boundary silices one by one, in a suitable order, so that at each ste the hoology reains unchanged, using the following.

5 MULTIPLIER IDEALS OF SUMS VIA CELLULAR RESOLUTIONS 5 Lea 1.2. Fix a silicial colex and a chain of silicial subcolexes = 0 1 r = Γ. If the relative chain colex of each air ( i 1, i ) has vanishing hoology for all i = 1,..., r, then the hoology of Γ is canonically isoorhic to that of. Proof. Reeatedly aly the long exact sequence of hoology. Throughout the rest of this section, we use M to denote a silicial hoologyanifold-with-boundary of diension di(m) = d, defined as follows. Definition 1.3. A silicial hoology-anifold-with-boundary of diension d is a silicial colex whose axial faces all have diension d, and such that the link of each i-face has the hoology of either a ball or a shere of diension d i 1. Reark 1.4. If the olyhedral colex P in Exale 1.1 subdivides a hoologyanifold-with-boundary, then the conditions of Proosition 5, including the ried condition (2), are satisfied by the barycentric subdivision of P. Lea 1.5. The link of any i-face in M is a diension d i 1 silicial hoologyanifold-with-boundary. Proof. If σ M is a face of diension i, then Γ = link M (σ) is ure of diension d i 1 because M is ure of diension d. The condition on the hoology of the link in Γ of a face τ Γ holds sily because the condition holds for link M (σ τ), which equals link Γ (τ) by definition. The above lea required no secial hyotheses on M. Our final observation before the roof of Proosition 5 is that the condition (2) of being ried is inherited by links of boundary faces. Lea 1.6. Under the hyotheses of Proosition 5, the link of any boundary silex of M satisfies all of the hyotheses on M in Proosition 5, including being ried. Proof. Let σ be a boundary silex of M. In view of Lea 1.5, we need only show that link(σ) satisfies (2). This condition is a consequence of the equality (3) (link(σ)) = M link(σ), because for any silex τ in link(σ), the intersection τ (link(σ)) is forced to equal τ M, which is a face of τ by (2). To rove (3), we first show that if τ (link(σ)) then τ M; note that for this ilication, condition (2) is not necessary. Relacing τ by a face of (link(σ)) containing it, if necessary, we ay assue that di(σ τ) = d 1; equivalently, τ is a axial face of (link(σ)). In this case, there is only one axial face of link(σ) containing τ. But the faces of link(σ) containing τ are in bijection with the faces of M containing σ τ. Hence there is only one axial face of M containing σ τ.

6 6 SHIN-YAO JOW AND EZRA MILLER Therefore σ τ ust be a boundary face of M. We conclude that τ M, because τ is a face of σ τ. For the reverse containent, suose now that τ M link(σ), and let τ = τ σ M. The intersection τ M is a face of M by condition (2), and it contains all of the vertices of σ and τ, because both σ and τ are boundary faces of M. The only face of τ containing all of the vertices of σ and τ is τ itself; hence τ = τ M is a boundary face of M. It follows that link(τ ) has the hoology of a ball (rather than a shere). But link(τ ) = link M (σ τ) = link link(σ) (τ), so τ (link(σ)). Proof of Proosition 5. Use induction on the diension of M. For di M = 1 the stateent is an eleentary clai concerning subdivided intervals, so assue that di M = d 2. Let S be the collection of boundary faces to be deleted. Without loss of generality, we ay assue that S is a cocolex inside M, which eans that if σ S and τ M such that τ σ, then τ S. Totally order the silices in S in such a way that all of the silices of axial diension d 1 coe first, then those of diension d 2, and so on. Let σ 1, σ 2,... be this totally ordered sequence of silices. Let M i be the result of deleting σ 1,..., σ i fro M, with M = M 0. The desired result will follow fro Lea 1.2, with M =, as soon as we show that the relative chain colexes C.