LENGTH FORMULAS FOR THE HOMOLOGY OF GENERALIZED KOSZUL COMPLEXES

LENGTH FORMULAS FOR THE HOMOLOGY OF GENERALIZED KOSZUL COMPLEXES

BOGDAN ICHIM and UDO VETTER

Let M be a finite module over a noetherian ring R with a free resolution of length 1. We consider the generalized Koszul complexesCλ¯(t) associated with a map ¯λ:M→ Hinto a finite freeR-moduleH(see [8], Section 3), and investigate the homology of Cλ¯(t) in the special setup when gradeIM = rankM = dimR. (IM is the first non-vanishing Fitting ideal of M.) In this case the (interesting) homology ofC¯λ(t) has finite length, and we deduce some length formulas. As an application we give a short algebraic proof of an old theorem due to Greuel (see [5], Proposition 2.5). We refer to [6] where one can find another proof by similar methods.

AMS 2000 Subject Classification: 13D25.

Key words: Koszul complex, Koszul bicomplex, homology.

1. INTRODUCTION

LetR be a noetherian ring and M an R-module with a presentation 0 −−−−→ F −−−−→ G −−−−→^χ M −−−−→ 0,

whereF,Gare free modules of finite rank. We consider anR-homomorphism

¯λ:M → Hinto a finite freeR-moduleH. With ¯λwe associate the generalized Koszul complexes Cλ¯(t). The main object of our interest is the homology of these complexes in the special setup when gradeI_χ = rankM = dimR. Here I_χ denotes the ideal generated by the maximal minors of a matrix repre- senting χ.

In Section 2 we give a short survey of generalized Koszul complexes and Koszul bicomplexes. Section 3 contains some results concerning the grade sensitivity of these and some related complexes.

The main part of the paper is concentrated in Section 4. In case gradeI_χ has the greatest possible value rankM + 1, Theorem 6.10 in [8] provides a satisfactory description of the homology ofCλ¯(t) depending on the gradeh of I_λ, λ:G → H being the corresponding lifted map. In a sense, the situation described above is another extremal case. Here, the homology modulesHⁱ of Cλ¯(t) have finite length for i≤min(h−1,2 rankM), and they are connected

REV. ROUMAINE MATH. PURES APPL.,52(2007),2, 177–199

by some length formulas which mainly involve the symmetric powers of the cokernel ofχ^∗ (Theorem 4.1 and Corollary 4.3).

In Section 5 we treat the very special situation of a quasi-homogeneous isolated complete intersection singularity. We can extend the length formulas of Section 4, in order to give a purely algebraic proof of an old theorem due to Greuel (see [5], 2.5). The formulas and – at least implicitly – an algebraic proof of Greuel’s Theorem are also contained in [6].

We use [4] for a general reference in commutative algebra. As there, we denote by

M (_p

M) the exterior power algebra (pth exterior power) of an R-module M, by S(M) (S_p(M)) the symmetric power algebra (pth symmetric power) of M, and by D(M) (D_p(M)) the divided power algebra (pth divided power) of M. The reader should always have in mind that for a finite freeR-moduleF there are canonical isomorphismsD(F^∗)∼=S(F)^∗ and D(F) ∼= S(F^∗)^∗ which we shall use implicitly several times in the following.

HereS(F)^∗=⊕S_p(F)^∗ is the so called graded dual ofS(F).

2. KOSZUL COMPLEXES AND KOSZUL BICOMPLEXES Generalizations of the classical Koszul complex have been known since a long time, e.g. the complexes due to Eagon and Northcott or those introduced by Buchsbaum and Rim. We refer to [4], Chapter A2.6, for a comprehensive treatment. Some related material may also be found in [2] and [3].

So the following constructions, generalizing the classical Koszul complex and its dual, are not really new. They should be viewed as appropriate compo- nents of the Koszul bicomplex, we define at the end of this section and which is the main tool of our investigations.

Let R be a commutative ring and ψ : G → F a homomorphism of an R-moduleGinto a finite freeR-moduleF. Letf₁, . . . , f_m be a basis ofF and f₁^∗, . . . , f_m^∗ the dual basis of F^∗. Then we may consider ψ as an element of G^∗⊗S(F) with the presentation ψ=_m

j=1ψ^∗(f_j^∗)⊗f_j.Set

∂_ψ(y₁∧ · · · ∧y_n⊗z) =

j

y₁∧ · · · ∧y_n ψ^∗(f_j^∗)

⊗z·f_j.

for all y_i ∈ G and z ