On the Regularity of Local Cohomology of Bigraded Algebras

The Hilbert functions and the regularity of the graded components of local cohomology of a bigraded algebra are considered. Explicit bounds for these invariants are obtained for bigraded hypersurface rings. Introduction In this paper we study algebraic properties of the graded components of local cohomology of a bigradedK-algebra. Let P0 be a Noetherian ring, P = P0[y1, . . . , yn] be the polynomial ring over P0 with the standard grading and P+ = (y1, . . . , yn) the irrelevant graded ideal of P . Then for any finitely generated graded P -module M , the local cohomology modules H i P+(M) are naturally graded P -module and each graded component H i P+(M)j is a finitely generated P0-module. In case P0 = K[x1, . . . , xm] is a polynomial ring, the K-algebra P is naturally bigraded with deg xi = (1, 0) and deg yi = (0, 1). In this situation, if M is a finitely generated bigraded P -module, then each of the modulesH i P+(M)j is a finitely generated graded P0-module. We are interested in the Hilbert functions and the Castelnuovo-Mumford regularity of these modules. In Section 1 we introduce the basic facts concerning graded and bigraded local cohomology and give a description of the local cohomology of a graded (bigraded) P -module from its graded (bigraded) P -resolution. In Section 2 we use a result of Gruson, Lazarsfeld and Peskine on the regularity of reduced curves, in order to show that the regularity of H i P+(M)j as a function in j is bounded provided that dimP0 M/P+M ≤ 1. The rest of the paper is devoted to study of the local cohomology of a hypersurface ring R = P/fP where f ∈ P is a bihomogeneous polynomial. In Section 3 we prove that the Hilbert function of the top local cohomology H P+(R)j is a nonincreasing function in j. If moreover, the ideal I(f) generated by all coefficients of f is m-primary where m is the graded maximal ideal of P0, then by a result of Katzman and Sharp the P0module H i P+ (R)j is of finite length. In particular, in this case the regularity of H i P+(R)j is also a nonincreasing function in j. In the following section we compute the regularity of H i P+(R)j for a special class of hypersurfaces. For the computation we use in an essential way a result of Stanley 1991 Mathematics Subject Classification. 13D45, 13D40, 13D02, 13P10. 1 and J. Watanabe. They showed that a monomial complete intersection has the strong Lefschetz property. Stanley used the hard Lefschetz theorem, while Watanabe representation theory of Lie algebras to prove this result. Using these facts the regularity and the Hilbert function of H i P+(P/f r λP )j can be computed explicitly. Here r ∈ N and fλ = ∑n i=1 λixiyi with λi ∈ K. As a consequence we are able to show that H P+ (P/f P )j has a linear resolution and its Betti numbers can be computed. We use these results in the last section to show that for any bigraded hypersurface ring R = P/fP for which I(f) is m-primary, the regularity of H i P+(R)j is linearly bounded in j. I would like to thank Professor Jürgen Herzog for many helpful comments and discussions. 1. Basic facts about graded and bigraded local cohomology Let P0 be a Noetherian ring, and let P = P0[y1, . . . , yn] be the polynomial ring over P0 in the variables y1, . . . , yn. We let Pj = ⊕ |b|=j P0y b where y = y1 1 . . . y bn n for b = (b1, . . . , bn), and where |b| = ∑ i bi. Then P be a standard graded P0-algebra and Pj is a free P0-module of rank ( n+j−1 n−1 ) . In most cases we assume that P0 is either a local ring with residue class field K, or P0 = K[x1, . . . , xm] is the polynomial ring over the field K in the variables x1, . . . , xm. We always assume that all P -modules considered here are finitely generated and graded. In case that P0 is a polynomial ring, then P itself is bigraded, if we assign to each xi the bidegree (1, 0) and to each yj the bidegree (0, 1). In this case we assume that all P -modules are even bigraded. Observe that if M is bigraded, and if we set Mj = ⊕ i M(i,j) Then M = ⊕ j Mj is a graded P -module and each graded component Mj is a finitely generated graded P0-module, with grading (Mj)i = M(i,j) for all i and j. Now let S = K[y1, . . . , yn]. Then P = P0 ⊗K K[y1, . . . , yn] = P0 ⊗K S. Let P+ := ⊕ j>0 Pj be the irrelevant graded ideal of the P0-algebra P . Next we want to compute the graded P -modules H i P+(P ). Observe that there are isomorphisms of graded R-modules H i P+(P ) ∼= lim −→k≥0 Ext i P (P/(P+) , P ) ∼= lim −→k≥0 Ext i P0⊗KS (P0 ⊗K S/(y), P0 ⊗K S) ∼= P0 ⊗K lim −→k≥0 Ext i P (S/(y) , S) ∼= P0 ⊗K H i (y)(S). 