(M i 1, M i ) are all acyclic. Let Γ be the silicial colex obtained by deleting the boundary fro link M (σ i ), which is acyclic by induction, via Lea 1.6. We clai that C.(M i 1, M i ) is (noncanonically) isoorhic to the reduced chain colex C.(Γ). Indeed, recall the bijective corresondence between the faces of link M (σ i ) and the faces of M containing σ i. Under this corresondence, Γ is aed bijectively to the set {τ M τ M = σ i } = {τ M i 1 τ σ i }. The faces of this cocolex inside M i 1 constitute a free basis of C.(M i 1, M i ), and this induces an isoorhis fro C.(Γ) to C.(M i 1, M i ). For future reference, here is the secial case of Proosition 5 that we aly later. Corollary 1.7. If M is a contractible ried silicial anifold-with-boundary, then deleting any collection of boundary faces fro M results in a silicial subcolex with vanishing reduced hoology. Reark 1.8. It would be referable to conclude in Corollary 1.7 that the subcolex is contractible, and ore generally that the subcolex in Proosition 5 is hootoyequivalent to M, using [Bjö84, Lea 10.3(i)]: if i = i 1 Γ i is a union of two silicial subcolexes such that Γ i and i 1 Γ i are both contractible, then i is hootoy-equivalent to i 1. The notation here is consistent with Lea 1.2; in our case i 1 is the result of deleting a boundary silex σ i fro i, and Γ i is the cone fro σ i over i 1 Γ i (thus Γ i is the closed star of σ i in i ). The roble is that, while Γ i is always contractible (it is a cone), the subcolex i 1 Γ i = link i (σ i ) need not be sily-connected, even though it has vanishing hoology.

7 MULTIPLIER IDEALS OF SUMS VIA CELLULAR RESOLUTIONS 7 2. A cellular free resolution Let A Z n r be an integer atrix with n rows and r coluns, and fix a real colun vector b R n with coordinates b 1,..., b n. (When alying these results in Section 3, all entries in the atrix A will be nonnegative.) Viewing the rows A 1,..., A n as functionals on R r, we get an (infinite but locally finite) affine hyerlane arrangeent A = {λ R r A j λ + b j = z j }. z Z n 1 j n The arrangeent A induces a olyhedral subdivision of R r. Fixing a nonnegative real nuber α R 0, the restriction of this olyhedral subdivision to the silex α = {(λ 1,..., λ r ) R r 0 λ λ r = α} is a olyhedral subdivision A α of α. Two oints λ and µ lie in the sae (relatively oen) cell of A α if and only if for each j = 1,..., n, there is an integer z j such that either A j λ + b j = A j µ + b j = z j, or else A j λ + b j and A j µ + b j both lie in the oen interval (z j, z j + 1). For the duration of this aer, fix any olyhedral subdivision P of α that refines A α, eaning that every face of P is contained in a face of A α. As in Exale 1.1, denote by the barycentric subdivision of P, realized so that the vertices of are the barycenters of the cells of P. We write σ(p ) for the vertex of that is the barycenter of the olytoe P P. Construct a odule M C[x ±1 1,..., x ±1 n ] generated by (Laurent) onoials over the olynoial ring C[x 1,..., x n ], so M is a Laurent onoial odule [MS05, Definition 9.6], as follows. For each olytoe P P, set P = x A1 λ+b 1 1 x An λ+b n n for any λ P, where z is the greatest integer less than or equal to z, and the cell P is the relative interior of P ; the onoial P is well-defined because the greatest integer arts defining it are equal for all vectors in the sae cell of P. Now define M = P P P. We can label each silex in with a onoial as follows. By virtue of the fact that each olytoe in P indexes a onoial P in the Laurent olynoial ring, the vertices of the silicial colex, which are the barycenters σ(p ) of the olytoes P P, are labeled with onoials. For each such vertex σ(p ), let us set σ(p ) = P. More generally, for each silex σ, if σ = (P 1 < < P l ) then set σ = P1 ; i.e., we label σ by the onoial of the sallest face in the chain. For (Laurent) onoials and, we say that divides if / is a onoial in C[x 1,..., x n ].

