Regularity of ideals and their powers

Transcription

1 Regularity of ideals and their powers Marc Chardin Introduction If R is a polynomial ring over a field k, M is a finitely generated graded R-module (for the standard grading of R) and m := R >0, the invariants and a i (M) := inf{µ H i m(m) >µ = 0} b i (M) := inf{µ Tor R i (M, R/m) >µ = 0} both give a definition of the Castelnuovo-Mumford regularity of M (over R): reg(m) := max{a i (M) + i} = max{b i (M) i}. i i We will particularly focus in this article on the control of the invariants a i (M) for i close to the dimension of the support of M (recall the a i (M) = for i > dim M). If X P n k = Proj(R) is a non empty projective scheme, reg(x) is the regularity of the unique homogeneous quotient B of R with positive depth such that X = Proj(B), it is also easily seen that reg(x) = max i>0 {a i (B) + i} for any homogeneous quotient B of R such that X = Proj(B). The regularity of a projective scheme depends on its embedding. It may also be defined in terms of the ideal sheaf I X defining X in P n k ; one has reg(x) = reg(i X) 1, with the standard definition of regularity for sheaves (see e.g. [Ei, 20.20]). The motivations for this work came from several sides. One was the search of bounds for the regularity of powers of an ideal (or of an ideal sheaf) that generalizes the ones obtained by Bertram, Ein and Lazarsfeld for ideal sheaves defining smooth equidimensionnal schemes in [BEL]. By the work of Kodiyalam and Cutkosky, Herzog and Trung [Ko, CHT] we know that, if I is a homogeneous ideal in a polynomial ring over a field, then reg(i m ) is eventually a linear function in m whose leading coefficient is at most the maximal degree of a generator of I; our goal was then to find a bound of that type for any power of a homogeneous ideal under geometric hypotheses on the corresponding scheme. Another interesting challenge is the conjecture of Katzman on the regularity of Frobenius powers of a homogenous ideal, or of a graded module, in positive characteristic. It turned out that this problem may also be attacked by the argument we use for ordinary powers. The works on regularity bounds in terms of degrees of defining equations and on regularity of the tensor product of two modules also appeared to be candidates for a common treatment, by considering the tensor product of any number of free resolutions and looking at the corresponding acyclicity defects : the (multiple) Tor modules. One central idea in our study is that from a complex D which is not too far from being acyclic, one can derive non trivial information on the cohomology of H 0 (D ) from a control on the cohomology of the D i s. This idea was a key point in the proof of a regularity bound for reduced curves by Gruson, Lazarsfeld and Peskine; our lemmas in the first section are variations from Lemma 1.6 in [GLP]. We state them in a ring theoretic version. Let us point out that the simplest of these results, Lemma 1.1, is in many cases sufficient to prove new interesting bounds on regularity. A direct application is to bound the regularity of Frobenius powers of a module. Our result, which solves in a non trivial series of cases conjectures of Katzman in [Ka], is the following: Theorem 0.1. Let S be a standard graded ring over a field of characteristic p > 0 and M a finitely generated graded S-module. Assume that dim(nonreg(s) Supp(M)) 1 and set b S i (M) := max{j Tor S i (M, k) j 0} if Tor S i (M, k) 0, and b S i (M) := else. Then, denoting by F the Frobenius functor, one has reg(f e M) max 0 i j dim S {pe b S i (M) + a j (S) + j i} reg(s) + max 0 i dim S {pe b S i (M) i}. 1

2 Another application given in section 3 is to provide bounds on the regularity of tensor product of modules in terms of the regularity of the modules. A useful ingredient in the study of tensor product of modules is the Tor modules of a s-tuple of modules. We give in an appendix several properties of these modules. The results are classical in the case of two modules, we needed some extensions for an arbitrary number of modules. In particular, we prove the following generalization of results of Auslander and Serre, Theorem 0.2. Let R be a regular local ring containing a field and M 1,..., M s be finitely generated R-modules. Then, (i) Tor R i (M 1,..., M s ) = 0 implies Tor R j (M 1,..., M s ) = 0 for all j i. (ii) Let j := max{i Tor R i (M 1,..., M s ) 0}. Then pdimm pdimm s = dim R + j ε, with 0 ε dim Tor R j (M 1,..., M s ). Moreover, ε ε 0 := min i {depth Tor R j i(m 1,..., M s ) + i} and the equality holds if ε 0 = depth Tor R j (M 1,..., M s ). As compared to earlier work of Sidman and others, we obtain bounds that mixes the invariants a i of one module with the b i s of the others (see Proposition 3.1). This type of bounds is in several cases sharper than the previously known ones. For example Proposition 3.1 (i) implies the following (see also Corollary 3.2), Theorem 0.3. Let k be a field, Z 1,..., Z s be closed subschemes of a projective k-scheme Z of dimension d. Assume that reg(z 1 ) reg(z s ). If the intersection of the Z i s is of dimension at most 1, then min{d,s} reg(z 1 Z s ) reg(z) + reg(z i ). i=1 Even in the case where Z is a projective space and the Z i are hypersurfaces, such a bound was not known without restrictions on the local number of defining equations of the intersection. As another illustration we derive the following bound for powers of ideals that refines the results of [Chan] and [GGP] (recall that b 0 (I) is the maximal degree of the minimal generators of I and b 1 (I) is the maximal degree of minimal first syzygies), Theorem 0.4. Let I be an homogeneous ideal of R := k[x 1,..., X n ] such that dim(r/i) 1. Then, for any m 1, a 1 (R/I m+1 ) max{a 0 (R/I), a 1 (R/I) + mb 0 (I)} and a 0 (R/I m+1 ) max{a 0 (R/I) + b 0 (I), a 1 (R/I) + b 1 (I)} + (m 1)b 0 (I), in particular, reg(i m+1 ) reg(i) + mb 0 (I) if I is linearly presented and reg(i m+1 ) 2reg(I) + (m 1)b 0 (I) in any case. We also use a natural map from the exterior algebra of the conormal module to the Tor algebra of the quotient ring to give estimates for the regularity of powers of generically complete intersection ideals defining a quotient of dimension 2, we get for example the following Theorem 0.5. Let I be an homogeneous ideal of R := k[x 1,..., X n ] such that dim(r/i) = 2 and d 1 d s be the degrees of a minimal system of generators of I. Assume that I p R p is a complete intersection for every prime p such that dim R/p = 2. Then, reg(i/i 2 ) max{reg(r/i) + max{b 0 (I), b 1 (I) 1, b 2 (I) 2}, a 2 (R/I) + d 1 + d 2 }. In section 5 we study quite carefully the regularity of two dimensional quotients of a Gorenstein standard graded algebra over a field. We first establish several lemmas that are useful for bounding the regularity of ideals by inducting on the dimension and/or the number of generators. We next turn to some duality 2

3 results on the Koszul homology of ideals, that we use, in the case of two dimensional quotients, to give quite sharp estimates of the regularity in terms of the degrees of generators and the minimal degree of a section of the unmixed component of the corresponding scheme (a projective curve). In case this degree is zero, for instance if the curve is reduced, this bound is the expected one. We also show by examples that there are cases where the bad behavior described for curves with very negative sections do occur. Proposition 5.10 gives a geometric description of the local cohomology in the case of an almost complete intersection of dimension 2 in a Gorenstein standard graded ring and explains why sections of negative degrees do influence the regularity : Theorem 0.6. Let S be a standard graded Gorenstein algebra of dimension n over a field k, a S be its a-invariant, I = (f 1,..., f n 1 ) be a graded S-ideal with dim(s/i) = 2, where f i is a form of degree d i. Assume that J := (f 1,..., f n 2 ) is a complete intersection strictly contained in I. Let C be the one dimensional component of Proj(S/I) and C be its residual in Proj(S/J). Then, (with indeg(0) = + ). a 0 (S/I) = a S + d d n 1 indeg(h 1 m(s/i C )), a 1 (S/I) = a S + d d n 1 indeg(i C /I), a 2 (S/I) = a S + d d n 2 indeg(i C /J), Our results on the regularity of two dimensional quotients (Proposition 5.12, Theorem 5.13 and Proposition 5.15) have as corollaries estimates on the regularity of schemes of dimension two, for example we have the following, Theorem 0.7. Let S be a standard graded Gorenstein ring, Z := Proj(S), n := dim Z and I be a graded S-ideal generated by forms of degrees d 1 d s. Set X := Proj(S/I) and l := min{s, n}. Assume that dim X = 2, the component of dimension two of X is a reduced surface S and µ(i X,x ) dim O Z,x for x S except at most at finitely many points, then reg(x) reg(s) + d 1 + d d l l. Without the hypothesis on the local number of generators, we are able to get a bound which is essentially twice the bound above (see Proposition 5.12). We then turn to another type of estimates for the regularity of powers of ideals that come from the acyclicity properties of the Z-complex. As a preliminary, we have to study the cohomology of the Koszul cycles, this is the subject of section 6. The Z-complex is, or approximates, a resolution by Koszul cycles of the symmetric algebra of an ideal. We therefore get results on the local cohomology of the symmetric powers. As a consequence, we also get results on the cohomology of powers. We prove, for instance, Theorem 0.8. Let I R = k[x 0,..., X n ] be an ideal generated in degrees d 1 d s. Set X := Proj(R/I) P n k, Xj := Proj(R/I j ) for j 2 and l := min{s, n}. Assume that dim X 3, µ(i p ) dim R p for all p Supp(R/I) of codimension at most n 2, and further X is generically reduced if dim X = 3, then reg(x j ) (j 1)d 1 + max{reg(x), d d l l}. In the second part of section 7, we combine these results with the estimates of section 5, or other estimates on the regularity, to give new results on the regularity of powers of ideal sheaves that corresponds to projective schemes of dimension at most 3. Propostion 7.13 and Propostion 7.14 extends in small dimension the estimate of [BEL] to schemes that are neither smooth nor unmixed, they hold in any characteristic. In the last section we collect what we proved on the torsion in the symmetric algebra of an ideal in the particular case of a polynomial ring, and derive an application to the implicitization problem, in the spirit of the work of Busé and Jouanolou [BJ]. We treat cases where the base locus is of dimension one (see Section 8 for the definition of the graded pieces Z [µ] in the result below) : 3

4 Theorem 0.9. Let φ : P n 1 P n be a rational map defined by n + 1 polynomials of degree d, I be the ideal generated by these polynomials and X P n 1 be the scheme defined by I. Assume that dim X = 1 and let C be the one dimensionnal component of X. If C has no section of degree < d and for any closed point x X { n 1 if x C µ(i X,x ) n else, then det(z [µ]) is a non-zero multiple of the equation of the closure of the image of φ in P n for µ (n 1)(d 1). Section 1. Some key lemmas In this section, R is a polynomial ring over a field and D a graded complex of finitely generated R-modules with D i = 0 for i < 0. The following lemmas give ways for controlling the local cohomology of H 0 (D ) in terms of local cohomologies of the D i s under restrictions on the dimension of the modules H i (D ) for i > 0. The quantities δ p := max i 0 {a p+i (D i )} and ε q := max i 0 {a i (D q+i )} are useful in these estimates. Lemmas below all derive from the study of the two spectral sequences coming from the double complex C md. For clarifying later references, we first state particular cases that are often used in the sequel before giving more refined statements. The proofs of the six lemmas will be given after they are all stated. We set H i := H i (D ) and introduce the following conditions on the dimension of the H i s : D l (τ) : dim H i max{τ, τ l + i}, i > 0. Lemma 1.1. a p (H 0 ) δ p for p τ 1 if D 1 (τ) is satisfied. Lemma 1.2. If D 1 (τ) is satisfied, there exists a natural map ξ : Hm τ 2 (H 0 ) H m(h τ 1 ) such that ξ µ is onto for µ > δ τ+1 and ξ µ is into if D 2 (τ) is satisfied and µ > δ τ. Lemma 1.3. If D 2 (τ) is satisfied, there exists a diagram of natural maps, 0 K H τ 3 m (H 0 ) α H τ 1 m (H 1 ) H τ 4 m (H 0 ) γ H τ 2 m (H 1 ) β H τ m(h 2 ) C 0 θ with K = ker(α) and C = coker(β), such that, (a) α µ and θ µ are onto for µ > δ τ, (b) θ µ is into and Im(γ µ ) = ker(β µ ) if D 3 (τ) is satisfied and µ > δ τ 1. Lemma 1.4. Assume that dim H i 1 for i 1. Then a p (H 0 ) δ p for p 0. Also a 0 (H q ) ε q and a 1 (H q ) ε q 1 for q > 0. Lemma 1.5. If dim H i 2 for i 2, then a 1 (H q ) ε q 1 for q > 0 and there exists natural maps for p 0 and for q > 0 such that α p : H p m(h 0 ) H p+2 m (H 1 ) and φ q : H 0 m(h q ) H 2 m(h q+1 ) (a) α p µ is onto for µ > δ p+1 and is into if µ > δ p, (b) φ q µ is onto for µ > ε q 1 and is into if µ > ε q. Lemma 1.6. If dim H i 3 for i 3, there exists complexes of natural maps 0 K p Hm(H p 0 ) αp H p+2 (H 1 ) βp H p+4 (H 2 ) C p 0 m 4 m

5 for p 1 that are exact except possibly in the middle, maps θ p : K p C p 1 for p 0, and for q > 0 a collection of maps Hm(H 0 q ) φq Hm(H 2 q+1 ), Hm(H 1 q ) ψq Hm(H 3 q+1 ) and ker(φ q ) ηq coker(ψ q+1 ) (with β 1 = ψ 1 ) such that: (a) ker(β p µ) = im(α p µ) for µ > δ p+1, (b) θ p µ is onto for µ > δ p+1 and is into if µ > δ p, (c) φ q µ is onto for µ > ε q 1, (d) ψ q µ is into for µ > ε q 1, (e) η q µ is onto for µ > ε q 1 and is into if µ > ε q. We now turn to the proof of these statements. Notice that Lemma 1.6 implies Lemma 1.5 that in turn implies Lemma 1.4. Proof of the lemmas 1.1 to 1.6. We choose the cohomological index of C md as line index and the homological index as column index. The two spectral sequences coming from the horizontal and vertical filtrations of this double complex have first terms: h 1Ej i = Ci m(h j ), h 2Ej i = Hi m(h j ) and v 1Ej i = Hi m(d j ). We therefore note that ( v 1Ej i) µ = 0 for µ > δ p if i j = p 0 and for µ > ε q if i j = q 0. This implies that H p (Tot(C md )) µ = 0 for p 0 and µ > δ p and H q (Tot(C md )) µ = 0 for q 0 and µ > ε q (with the convention Tot(C ) l = i j=l Ci j ). We will now combine these vanishing with the study of the further steps in the spectral sequences coming from the horizontal filtration to derive the six lemmas. For the first three lemmas, we assume that condition D 1 (τ) is satisfied and concentrate on the bottom right part of the diagram at step 2 and relabel the second differentials that we will use by setting ξ := h 2d τ 2 0, α := h 2d τ 3 0, β := h 2d τ 2 1, γ := h 2d τ 4 0 : h 2 E0 τ 4 h γ 2 E0 τ 3 h 2 E1 τ 2 β h 2 E1 τ 1 α ξ h 2E0 τ 2 h 2E0 τ 1 h 2 E τ h 2 2E τ h 1 2E0 τ h 2 E τ+1 h 3 2E τ+1 h E0 τ+1 h 2 E3 τ+2 h E0 τ