2 Since H i S+(S) = 0 for i 6= n, we get H i P+(P ) = { P0 ⊗k H (y)(S) for i = n, 0 for i 6= n. Let M be a graded S-module. We write M = HomK(M,K) and consider M ∨ a graded S-module as follows: for φ ∈ M and f ∈ S we let fφ be the element in M with fφ(m) = φ(fm) for all m ∈ M, and define the grading by setting (M)j := HomK(M−j , K) for all j ∈ Z. Let ωS be the canonical module of S. Note that ωS = S(−n), since S is a polynomial ring in n indeterminates. By the graded version of the local duality theorem, see [1, Example 13.4.6] we have H S+(S) ∨ = S(−n) and H i S+(S) = 0 for i 6= n. Applying again the functor ( ) we obtain H S+(S) = HomK(S(−n), K) = HomK(S,K)(n). We can thus conclude that H S+(S)j = Homk(S,K)n+j = HomK(S−n−j, K) for all j ∈ Z. Let Sl = ⊕ |a|=l Ky . Then HomK(S−n−j, K) = ⊕ |a|=−n−j Kz, where z ∈ HomK(S−n−j, K) is the K-linear map with z(y) = { z, if b ≤ a, 0, if b 6≤ a. Here we write b ≤ a if bi ≤ ai for i = 1, . . . , n. Therefore H S+(S)j = ⊕ |a|=−n−j Kz , and this implies that H P+(P )j = P0 ⊗K H n (y)(S)j = ⊕ |a|=−n−j P0z . (1) Hence we see that H P+(P )j is free P0-module of rank ( −j−1 n−1 ) . Moreover, if P0 is graded H P+(P )(i,j) = ⊕ |a|=−n−j (P0)iz a = ⊕ |a|=i |b|=−n−j Kxz. The next theorem describes how the local cohomology of a graded P -module can be computed from its graded free P -resolution Theorem 1.1. Let M be a finitely generated graded P -module. Let F be a graded free P -resolution of M . Then we have graded isomorphisms H P+ (M) ∼= Hi(H P+(F)). 3 Proof. Let F : · · · → F2 → F1 → F0 → 0. Applying the functor H P+ to F , we obtain the complex H P+(F) : · · · → H P+(F2) → H P+(F1) → H P+(F0) → 0. We see that H P+(M) = Coker(H n P+ (F1) → H P+(F0)) = H0(H P+(F)), since H i P+(N) = 0 for each i > n and all finitely generated P -modules N . We define the functors: F(M) := H P+(M) and Fi(M) := H P+ (M). The functors Fi are additive, covariant and strongly connected, i.e. for each short exact sequence 0 → U → V → W → 0 one has the long exact sequence 0 · · · → Fi(U) → Fi(V ) → Fi(W ) → Fi−1(U) → · · · → F0(V ) → F0(W ) → 0. Moreover, F0 = F and Fi(F ) = H P+ (F ) = 0 for all i > 0 and all free P -modules F . Therefore, the theorem follows from the dual version of [1, Theorem 1.3.5]. Note that if M is a finitely generated bigraded P -module. Then H P+(M) with natural grading is also a finitely generated bigraded P -module, and hence in Theorem 1.1 we have bigraded isomorphisms H P+ (M) ∼= Hi(H P+(F)). 2. regularity of the graded components of local cohomology for modules of small dimension Let P0 = K[x1, . . . , xm], and M be a finitely generated graded P0-module. By Hilbert’s syzygy theorem, M has a graded free resolution over P0 of the form 0 → Fk → · · · → F1 → F0 → M → 0, where Fi = ⊕ti j=1 P0(−aij) for some integers aij . Then the Castelnuovo-Mumford regularity reg(M) of M is the nonnegative integer regM ≤ max i,j {aij − i} with equality holding if the resolution is minimal. If M is an Artinian graded P0module, then reg(M) = max{j : Mj 6= 0}. We also use the following characterization of regularity reg(M) = min{μ : M≥μ has a linear resolution}. Let M be a finitely generated bigraded P -module, thus H i P+(M)j is a finitely generated graded P0-module. Let fi,M be the numerical function given by fi,M(j) = regH i P+ (M)j 4 for all j. In this section we show that fi,M is bounded provided that M/P+M has Krull dimension ≤ 1. There are some explicit examples which show that the condition dimP0 M/P+M ≤ 1 is indispensable. We postpone the example to Section 4. First one has the following Lemma 2.1. Let M be a finitely generated graded P -module. Then dimP0 Mi ≤ dimP0 M/P+M for all i. Proof. Let r = min{j : Mj 6= 0}. We prove the lemma by induction on i ≥ r. Let i = r. Note that M/P+M = Mr ⊕Mr+1/P1Mr ⊕ · · · . It follows thatMr is a direct summand of the P0-moduleM/P+M , so that dimP0 Mr ≤ dimP0 M/P+M . We now assume that i > r and dimP0 Mj ≤ dimP0 M/P+M , for j = r, . . . , i − 1. We will show that dimP0 Mi ≤ dimP0 M/P+M . We consider the exact sequence of P0-modules 0 → P1Mi−1 + · · ·+ Pi−rMr → Mi φ → (M/P+M)i → 0. By the induction hypothesis, one easily deduces that dimP0 i−r ∑ j=1 PjMi−j ≤ dimP0 M/P+M, and since (M/P+M)i is a direct summand of M/P+M it also has dimension ≤ dimP0 M/P+M . Therefore, by the above exact sequence, dimMi ≤ dimP0 M/P+M , too. The following lemma is needed for the proof next proposition. Lemma 2.2. Let M be a finitely generated graded P -module. Then there exists an integer i0 such that AnnP0 Mi = AnnP0 Mi+1 for all i ≥ i0 Proof. Since P1Mi ⊆ Mi+1 for all i and M is a finitely generated P -module, there exists an integer t such that P1Mi = Mi+1 for all i ≥ t. This implies that AnnP0 Mt ⊆ AnnP0 Mt+1 ⊆ . . . . Since P0 is noetherian, there exists an integer k such that AnnP0 Mt+k = AnnP0 Mi for all i ≥ t+ k = i0. Proposition 2.3. Let M be a finitely generated graded P -module. Then dimP0 H i P+ (M)j ≤ dimP0 Mj for all i and j ≫ 0. Proof. Let P+ = (y1, . . . , yn). Then by [1, Theorem 5.1.19] we have H i P+(M) ∼= H i(C(M).) for all i ≥ 0 where C(M). denote the (extended) Čech complex of M with respect to y1, . . . , yn defined as follows: C(M). : 0 → C(M) → C(M) → · · · → C(M) → 0 5