8 8 SHIN-YAO JOW AND EZRA MILLER Lea 2.1. If P < Q are olytoes in P, then Q divides P. Consequently, if σ is a silex, then σ is the least coon ultile of { τ τ is a vertex of σ}. Proof. We rove only the first sentence, as the second follows easily. Fro the ersective of the greatest integer arts used in constructing P and Q, the only difference between the barycenters of P and Q is that ore of the affine functions λ A j λ+b j can take integer values on σ(p ) than on σ(q). In articular, if A j σ(q) + b j = z j, then A j σ(p ) + b j equals either z j or z j + 1. This being true for all j, we conclude that Q divides P (in fact, P / Q is a squarefree onoial in C[x 1,..., x n ]). Lea 2.1 is recisely the condition under which the onoial labels σ for σ define a cellular free colex on [BS98, Section 1] (see also Chaters 4 and 9 in [MS05] for background). Briefly, this is the colex 0 σ σ σ 0 σ r 1 σ i σ 0 of free C[x 1,..., x n ]-odules where the orhis σ τ of rincial Laurent onoial odules is, for each codiension 1 face τ σ, the natural inclusion ties ±1. The sign is deterined by arbitrary but fixed orientations for the silices in. Proosition 2.2. The cellular free colex suorted on, in which σ is labeled by σ, is a cellular free resolution of the Laurent onoial odule M. Proof. By [BS98, Proosition 1.2], we need to show that the silicial subcolex = {σ σ divides } is acyclic for all Laurent onoials. Let = x e 1 1 xen n. For σ = (P 1 < < P l ), the onoial σ divides if and only if the barycenter τ = σ(p 1 ) satisfies the conditions A j τ + b j < e j + 1 for all j. Coare the subcolex Γ consisting of those oints λ such that A j λ + b j e j + 1 for all j. This subcolex is a (coact convex) olytoe, and hence Γ is a contractible silicial anifold-with-boundary. But is obtained fro Γ by deleting those vertices τ, necessarily on the boundary of Γ, having A j τ + b j = e j + 1 for soe j. The desired result is therefore a consequence of Corollary 1.7, which alies by Reark Log resolution to cellular resolution For the duration of this section, fix a noral Q-Gorenstein colex variety X and ideal sheaves a 1,..., a r, b on X. In addition, fix a log resolution π : X X of b and a i +a j for all i, j {1,..., r}. By definition, this eans that b O X and (a i +a j ) O X are locally rincial ideal sheaves on X. The secial case a i = a j ilies that a i O X = O X ( A i ) for i {1,..., r} and b O X = O X ( B)

9 MULTIPLIER IDEALS OF SUMS VIA CELLULAR RESOLUTIONS 9 are the ideal sheaves associated to effective divisors A i and B on X. We wish to calculate the ultilier ideals J ((a 1 + +a r ) α b β ) for nonnegative real nubers α and β. This requires a log resolution of a = a a r. Fortunately, the orhis π : X X is one, as can be seen by alying the following locally on X. Lea 3.1. Let R be a local integral doain. 1. If f 1,..., f n is a nonzero rincial ideal of R, then there is an index i {1,..., n} such that f i divides f j for all j {1,..., n}. 2. If f 1,..., f r R are eleents such that f i, f j is a rincial ideal for all i, j {1,..., r}, then there exists a erutation ξ S r such that f ξ(1) f ξ(2) f ξ(r). In articular f 1,..., f r = f ξ(1) is a rincial ideal. Proof. Suose f 1,..., f n is generated by f R. Then each f j = f j f is a ultile of f. On the other hand, f = c j f j = f c j f j with c j R. Since R is a doain, this ilies that 1 = c j f j. Since R is local, we conclude that soe f i is a unit. This roves the first clai. The second is an iediate consequence of the first. Siilar to a i and b fro before, the ideal sheaf a O X = O X ( C) is deterined by an effective divisor C on X. Let D 1,..., D n be the set of rie divisors aearing in A 1,..., A r, B, C, and the relative canonical divisor K X /X. Thus n A j = a ij D i for j {1,..., r}, βb K X /X = C = n b i D i, and n γ i D i in ters of D 1,..., D n, where the coefficients a ij and γ i are integers, but b i can be real nubers. Recall that the ultilier ideal of a with weighting coefficient α is J (a α ) = π O X ( αc K X /X ) O X, where n δ id i = n δ i D i is the greatest integer divisor less than or equal to n δ id i. More generally, for λ R r 0 and β 0, set J (a λ 1 1 aλr r bβ ) = π O X ( λ 1 A λ r A r + βb K X /X ) O X. How do these ultilier ideals deend on the choices of λ R r and β 0? Given the decoositions into rie divisors D 1,..., D n with integer coefficient atrix A = (a ij ) and real vector b = (b i ), we are verbati in the situation fro Section 2, whose notation we assue henceforth. Lea 3.2. For each olytoe P P, there is a single ideal sheaf J P O X that coincides with the ultilier ideals J (a λ 1 r bβ ) for all λ P in the interior of P.