6 The dotted arrows show direction of differentials at step 3. We set θ := h 3d τ 3 0. As for i 0 and l 2, h 2E τ+l+i 1 l 1 = Hm τ+l+i 1 case, and therefore h l dτ+i 1 0 : h l Eτ+i 1 0 h l Eτ+l+i 1 l 1 (H l 1 ) = 0 it follows that also h l Eτ+l+i 1 l 1 is the zero map. = 0 in this This implies that h l Eτ+i 1 0 h 2E0 τ+i 1 = Hm τ+i 1 (H 0 ) for any l 2 and i 0, and proves Lemma 1.1. For Lemma 1.2 first notice that coker(ξ) h 3E1 τ h E1 τ. Moreover if D 2 (τ) is satisfied, for l 2, h 2E τ+l 1 l = Hm τ+l 1 (H l ) = 0 and therefore h 3E0 τ 2 h E0 τ 2, h 3E1 τ 1 h E1 τ 1 and h 4E2 τ h E2 τ. The first isomorphism finishes to prove Lemma 1.2, and the last two isomorphisms imply Lemma 1.3 (a). Furthermore, if D 3 (τ) is satisfied then one also has h 4E0 τ 3 h E0 τ 3 and h 3E1 τ 2 h E1 τ 2 because l+1 = Hm τ+l 1 (H l+1 ) = 0 for l 2, which implies Lemma 1.3 (b). h 2E τ+l 1 We now turn to Lemma 1.6. At step 2 we have, after relabeling the maps : Hm(H 0 3 ) H 0 m(h 2 ) H 0 m(h 1 ) H 0 m(h 0 ) H 1 φ m(h 3 3 ) H 1 φ m(h 2 2 ) H 1 φ m(h 1 1 ) H 1 α m(h 0 0 ) ψ 3 ψ 2 ψ Hm(H ) Hm(H 2 3 ) Hm(H 2 2 ) H 2 m(h 1 ) H 2 α m(h 1 0 ) β 0 Hm(H 3 4 ) Hm(H 3 3 ) Hm(H 3 2 ) H 3 m(h 1 ) H 3 α m(h 2 0 ) β Hm(H 4 2 ) Hm(H 4 1 ) Hm(H 4 0 ) α 3 β Hm(H 5 2 ) Hm(H 5 1 ). α Hm(H 6 2 ) Hm(H 6 1 ) The dotted arrows shows direction of maps at step 3. This spectral sequence abouts at step 4 because h l di j = 0 for any i, j if l 4. The estimates we proved on the degree where the graded components of the corresponding total complex vanishes therefore implies Lemma 1.6. These lemmas will be used frequently in the case where D is of the form F R M, where F is a graded complex of finitely generated free R-modules, with F i = 0 for i < 0 and M is a graded R-module. Each F i is of the form j R[ j] β ij and we set b i (F ) := max{j β ij 0} if F i 0 and b i (F ) := else. In this context, the following two easy lemmas will be of use: Lemma 1.7. Let F be a graded complex of finitely generated free R-modules, with F i = 0 for i < 0 and M a finitely generated graded R-module. Then a p (F i R M) = a p (M) + b i (F ). 6

7 Lemma 1.8. Let F 1,..., F s be graded complexes of finitely generated free R-modules, with F j i = 0 for i < 0 and any j, and F be the tensor product of these complexes. Then b l (F ) = max {b i 1 (F 1 ) + + b is (F s )}. i 1+ +i s=l Section 2. Regularity of Frobenius powers In all this section R is a Noetherian ring of characteristic p > 0, F is the Frobenius functor and NonReg(R) denotes the set of prime ideals of R such that R p is not regular. If R is a graded ring with a unique graded maximal ideal m (for instance R is local or standard graded over a field), M is a finitely generated graded R-module and F is a minimal graded free R-resolution of M, then any graded free R-resolution F of M is isomorphic to F T, where T is a direct sum of trivial exact sequences 0 R id R 0. Therefore because F preserves trivial exact sequences. H i (F e F ) H i (F e F ). Notation 2.1. The isomorphism class of H i (F e F ) for F a graded free R-resolution of the finitely generated graded R-module M is denoted by D i,e S (M). Notice that D 0,e S (M) = F e (M), and we have the following result, Lemma 2.2. Let S be a standard graded ring over a field of characteristic p > 0 and M a finitely generated graded S-module. Then D i,e S (M) is supported in NonReg(S) Supp(M) for any i > 0 and any e. Proof. Let F be a minimal graded free S-resolution of M. As F commutes with localization (see e.g. [BH, 8.2.5]) and localization is flat, H i (F e F ) p H i (F e (F S S p )) D i,e S p (M p ). If p is not in NonReg(S), F is exact in S p and therefore H i (F e (F S S p )) = 0 for i > 0. On the other hand D i,e S p (M p ) = 0 for any i if p is not in the support of M. Theorem 2.3. Let S be a standard graded ring over a field of characteristic p > 0 and M a finitely generated graded S-module. Assume that dim(nonreg(s) Supp(M)) 1 and set b S i (M) := max{j Tor S i (M, k) j 0} if Tor S i (M, k) 0, and b S i (M) := else. Then, reg(f e M) max 0 i j dim S {pe b S i (M) + a j (S) + j i} reg(s) + max 0 i dim S {pe b S i (M) i}. Proof. Let F be a minimal graded free S-resolution of M, therefore F i = j S[ j] β ij, where β ij = dim k Tor S i (M, k) j. The complex F e F is a complex of graded free S-modules and F e F i = j S[ jp e ] βij. The hypothesis implies that dim H i (F e F ) 1 for i > 0 by Lemma 2.2. Therefore it follows from Lemma 1.7 and Lemma 1.1 (or Lemma 1.4) that which proves our claim. a l (F e M) max {a i+l(s) + p e b S i (M)}, l 0, 0 i dim S l This theorem gives a positive answer to question (2) in the introduction of [Ka] when dim(nonreg(s) Supp(M)) 1 by providing the bound reg(f e M) reg(s) + p e max 0 i dim S {bs i (M)}. Notice that Lemma 1.4 also shows that reg(d i,e S (M)) pe for any i under the hypotheses of Theorem 2.3, and Lemma 1.5 tells us that reg(f e M) p e in the case where NonReg(S) Supp(M) is of dimension two if and only if a 2 (D 1,e S (M)) pe in this case. The question of Katzman may naturally be extended by asking: 7

8 If F is a graded complex of finite free S-modules, does one have reg(h i (F e F )) p e for any i? The theorem also gives an affirmative answer to Conjecture 4 in [Ka] in the special case below, Corollary 2.4. Let I, J R be two homogeneous ideals and L i be the ideal generated by X n,..., X n i+1 for i 1. Assume that (1) Proj(R/J) is regular outside finitely many points, (2) for i 1, Proj(R/(J + L i )) is regular and X n i is not a zero divisor in R/(J + I + L i ) sat. Then there exists C I such that the Gröbner basis of I [pe] + J for the reverse lexicographic order is generated in degrees at most p e C I for any e > 0. Proof. Condition (2) implies that reg(i [pe] +J) = reg(in rev lex (I [pe] +J)) so that we may apply Theorem 2.3 to S := R/J to get our claim. Notice that, when condition (1) is satisfied, Bertini theorem implies that, if k is infinite, there exists L i s such that Proj(R/(J + L i )) is regular for i 1. These L i s being chosen, there exists a non empty open subset of the linear group such that condition (2) is satisfied after applying a linear transformation corresponding to a point in this subset. Section 3. Regularity of tensor products of modules To a collection of R-modules M 1,..., M s is associated a collection of R-modules Tor R i (M 1,..., M s ), which are supported in Tor R 0 (M 1,..., M s ) = M 1 R R M s, some properties of these modules are collected in the appendix. Proposition 3.1. Let R be a polynomial ring over a field and M, M 1,..., M s be finitely generated graded R-modules. Set T i := Tor R i (M, M 1,..., M s ), d := dim M, τ := dim T 1 and b l := max {b i 1 (M 1 ) + + b is (M s )}. i 1 + +i s =l Then, (i) For p τ 1, a p (T 0 ) max {a p+i(m) + b i }. 0 i d p (ii) Hm τ 2 (T 0 ) µ Hm(T τ 1 ) µ for µ > max 0 i d τ+2 {a τ+i 2(M) + b i }. (iii) For there is an exact sequence, µ > max 0 i d τ+3 {a τ+i 3(M) + b i }, H τ 4 m (T 0 ) µ H τ 3 m (T 1 ) µ H τ m(t 2 ) µ H τ 3 m (iv) If τ 1, a p (T 0 ) max 0 i d p {a p+i (M) + b i } for p 0, a 0 (T q ) max 0 i d {a i (M) + b q+i }, for q 0, a 1 (T q ) max 0 i d {a i (M) + b q+i 1 }, for q 1. (T 0 ) µ H m τ 1 (T 1 ) µ 0. Proof. Let F i be a minimal finite free R-resolution of M i and F be the tensor product over R of the complexes F. i We now apply Lemma 1.7 and lemmas 1.1, 1.2, 1.3 and 1.4 to F R M to get respectively (i), (ii), (iii) and (iv), noticing that H q (F R M) T q by Lemma A.3 and condition D l (τ) is satisfied for any l 1 by Theorem A.7 (i). 8

9 Corollary 3.2. Let k be a field, Z 1,..., Z s be closed subschemes of P n k and let Z := Z 1 Z s. Assume that reg(z 1 ) reg(z s ) and there exists S Z of dimension at most one such that, locally at each point of Z S, Z is a proper intersection of Cohen-Macaulay schemes. Then, reg(z) min{n,s} i=1 reg(z i ). Proof. Let R be the polynomial ring in n+1 variables over k and I 1,..., I s be the defining ideals Z 1,..., Z s, respectively. Corollary A.10 shows that the hypothesis implies that dim Tor 1 (R/I 1,..., R/I s ) 2. Therefore by Proposition 3.1 (with M = R and M i = R/I i ) reg(z) = max p>0 {a p(r/(i I s )) + p} max p>0 max p>0 max {b i 1 (R/I 1 ) + + b is (R/I s ) + p n 1} i 1+ +i s=n+1 p max {ɛ i 1 reg(r/i 1 ) + + ɛ is reg(r/i s )} i 1 + +i s =n+1 p where ɛ j = 0 if j 0 and ɛ j = 1 else (note that b 0 (R/I i ) = 0 for any i). The bound follows. Corollary 3.3. Let R be a polynomial ring over a field, M a finitely generated graded R-module and f 1,..., f s be forms of degrees d 1 d s. Set M := M/(f 1,..., f s )M, δ := dim M and δ := dim M, then a δ (M ) max{a δ (M), a δ +1(M) + d 1,..., a δ (M) + d d δ δ }, and For instance, if dim M = 1, and if dim M = 2, a δ 1(M ) max{a δ 1(M), a δ (M) + d 1,..., a δ (M) + d d δ δ +1}. reg(m ) max{a 0 (M), a 1 (M) + d 1,..., a δ (M) + d d δ }. reg(m /H 0 m(m )) max{a 1 (M), a 2 (M) + d 1,..., a δ (M) + d d δ 1 } + 1. Proof. Note that Tor R i (M, R/(f 1 ),..., R/(f s )) H i (K (f 1,..., f s ; M)) by Corollary A.3 and [BH, 1.6.6], and apply Lemma 1.1. Example 3.4. Let I and J be two homogeneous ideals of R := k[x 1,..., X n ] with codim(i + J) min{n 1, codimi + codimj}. Assume that R p /I p and R p /J p are Cohen-Macaulay for every homogeneous prime p supported in V (I + J) such that dim R/p 2. Then the estimate given by Corollary 3.2 gives for any p a p (R/(I + J)) max {b i(r/i) + b n p i (R/J)} n 0 i n p and Corollary 3.3 gives a p (R/(I + J)) max {b i(r/i) + a p+i (R/J)}. 0 i n p The first formula is more symmetric, but on the other hand in the second the roles of I and J may be permuted to have another estimate, and there is fewer terms in the second maximum as a p+i (R/J) = 0 for i > dim R/J p. More significatively for proving bounds, if a 0 (M) > a 1 (M) > > a dim M (M) then b n p i (M) n a p+i (M) for any i, so that in this case the second estimate is always sharper then the first. Note also that the first estimate for p = 0 is at least max{a 0 (R/I), a 0 (R/J)} and in the second we can take the minimum of these two terms as the only term concerning the m-primary components of I and J. Section 4. Regularity of powers of ideals 9

10 We will here apply the results of section 3 to control the regularity of I m /I m+1 = Tor R 1 (R/I, R/I m ), for m 1 in terms of invariants attached to I. We start with an example which refines previous results in [Chan] and [GGP]. Example 4.1. Let I be an homogeneous ideal of R := k[x 1,..., X n ] such that dim(r/i) 1. Note that b 0 (R/I) = 0, b 1 (R/I) = b 0 (I) and b 2 (R/I) = b 1 (I). By Proposition 3.1 (iv) applied to M := R/I m and M 1 := R/I, one has a 0 (I m /I m+1 ) max{a 0 (R/I m ) + b 0 (I), a 1 (R/I m ) + b 1 (I)}. Now one also has a 1 (I m /I m+1 ) max{a 0 (R/I), a 1 (R/I) + mb 0 (I)}, by Proposition 3.1 (iv) applied to M := R/I and M 1 := R/I m. Therefore by an immediate recursion on m, a 0 (R/I m+1 ) max{a 0 (R/I) + b 0 (I), a 1 (R/I) + b 1 (I)} + (m 1)b 0 (I), in particular, reg(i m+1 ) reg(i) + mb 0 (I) if I is linearly presented and reg(i m+1 ) 2reg(I) + (m 1)b 0 (I) in any case. The next lemma will be useful to study the cases of dimension two and three. Lemma 4.2. Let I be an homogeneous ideal of R := k[x 1,..., X n ] generated in degrees d 1 d s and d := dim(r/i). Assume that I p R p is a complete intersection for every prime p Supp(R/I) of maximal dimension. Then, a d (Tor R m(r/i, R/I)) a d ( m (I/I 2 )) a d (R/I) + d d m. Proof. Set B := R/I. The alternating algebra structure on Tor R (B, B) and the identification I/I 2 ψ Tor R 1 (B, B) gives rise to a B-algebra homomorphism: ψ : (I/I 2 ) Tor R (B, B) which is an isomorphism locally at every prime p such that dim R/p = d by [BH, 2.3.9]. Therefore a d (Tor R m(b, B)) a d ( m (I/I 2 )). Also if I = (f 1,..., f s ) with deg f i = d i, there is a natural onto map T m := m m (f i,..., f s )/I(f i,..., f s ) (I/I 2 ) i=1 of graded modules supported in dimension at most d, which shows that a d ( m (I/I 2 )) a d (T m ). By Proposition 3.1 (i) we have a d (T m ) a d (R/I) + d d m, because (f i,..., f s )/I(f i,..., f s ) is generated in degree at most d i. This concludes the proof. Theorem 4.3. Let I be an homogeneous ideal of R := k[x 1,..., X n ] such that dim(r/i) = 2. Assume that I p R p is a complete intersection for every prime p such that dim R/p = 2. Set a i := a i (R/I), b i := b i (I) and let d 1 d s be the degrees of a minimal system of generators of I. Then a 0 (Tor R n (R/I, R/I)) a 0 + b n 1 and for any m > 0, a 0 (Tor R m(r/i, R/I)) max{a 0 + b m 1, a 1 + b m, a 2 + b m+1, a 2 (Tor R m+1(r/i, R/I))} max{a 0 + b m 1, a 1 + b m, a 2 + b m+1, a 2 + d d m+1 }, a 1 (Tor R m(r/i, R/I)) max{a 1 + b m 1, a 2 + b m } and a 2 (Tor R m(r/i, R/I)) a 2 + d d m. 10