10 10 SHIN-YAO JOW AND EZRA MILLER Proof. In fact, the greatest integer divisors λ 1 A λ r A r + βb K X /X on X all coincide for λ P, by construction. Therefore we can define I P = O X ( λ 1 A λ r A r + βb K X /X ) for any λ P, and J P = π I P does not deend on λ P. As we did for onoials in Section 2 (before Lea 2.1), having defined objects I P and J P indexed by olytoes in P, we can define objects I σ and J σ indexed by silices in : if σ = (P 1 < < P l ) then set I σ = I P1, and set J σ = π I σ. Reark 3.3. Unwinding the definitions, we find that J σ = J (a λ 1 r b β ), where λ is the barycenter of the sallest olytoe P 1 in the chain P 1 < < P l corresonding to the silex σ. (Recall that is the barycentric subdivision of the olyhedral colex P fro the beginning of Section 2.) The analogue of Lea 2.1 holds here, as well. Lea 3.4. If P < Q are olytoes in P, then I Q I P. Consequently, if σ, then I σ = {I τ τ is a vertex of σ}. The sae is true with J in lace of I. Proof. The arguent is essentially the sae as the roof of Lea 2.1. Lea 3.5. Resue the notation involving a = a a r fro after Lea 3.1, so a O X = O X ( C). Then, as ideal sheaves over X, O X ( αc + βb K X /X ) = P P I P. Proof. Let X be an arbitrary oint. By Lea 3.1, there is an oen neighborhood U of and a erutation ξ S r such that after restricting to U, we have A ξ(1) A ξ(r), where E F for divisors E and F eans that F E is effective. It follows that on U we have C = A ξ(1), and also αa ξ(1) λ 1 A λ r A r whenever λ λ r = α. Hence both sides of the desired equality are equal to O X ( αa ξ(1) + βb K X /X ) on U. Equality holds on X because is arbitrary. We have arrived at our ain result. For notational uroses, choose an arbitrary but fixed collection of orientations for the silices in. There results an incidence function that assigns to each codiension 1 face τ of each silex σ a sign ( 1) σ,τ.