11 Note that Tor R m(r/i, R/I) = 0 and b m = 0 if m > n. Proof. By Lemma 4.2, a 2 (Tor R m(b, B)) a 2 + d d m and the result follows from Lemma 1.5 (b). Corollary 4.4. Let I be an homogeneous ideal of R := k[x 1,..., X n ] such that dim(r/i) = 2 and d 1 d s be the degrees of a minimal system of generators of I. Assume that I p R p is a complete intersection for every prime p such that dim R/p = 2. Then, reg(i/i 2 ) max{a 0 (R/I) + b 0 (I), a 1 (R/I) + b 1(I), a 2 (R/I) + b 2(I), a 2 (R/I) + d 1 + d 2 } where b i (I) := max j i b j (I). max{reg(r/i) + max{b 0 (I), b 1 (I) 1, b 2 (I) 2}, a 2 (R/I) + d 1 + d 2 } Proof. Recall that I/I 2 Tor 1 (R/I, R/I) and apply Theorem 4.3. Theorem 4.5. Let I be an homogeneous ideal of R := k[x 1,..., X n ] such that dim(r/i) = 3. Assume that I p R p is a complete intersection for every prime p such that dim R/p = 3. Set a i := a i (R/I), b i := b i (I) and let d 1 d s be the degrees of a minimal system of generators of I. Then, (i) there exists a natural map ψ : H 0 m(i/i 2 ) H 2 m(tor 2 (R/I, R/I)) such that ψ µ is onto for µ > max{a 1 + b 0, a 2 + b 1, a 3 + b 2 } and into for µ > max{a 0 + b 0, a 1 + b 1, a 2 + b 2, a 3 + b 3, a 3 + d 1 + d 2 + d 3 }, (ii) H 1 m(i/i 2 ) µ = 0 for µ > max{a 1 + b 0, a 2 + b 1, a 3 + b 2, a 3 + d 1 + d 2 }, (iii) H 2 m(i/i 2 ) µ = 0 for µ > max{a 2 + b 0, a 3 + b 1 }, (iv) H 3 m(i/i 2 ) µ = 0 for µ > a 3 + b 0. Proof. Let F be a minimal free resolution of R/I. As I R is a generically a complete intersection, a 3 (Tor R m(r/i, R/I) a 3 + d d m by Lemma 4.2. The result then follows from Lemma 1.6. Corollary 4.6. Let I be an homogeneous ideal of R := k[x 1,..., X n ], B := R/I and S := Proj(B) be the corresponding projective scheme. Assume that S is of dimension two and generically a complete intersection. Set a i := a i (B), b i := b i (I) and let d 1 d s be the degrees of a minimal system of generators of I. Then if Ω B is the module of Kähler differentials of B (over k), one has (i) a 3 (Ω B ) a 3 + 1, a 2 (Ω B ) max{a 2 + 1, a 3 + b 0 }, (ii) a 1 (Ω B ) max{a 1 + 1, a 2 + b 0, a 3 + b 1 } if S is generically reduced, (iii) a 0 (Ω B ) max{a 0 + 1, a 1 + b 0, a 2 + b 1, a 3 + b 2, a 3 + d 1 + d 2 } if S is a reduced complete intersection outside finitely many points. Proof. We consider the exact sequence presenting Ω B and defining K : 0 K I/I 2 ψ B[ 1] n Ω B 0, where ψ is given by the jacobian matrix on a minimal system of generators of I. Taking cohomology on the sequence we get (i) from Theorem 4.5 (iv). If S is generically reduced, then K is supported in dimension at most two, and therefore (ii) follows from Theorem 4.5 (iii). In the situation of (iii) dim K 1, and Theorem 4.5 (ii) proves the claim. Section 5. Regularity results in small dimensions In this section S is a standard graded ring over a field. The following lemmas are useful for bounding regularity of ideals by inducting on the dimension of the quotient ring and/or number of generators: Lemma 5.1. Let I be a graded S-ideal and f S be a form such that depth (S/(I : (f))) > 0. Then, H 0 m(s/i) H 0 m(s/i + (f)). 11

12 Proof. I (f) = (f).(i : (f)) so that depth (S/I (f)) > 0. Therefore the exact sequence 0 S/I (f) S/I S/(f) S/I + (f) 0 shows that Hm(S/I) 0 Hm(S/(f)) 0 Hm(S/I 0 + (f)) and proves the assertion. Lemma 5.2. Let I be a graded S-ideal and f a form of degree δ in S and i an integer. Then, a i (S/I + (f)) max{a i (S/I), a i+1 (S/I) + δ, a i+2 (S/I) + δ, a i+1 (S/I : (f)) + δ}. Proof. We first consider the exact sequence, 0 ((I : (f))/i)[ δ] S/I[ δ] f S/I S/(I + (f)) 0. Taking cohomology, it implies that a i (S/I + (f)) max{a i (S/I), a i+1 (S/I) + δ, a i+2 ((I : (f))/i) + δ} and taking cohomology on the canonical exact sequence 0 (I : (f))/i S/I S/(I : (f)) 0 implies that and concludes the proof. a i+2 ((I : (f))/i) max{a i+1 (S/I : (f)), a i+2 (S/I)}, Lemma 5.3. Let f 1,..., f s be a sequence of forms in S, f i := (f 1,..., f i ) and let I i be the S-ideal generated by f 1,..., f i. Assume that I i 1 and I i coincide locally in dimension l. Then if p > l, a p (H j (f i ; S)) max{a p (H j (f i 1 ; S)), a p (H j 1 (f i 1 ; S)) + d i }, j where d i := deg f i. Proof. The exact sequence of [BH, ] gives rise to exact sequences 0 M j H j (f i 1 ; S) H j (f i ; S) H j 1 (f i 1 ; S)[ d i ] N j 0 where M j and N j have a support of dimension < l by [BH, ]. Therefore we get an exact sequence Hm l+1 (H j (f i 1 ; S)) Hm l+1 (H j (f i ; S)) Hm l+1 (H j 1 (f i 1 ; S))[ d i ] Hm l+2 (H j (f i 1 ; S)) Hm l+2 (H j (f i ; S)) Hm l+2 (H j 1 (f i 1 ; S))[ d i ] Hm(H d j (f i 1 ; S)) Hm(H d j (f i ; S)) Hm(H d j 1 (f i 1 ; S))[ d i ] 0. from which the result follows. If I is a S-ideal and t an integer, we will denote by I t the intersection of primary components of I of codimension at most t. We will need the following result on subideals generated by general elements of a fixed ideal: 12

13 Lemma 5.4. Assume that k is infinite and let I be a graded S-ideal of codimension r generated by forms of degrees d 1 d s. Consider the conditions: (1) µ(i p ) dim S p for every p I of codimension r, (2) µ(i p ) dim S p for every p I r of codimension r + 1, (3) µ(i p ) dim S p for every p I r+1 of codimension r + 2, (4) µ(i p ) dim S p + 1 for every p of codimension r + 1. (5) µ(i p ) dim S p + 1 for every p of codimension r + 2. For µ Z, let (H j,µ ) j Sµ be a k basis of the degree µ part of I, and set f i := j S di λ ij H j,di and I i := (f 1,..., f i ) for 1 i s. There exists Zariski open subset Ω l of Spec(k[λ ij ]) such that Ω l if condition (l) holds and: (1) λ Ω 1 implies I = I s and codim(i + (I r : I)) r + 1, (2) λ Ω 2 implies I = I s, codim(i r+1 : I) r + 1 and if λ Ω 2 Ω 4, codim(i + (I r+1 : I)) r + 2, (3) λ Ω 3 implies I = I s, codim(i r+2 : I) r + 2 and if λ Ω 3 Ω 5, codim(i + (I r+2 : I)) r + 3, (4) λ Ω 4 implies I = I s and codim(i r+2 : I) r + 2, (5) λ Ω 5 implies I = I s and codim(i r+3 : I) r + 3. Proof. See [U, 1.4] or [CEU, 2.5 (b)]. Corollary 5.5. Let f 1,..., f s be a sequence of forms in S of degrees d 1 d s and I the S-ideal they generate. Set d := dim S/I and r := codimi. If µ(i p ) dim S p + 1 for every p of codimension r + 2, and µ(i p ) dim S p for every p I r+1 of codimension r+2, then a d (H 1 (f; S))) max{a d 2 (S/I), σ r+1 +a S }, a d (H s r (f; S))) = a d (ω S/I ) + a S + σ s and a d (H j (f; S)) max{a d 2 (S/I) + σ r+j+1 σ r+2, a d (ω S/I ) + a S + σ r+j }, j 1. Proof. We induct on s. Note that a d (H 1 ) max{a d 2 (S/I), σ r+1 + a S } by Lemma 1.2 and H s ω S/I [σ s + a S ], so that we may assume s r + 3. We choose generators as in Lemma 5.4 (3) and we write I = I s 1 J where I s 1 is the ideal generated by the first s 1 r + 2 new generators of I. By Lemma 5.4 (3), as I r+2 I s 1, J is such that codimj r + 2 and codim(i s 1 + J) r + 3. We have an exact sequence, and as dim S/(I s 1 + J) d 3 we get an onto map 0 S/I S/I s 1 S/J S/(I s 1 + J) 0, H d 2 m (S/I) H d 2 m (S/I s 1 ) Hm d 2 (S/J) which shows that a d 2 (S/I s 1 ) a d 2 (S/I) and the result follows from Lemma 5.3 by recursion on s. Theorem 5.6. Let S be a standard graded algebra of dimension n and I be a graded S-ideal generated by s forms of degrees d 1 d s 1. Assume that dim S/I = 2 and set l := min{s, n 1}, then reg(s/i sat ) reg(s) + d d l l. Proof. Apply Corollary 3.3 to M := S and notice that reg(s) max i>0 {a i (S) + i}. Lemma 5.7. Let S be a standard graded Cohen-Macaulay algebra of dimension n over a field k, I = (f 1,..., f s ) be a graded S-ideal of codimension r. Set := Ext r S(, ω S ) and σ := d d s. Then, H i (f 1,..., f s ; ω S ) [σ] H s r i (f 1,..., f s ; S). 13

14 Proof. Set := Homgr S (, k) and f := (f 1,..., f s ). Comparing the two spectral sequences coming from C mk (f; S) gives rise to filtrations F0 i F1 i Fn r i = H s i (f; Hm(S)) n H i (f; ω S )[σ] such that (Fj i/f j 1 i ) is supported in codimension at least r + j (it is a subquotient of Hm n r j (H s r i j (f; S))) Ext r+j S (H s r i j(f; S), ω S )) and exact sequences 0 N i H m n r (H s r i (f; S))) F 0 0 i such that N i is supported in codimension at least r + 2. Taking graded k-duals, we get exact sequences and 0 (F i 0) H s r i (f; S) N i 0 0 M i H i (f; ω S )[σ] (F i 0) 0 where M i is supported in codimension at least r + 1. We therefore get which proves our claim. H s r i (f; S) ((F i 0) ) H i (f; ω S )[σ] If further S is Gorenstein, then this can be rewritten H i (f 1,..., f s ; S) [σ + a S ] H s r i (f 1,..., f s ; S). In the case r = n 1, we have Lemma 5.8. Let S be a standard graded Cohen-Macaulay algebra of dimension n over a field k, I = (f 1,..., f s ) be a graded S-ideal of codimension n 1. Set := Homgr S (, k), := Ext n 1 S (, ω S ) and σ := d d s. Then, H i (f 1,..., f s ; ω S ) [σ] H s n i+1 (f 1,..., f s ; S) and H 0 m(h i (f 1,..., f s ; ω S ))[σ] H 0 m(h s n i (f 1,..., f s ; S)). Proof. Following the steps of the proof of Lemma 5.7, we notice that in this case N i = 0 and M i Hm(H 0 s n i (f; S)), from which the result follows. In the case of quotients of dimension 2, the situation is fairly more complicated, we will restric ourselves to the Gorenstein case for simplicity. If M is a finitely generated S-module, we will set M unm := M/Hb 0 (M), where b is the intersection of prime ideals associated to M that are not of minimal codimension. Therefore M unm has all its associated primes of codimension equal to the codimension of the support of M. If M = S/I then M unm = S/I top, where I top is the intersection of primary components of I of minimal codimension. Lemma 5.9. Let S be a standard graded Gorenstein algebra of dimension n over a field k, a S be its a-invariant, I = (f 1,..., f s ) be a graded S-ideal with dim(s/i) = 2, where f i is a form of degree d i. Set := Homgr S (, k) and σ := d d s, then and there is exact sequences H s n+2 (f 1,..., f s ; S)[a S + σ] ω S/I H 2 m(s/i), H 0 m(h s n+1 (f 1,..., f s ; S))[a S + σ] H 1 m(s/i top ), 0 H 1 m(s/i) H s n+1 (f 1,..., f s ; S)[a S + σ] H s n+1 (f 1,..., f s ; S) unm [a S + σ] 0 14