11 MULTIPLIER IDEALS OF SUMS VIA CELLULAR RESOLUTIONS 11 Theore 3.6. With J σ as in Reark 3.3, there is an exact sequence 0 σ r 1 J σ σ i J σ σ 0 J σ J ((a a r ) α b β ) 0 of sheaves on X, in which the orhis J σ J τ is inclusion ties ( 1) σ,τ. Here, i = {σ di(σ) = i} is the set of i-diensional silices in. Proof. We shall first verify the exactness of the colex 0 σ r 1 I σ σ i I σ σ 0 I σ σ 0 I σ 0 on X, where it should be noted that the su at the far right is not a direct su. It is enough to verify exactness at the stalk of an arbitrary oint X. Reordering the rie divisors D 1,..., D n aearing in A 1,..., A r, B, and K X /X if necessary, we assue that only D 1,..., D k ass through. Since D 1,..., D n intersect transversally, if we ass to an analytic neighborhood or coletion at, then we can choose local coordinates x 1,..., x k so that D i is given by the vanishing of x i for i = 1,..., k. In these coordinates, I σ becoes the rincial Laurent onoial odule in C[x ±1 1,..., x ±1 generated by the onoial σ fro before Lea 2.1, excet that all of the variables x k+1,..., x n have been set equal to 1 in σ. The exactness thus follows fro Proosition 2.2, given that our olyhedral colex P refines the subdivision induced by the linear functionals A j λ + b j for j = 1,..., k, instead of j = 1,..., n. (Another reason the exactness follows fro Proosition 2.2 is that setting the variables equal to 1 is the exact oeration of hoogeneous localization ; see [Mil00, Section 3.6], articularly Proosition there.) Pushing forward under π coletes the roof. Indeed, Lea 3.5 ilies that I σ = I P = O X ( αc + βb K X /X ) σ 0 P P ushes forward to yield J ((a a r ) α b β ), while local vanishing [Laz04, Theore (i)] guarantees that the higher direct iages vanish on all I σ. Reark 3.7. In the secial case where X = Sec C[x 1,..., x n ] and a 1,..., a r, b are all rincial onoial ideals, all of the J σ are rincial onoial ideals as well, so the exact sequence in Theore 3.6 becoes a cellular resolution of the onoial ideal J ((a a r ) α b β ) in the sense of [MS05, Definition 4.3]. Exale 3.8. Let us illustrate Theore 3.6 by an exale. Take X = Sec C[x, y], r = α = 2, a 1 = xy, a 2 = x + y, and b = O X. A direct coutation shows that k ]

12 12 SHIN-YAO JOW AND EZRA MILLER on the doubled 1-silex {(λ 1, λ 2 ) R 0 λ 1 + λ 2 = 2}, we have x 2 y 2 if λ 2 = 0 xy if 0 < λ 2 < 1 J (a λ 1 1 aλ 2 2 ) = xy(x + y) if λ 2 = 1 x + y if 1 < λ 2 < 2 (x + y) 2 if λ 2 = 2. These regions where J (a λ 1 1 aλ 2 2 ) stays constant yield a olyhedral subdivision P of the doubled 1-silex, and we let be the barycentric subdivision of P. Exlicitly, consists of five vertices and four edges v 1 = (2, 0), v 2 = ( 3 2, 1 2 ), v 3 = (1, 1), v 4 = ( 1 2, 3 2 ), v 5 = (0, 2) l i = the line segent between v i and v i+1, for i = 1, 2, 3, 4. The vertices in are naturally labeled with the ultilier ideals J v1 = J (a 2 1 a0 2 ) = x2 y 2, J v2 = J (a 3/2 1 a 1/2 2 ) = xy, J v3 = J (a 1 1a 1 2) = xy(x + y), J v4 = J (a 1/2 1 a 3/2 2 ) = x + y, J v5 = J (a 0 1 a2 2 ) = (x + y)2. The edges of are labeled by J li, which for i = 1, 2, 3, 4 is defined to be the least coon ultile of J vi and J vi+1. It is easily verified that J li equals the saller one of J vi and J vi+1. The exact sequence of Theore 3.6 is or ore exlicitly, J li x 2 y 2 xy(x + y) xy(x + y) (x + y) 2 5 J vi J ((a 1 + a 2 ) 2 ) 0, x 2 y 2 xy xy(x + y) x + y (x + y) 2 J ( xy, x + y 2 ) 0. After canceling redundant ters, this is essentially a Koszul resolution 0 xy(x + y) xy x + y J ( xy, x + y 2 ) 0.

13 MULTIPLIER IDEALS OF SUMS VIA CELLULAR RESOLUTIONS Alications The Introduction already contains soe iediate alications of Theore 3.6, naely the Takagi-style suation forula in Corollary 2 and the derivation of Howald s onoial ultilier ideal forula in Corollary 3. In this section, we collect further alications: a new exactness roof for the Skoda colex, and additional suation forulas for ultilier ideals, including the graded syste case. We begin with the suation forulas. First, we note the following. Reark 4.1. Theore 3.6 still holds when b β is relaced by b β 1 1 b βs s, since the contants β k erely translate the affine hyerlane arrangeent A in Section 2. Corollary 4.2. Let a 1,..., a r, b 1,..., b s O X be nonzero ideal sheaves on a noral Q-Gorenstein variety X and let α, β 1,..., β s be ositive real nubers. Then J (X, (a a r ) α b β 1 1 b βs s ) = J (X, a λ 1 r b β 1 1 b βs s ). λ 1 + +λ r=α Proof. This follows iediately fro the surjectivity on the right of the exact sequence in Theore 3.6. Our next result concerns the notion of graded syste of ideals. For the definition of this and the ultilier ideal associated to it, see [Laz04, Section 11.1.B]. A secial case has already aeared in [Tak05, Proosition 4.10]. Corollary 4.3. Corollary 4.2 still holds when the ideal sheaves a i and b j are all relaced by graded systes of ideals. Proof. The forula for r > 2 can be obtained by reeatedly alying the forula for r = 2, so it suffices to rove this case, i.e., J (X, (a. + b.) α c β 1 1,. c βs s,.) = 0 t α J (X, a. α t b ṭ c β 1 1,. c βs s,.). First let us verify, i.e., J (X, a. α t b ṭ c β 1 1,. c βs s,.) J (X, (a.+b.) α c β 1 1,. c βs s,.) for any fixed t [0, α]. By definition, the left-hand side is equal to J (X, a α t for soe large, and we see that J (X, a α t b t β 1 c β 1 b t βs c1, c βs 1, c s,) J (X, (a + b ) α β1 βs c 1, c s,) (by Corollary 4.2) J (X, (a. + b.) α β 1 βs c 1, c J (X, (a. + b.) α c β 1 1,. c βs s,.) (by definition). To rove the reverse inclusion, first by definition there exists soe large such that J (X, (a. + b.) α c β 1 1,. c βs s,.) = J (X, ( a i b i ) α β 1 βs c 1, c s,), i=0 s,) s,)

14 14 SHIN-YAO JOW AND EZRA MILLER and by Corollary 4.2 this right-hand side can be exressed as J (X, ( a λ i b λ β1 i i ) c 1, c λ 0 + +λ = α 1 i )( i=0 βs s,). Now take a ositive integer which is divisible by 1, 2,...,. Since a., b. and c j,. are graded systes, we have so a λ i i = a i ( i iλ i a iλi, 1 a λ i i )( b λ i i=0 b λ i i = b ( i)λ i i i β1 βs i ) c 1, c s, a P b ( i)λi iλ i β j, cj, = c 1 P ( i)λ i j, β j i=0 b c β 1 1, c βs β j cj,, and since λ λ = α 1, if we let t := ( i)λ i, then iλ i = α t, hence J (X, ( 1 a λ i i )( b λ i i=0 i=0 s,, β1 βs i ) c 1, c s,) J (X, a α t b t c β 1 1, c βs s,) J (X, a. α t b ṭ c β 1 1,. c βs s,.). All of the corollaries of Theore 3.6 we have given so far are Takagi-style suation forulas which ay also be obtained by straightforwardly generalizing the roof of Takagi s original forula (1) (see [Tak05, Theore 3.2]). However, in our final alication we want to deonstrate that Theore 3.6 ay be otentially ore useful than (1) by deriving the exactness of the Skoda colex fro the forer. For this we assue that the ideal sheaves a 1,..., a r are locally rincial on the Q-Gorenstein variety X. For every nonnegative integer α < r, the inclusion a i O X induces a natural a a i J (a α b β ) J (a α+1 b β ). Writing