15 and 0 H 1 m(h s n+1 (f 1,..., f s ; S) unm )[a S + σ] H 0 m(s/i) H s n (f 1,..., f s ; S)[a S + σ]. Proof. The first isomorphimes are classical, and follows from Lemma 5.7. We set := Ext n 2 S (, ω S ). First notice that if s = n 2 the other assertions are trivial. Therefore we may assume that s n 1 and set f := (f 1,..., f s ). Comparing the two spectral sequences coming from C mk (f; S) gives rise to a filtration F 0 F 1 F 2 = H s (f; Hm(S)) n (S/I)[σ + a S ] with F 0 coker(hm(h 0 s n+1 (f; S)) Hm(H 2 s n+2 (f; S))), F 1 /F 0 Hm(H 1 s n+1 (f; S)) and F 2 /F 1 ker(hm(h 0 s n (f; S)) Hm(H 2 s n+1 (f; S))). This shows first that Hm(H 2 s n+2 (f; S)) (S/I) [σ + a S ], and provides, by duality, three exact sequences: 0 (F 2 /F 1 ) (S/I)[σ + a S ] F1 0 0 H 1 m(h s n+1 (f; S)) F 1 F F 0 (S/I) [σ + a S ] H 0 m(h s n+1 (f; S)) 0. These in turn identifies Hm(H 0 s n+1 (f; S)) as the cokernel of the map (S/I) (S/I) (S/I top ) shifted in degree by σ + a S and proves the second isomorphism. We also have a filtration G 0 G 1 = G 2 = H n+1 (f; Hm(S)) n Hs n 1(f; S)[σ + a S ] with G 0 = coker(hm(h 0 0 (f; S)) Hm(H 2 1 (f; S))). This implies that Hm(H 2 1 (f; S)) (H s n 1 (f; S)) [σ + a S ], and provides the exact sequences 0 G 0 H s n 1 (f; S) [σ + a S ] H 0 m(s/i) and 0 H 1 m(s/i) H s n 1 (f; S)[σ + a S ] G 0 0 therefore proving that G 0 H unm s n 1[σ + a S ] and thereby giving the first exact sequence. For the last exact sequence we consider the filtration G 0 G 1 G 2 = H n (f; H n m(s)) H s n(f; S)[σ + a S ] with G 2/G 1 = ker(h 0 m(h 0 (f; S)) H 2 m(h 1 (f; S))). By duality, and using the identification of G 0 above, it gives an exact sequence 0 H 1 m(h s n+1 (f 1,..., f s ; S) unm )[a S + σ] H 0 m(s/i) (G 2/G 1) 0 and the filtration gives a natural injection (G 2/G 1) H s n (f 1,..., f s ; S)[a S + σ]. This gives a fairly geometric descritption of the local cohomology in the case of an almost complete intersection of dimension 2, Proposition Let S be a standard graded Gorenstein algebra of dimension n over a field k, a S be its a-invariant, I = (f 1,..., f n 1 ) be a graded S-ideal with dim(s/i) = 2, where f i is a form of degree d i. Assume that J := (f 1,..., f n 2 ) is a complete intersection strictly contained in I. Let C be the one dimensionnal component of Proj(S/I) and C be its residual in Proj(S/J). Then, (with indeg(0) = + ). Therefore, a 0 (S/I) = a S + d d n 1 indeg(h 1 m(s/i C )), a 1 (S/I) = a S + d d n 1 indeg(i C /I), a 2 (S/I) = a S + d d n 2 indeg(i C /J), (i) H 0 m(s/i) = 0 if and only if C is arithmetically Cohen-Macaulay. (ii) H 1 m(s/i) = 0 if and only if I is unmixed. (iii) a 0 (S/I) a S + d d n 1 if C is reduced. Further if C is geometrically reduced, the inequality is strict if and only if C is geometrically connected. 15

16 (iv) If η := max{µ H 0 (C, O C ( µ)) 0} is positive or C is geometrically disconnected, then reg(s/i) = a 0 (S/I) = a S + d d n 1 + η. and Proof. The values for a 0 and a 1 directly follows from the isomorphismes H 0 m(s/i)[a S + d d s ] H 1 m(s/i top ) H 1 m(s/i) (I C /I)[a S + d d s ] given by Proposition 5.9, and the value of a 2 is given by liaison (see e.g. [CU, 4.1 (a)]). Properties (i) and (ii) are direct corollaries of the isomorphismes above. For (iii) and (iv) recall the exact sequence 0 (S/I C ) µ H 0 (C, O C (µ)) H 1 m(s/i C ) 0, note that (S/I C ) µ = 0 for µ < 0 and (S/I C ) 0 = k. Also µ H 0 (C, O C (µ)) is a finitely generated graded S-module and therefore has no element of negative degree if it is reduced, and if C is geometrically reduced, dim k H 0 (C, O C ) is the number of components of C over an algebraic closure of k. For (iv) notice that a 1 (S/I) + 1 d d n 1 + a S, a 2 (S/I) + 2 d d n 2 + a S + 1 d d n 1 + a S, and that Hm(S/I 1 C ) 0 0 if C is geometrically disconnected. Example Let b m := (y m z x m t, z m+1 xt m ) I m := (z m, t m, y m z x m t) S := k[x, y, z, t], C m := Proj(S/I m ) and J m b m be the defining ideal of the monomial curve (1, m 2, m(m + 1)). (1) Im top = Im sat = (z, t) m + (y m z x m t) = b m : (J m (x, z)) have regularity 2m 1, (2) reg(j m ) = m 2 and reg(j m (x, z)) = m 2 + 1, (3) Γ(C m, O Cm (l)) = 0 if and only if l < (m 1) 2, (4) reg(s/i j m) = (m + j 1)(m + 1) 2 (if m > 1). Proposition Let S be a standard graded Gorenstein algebra of dimension n over a field k, a S be its a-invariant, I = (f 1,..., f s ) be a graded S-ideal with dim(s/i) = 2, where s n 1 and f i is a form of degree d i, with d 1 d s. If the one dimensionnal component C of Proj(S/I) is generically a complete intersection, then, a 0 (S/I) 2(d d n 2 + a S + 1) + d n 1 + η, with η := max{µ H 0 (C, O C ( µ)) 0}, unless n = 2 and d 2 > a S + η + 2, in which case a 0 (S/I) d 1 + d 2 + a S. Proof. We may change the generators of I to assume that the f i s satisfies property (1) of Lemma 5.4. Set r := n 2. We may assume that I r I, because otherwise a 0 (S/I) =, and set J := I r : I. As S is Gorenstein, (I r : I)/I r ω S/I [ σ a S ] is perfect of dimension 2, σ = d d r and Hm(ω 2 S/I ) is the graded k-dual of End(ω S/I ) µ Z H 0 (C, O C (µ)). It follows that reg(j/i r ) = σ + a S + reg(ω S/I ) = σ + a S + a 2 (ω S/I ) + 2 = σ + a S + η + 2. This implies that J is generated modulo I r in degrees σ + a S + η + 2. As J and I C do not share any associated prime, we may choose f J of degree at most σ + a S + η + 2 which is not a zero divisor modulo I C so that I r : (f) = I : (f) have positive depth. By Lemma 5.1, a 0 (S/I) a 0 (S/I + (f)). Now σ + a S + η + 2 d n unless d 1 = 1 and S is a polynomial ring (in which case Hm(S/I) 0 = 0) or n = 2 and d n > a S + η + 2. Therefore, in any of the two remaining cases, Corollary 3.3 (with M := S as a quotient of a polynomial ring R) shows the asserted inequality. With a slightly stronger condition on the local number of generators of Proj(S/I), we have the following result, Theorem Let S be a standard graded Gorenstein algebra of dimension n over a field k, a S be its a-invariant, I = (f 1,..., f s ) be a graded S-ideal with dim(s/i) = 2, where s n and f i is a form of degree d i, 16