a τ = i τ a i for τ {0, 1} r R r, the Skoda colex is the cellular colex 0 a 1 a r J (b β ) r a τ J (a α b β ) a i J (a r 1 b β ) J (a r b β ) 0 τ =r α suorted on a silex whose faces are in bijection with the vectors τ {0, 1} r. Corollary 4.4. The Skoda colex is exact. A roof for sooth X aears in [Laz04, Section 9.6.C]. Here, we rovide an alternate roof, which works with no extra effort in the Q-Gorenstein setting. Proof. Consider the subdivisions A α of the silices α for α = 0,..., r, defined at the beginning of Section 2. Every vector τ {0, 1} r R r induces an ebedding α r, where α = r τ, by adding τ. Choose a refineent P of A r so fine

15 MULTIPLIER IDEALS OF SUMS VIA CELLULAR RESOLUTIONS 15 that the subdivision induced on α under every one of these inclusions, for all τ and all α = 0,..., r, refines A α. For exale, in the notation fro the beginning of Section 2, let P be the subdivision induced by the affine hyerlane arrangeent A + {0, 1} r = {λ + τ R r A j λ + b j = z j } z Z n τ {0,1} r 1 j n obtained by translating the arrangeent A u by every vector τ {0, 1} r. As in Theore 3.6, let be the barycentric subdivision of P. Recall the definition of J σ = J (a λ 1 r b β ) fro Reark 3.3, and write λ(σ) = λ = (λ 1,..., λ r ). Define a double colex J.,. in which J,q = J σ, di(σ)=q τ = τ λ(σ) where τ λ(σ) eans that λ(σ) τ r τ (equivalently, λ(σ) τ has nonnegative entries). Each horizontal colex J.,q is a direct su, over σ with di(σ) = q, of colexes J σ C C.(Γτ ), where τ λ(σ) is axial and C.(Γ τ ) is the reduced chain colex of a silex Γ τ with τ any vertices. The vertical differentials of J.,. are induced by the differentials of Theore 3.6, for each fixed τ. In articular, the vertical colex on the suands J σ indexed by a fixed τ λ = λ(σ) is a resolution of a τ J (a r τ b β ) by Theore 3.6, because J (a λ 1 r b β ) = a τ J (a λ 1 τ 1 τr r b β ). It follows that the vertical hoology of J.,. is the Skoda colex. On the other hand, the horizontal hoology of J.,. is identically zero because every silex Γ τ is contractible. The standard sectral sequence arguent shows the desired result. References [BS98] Dave Bayer and Bernd Sturfels, Cellular resolutions of onoial odules, J. Reine Angew. Math. 502 (1998), [Bjö84] Anders Björner, Toological ethods, Handbook of cobinatorics, Vol. 1, 2, , Elsevier, Asterda, [DEL00] Jean-Pierre Deailly, Lawrence Ein and Robert Lazarsfeld, A subadditivity roerty of ultilier ideals, Michigan Math. J. 48 (2000), [How01] Jason Howald, Multilier ideals of onoial ideals, Trans. Aer. Math. Soc. 353 (2001), no. 7, (electronic). [Laz04] Robert Lazarsfeld, Positivity in Algebraic Geoetry I II, Ergeb. Math. Grenzgeb., vols , Berlin: Sringer, [Mil00] Ezra Miller, The Alexander duality functors and local duality with onoial suort, J. Algebra 231 (2000), [MS05] Ezra Miller and Bernd Sturfels, Cobinatorial coutative algebra, Graduate Texts in Matheatics 227, New York: Sringer, [Mun84] Jaes R. Munkres, Eleents of algebraic toology, Addison Wesley, Menlo Park, CA, 1984.

16 16 SHIN-YAO JOW AND EZRA MILLER [Mus02] Mircea Mustaţǎ, The ultilier ideals of a su of ideals, Trans. Aer. Math. Soc. 354 (2002), no. 1, [Tak05] Shunsuke Takagi, Forulas for ultilier ideals on singular varieties, Aer. J. Math. 128 (2006), arxiv:ath.ag/ [Zie95] Günter Ziegler, Lectures on olytoes, Graduate Texts in Matheatics Vol. 152, Sringer Verlag, New York, 1995.