17 with d 1 d s. Let C be the one dimensionnal component Proj(S/I) and η := max{µ H 0 (C, O C ( µ)) 0}. If µ(i p ) dim S p for every homogeneous prime ideal p such that m p I C, then a 0 (S/I) a S + d d n 1 + max{d n, η}. Proof. We may assume that the f i s satisfies conditions (1) and (2) of Lemma 5.4. We set f i := (f 1,..., f i ), σ i := d d i, I i for the ideal generated by f 1,..., f i, and induct on i to bound a 0 (S/I i ). For i n 2, a 0 (S/I i ) = as I i is perfect, and a 0 (S/I n 1 ) a S + d d n 1 + η by Proposition If i n, a 0 (S/I i ) max{a 0 (S/I i 1 ), a 1 (S/I i 1 ) + d i, a 2 (S/I) + d i, a 1 (S/I i 1 : (f i )) + d i } max{σ n 1 + a S + max{η, d n }, σ n 1 + a S + d i, σ n 2 + a S + d i, a 1 (S/I i 1 : (f i )) + d i } max{σ n 1 + a S + max{η, d n }, a 1 (S/I i 1 : (f i )) + d n } by Lemma 5.2, the induction hypothesis, Theorem 5.6 and the inequality d i d n. It remains to show that a 1 (S/I i 1 : (f i )) σ n 1 + a S. As i n, Lemma 5.4 (2) implies that for every p I C of codimension n 1, (I n 1 ) p = I p, so that (I i 1 ) p = (I i ) p. If codim(i i 1 : (f i )) = n, a 1 (I i 1 : (f i )) =, else I i 1 : (f i ) = J K with J pure of dimension one and codim(i + J) = n. Now Hm(S/J 1 K) Hm(S/J) 1 and a 1 (S/J) σ n 1 + a S by [CP, 3, Corollaire 2 (i)]. For applications it may also be of interest to bound the local cohomology of all the Koszul homology modules. This is sometimes possible with the help of Corollary 5.5 and Lemma 1.5, Proposition Let S be a standard graded Gorenstein algebra of dimension n over a field k, a S be its a-invariant, I = (f 1,..., f s ) be a graded S-ideal with dim(s/i) = 2, where s n and f i is a form of degree d i, with d 1 d s. Then for 0 < i < s n + 2, a 1 (H i (f; S)) a S + d d n+i 1. Let C be the one dimensionnal component Proj(S/I) and η := max{µ H 0 (C, O C ( µ)) 0}. If µ(i p ) dim S p for every homogeneous prime ideal p such that m p I C and µ(i p ) dim S p + 1 for every homogeneous prime ideal p such that m p I, then a 2 (H 1 (f; S)) a S + d d n 1 + max{d n, η} and for 1 < i < s n + 2 and for 0 < i < s n + 1 a 2 (H i (f; S)) a S + d d n 1 + max{d n, η} + d n d n+i 1, a 0 (H i (f; S)) a S + d d n 1 + max{d n, η} + d n d n+i. Proof. The bound for a 1 (H i (f; S)) follows directly from Lemma 1.5. By Corollary 5.5, a 2 ((H 1 (f; S)) max{a 0 (S/I), a S + d d n 1 + η}, and for 1 < i < s n + 2, a 2 ((H i (f; S)) max{a 0 (S/I) + d n d n+i 1, a S + d d n+i 2 + η}, But by Theorem 5.13, a 0 (S/I) a S + d d n 1 + max{d n, η}. 17

18 which gives the estimate for a 2 ((H i (f; S)). Lemma 1.5 (b) then provides the bound on a 0 (H i (f; S)). By liaison we can give estimates on the regularity in the case where the one dimensionnal component is generically a complete intersection, Proposition Let S be a standard graded Gorenstein algebra of dimension n, multiplicity e S and a-invariant a S and I be a graded S-ideal generated by s forms of degrees d 1 d s. Let C be the component of dimension one of X := Proj(S/I) and let C be the residual of C in a complete intersection Y of degrees d 1,..., d n 2 and assume that C. Then η = max{0, reg(c ) reg(y ) + 1}. Moreover, if C is reduced, or if S is geometrically reduced and C is generically a complete intersection, then η max{0, deg(c ) reg(y )} = max{0, e S d 1 d n 2 deg(c) (d d n 2 + a S + 2)}. Proof. The first equality is [CU, 4.2]. To prove the second inequality, we may assume that k is algebraically closed and use Bertini theorem to find a complete intersection b I of degrees d 1,..., d n 2 such that J := b : I is radical (see for instance [CU, 4.4 (f)] with c = r and c = c 1 together with [CU, 4.5]). Now, as C or Proj(R/J) is reduced over a perfect field, and both have the same regularity and degree e S d 1 d n 2 deg(c), reg(c ) deg(c ) 1 by [GLP, 1.1 and Remark (1) p. 497], applied either to C or to Proj(R/J). The asserted inequality follows. Proposition Let S be a standard graded Gorenstein ring, Z := Proj(S), n := dim Z and I be a graded S-ideal generated by forms of degrees d 1 d s. Set X := Proj(S/I). Assume that dim X = 2 < n and the component of dimension two of X is a reduced surface S. Then, reg(x) 2(d d n 2 + a S + 2) + d n 1, if s n 2. If moreover µ(i X,x ) dim O Z,x for x S except at most at finitely many points, then reg(x) reg(s) + d 1 + d d l l, l := min{s, n}. Proof. We may assume that k is infinite and s n 1. Recall that reg(s) = a S + n + 1 because S is Cohen-Macaulay of dimension n + 1. Let h S 1 that is not in any associated prime of I except possibly m and such that S {h = 0} is reduced (see [FOV, ]). The ideal (I, h) satisfies the hypotheses of Proposition 5.12, and one has an exact sequence 0 (I : (h)/i)[ 1] H 0 m(s/i)[ 1] H 0 m(s/i) H 0 m(s/i + (h)) H 1 m(s/i)[ 1] H 1 m(s/i) so that a 1 (S/I) + 1 a 0 (S/I + (h)). Notice that a S/(h) = a S + 1. As the component of dimension 1 of X {h = 0} is S {h = 0}, which is reduced and therefore have no section of negative degree, Proposition 5.12 implies that a 0 (S/I + (h)) 2(d d n 2 + a S/(h) + 1) + d n 1, and if further the extra hypothesis on the local number of generators is satisfied, then by Theorem 5.10 or Theorem 5.13, a 0 (S/I + (h)) d d l + a S/(h). These estimates together with the inequality a 1 (S/I) + 1 a 0 (S/I + (h)) above proves our claim. 18

19 Proposition Let S be a standard graded Gorenstein algebra of a-invariant a S and multiplicity e S and I be a graded S-ideal generated in d 1 d 2 d s. Let X := Proj(S/I) and assume that dim X = 2 < n, X is generically a complete intersection and Proj(S) is geometrically reduced of dimension n. If s n 2, reg(x) d d n 1 + a S max{d d n 2 + a S + 2, e S d 1 d n 2 deg X}. Proof. The proof follows along the same lines as for the proof of Proposition 5.16, using Proposition 5.15 in place of Proposition Section 6. Local cohomology of Koszul cycles A very useful tool in our study of regularity of powers of an ideal will be the Z-complex introduced by Herzog, Simis and Vasconcelos that approximates a free resolution of the symmetric algebra of an ideal, in a sense that will be clarified in the next section. We will now explain results on the local cohomology of the Koszul cycles in the spirit of [BC, 4.1]. Proposition 6.1. Let S be standard graded algebra of dimension n over a field and M be a finitely generated graded S-module. Let K := K (f; M) be the Koszul complex on M given by a sequence of forms f := (f 1,..., f s ) of degrees d 1 d s and denote by Z i and H i the i-th cycles and homology modules of K. Set d := dim H 0 and σ i := i j=1 d j. Then H p m(z q ) µ = 0 if either, (i) µ > max 0 i min{s q 1,d p} {a p+i (M) + σ q+i+1 }, and Hm p+i (H q+i ) µ = 0 for i 0; or (ii) µ > max max{0,p d} i q {a p i (M) + σ q i } and H p i 1 m (H q i ) µ = 0 for 1 i p 1. Proof. We consider, as in [BC, 4.1], the double complex C mk >q where K >q : 0 K s K q+1 Z q 0. It gives rise to two spectral sequences, one of which have as first terms, Hm(K p i ) if s i > q, 1 E p i = Hm(Z p q ) if i = q, 0 else. As a p (K i ) = a p (M) + σ i, ( 1 E p+i 1 q+i ) µ = 0 if i 1 and µ > max 0 i min{s q 1,d p} {a p+i (M) + σ q+i+1 } so that for such a µ, H p m(k q ) µ ( E p q) µ. The other spectral sequence has as second terms: 2 E p i = { H p m (H i ) if s i q, 0 else, and (i) follows as 2 E p+i q+i = 0 if i < 0 and ( 2 E p+i q+i ) µ = 0 for i 0 by hypothesis. For (ii) we consider the complex K q : 0 K q K q 1 K 0 0. The first terms of the spectral sequences derived from C mk q are now 1 E p i = { H p m (K i ) if 0 i q, 0 else, 19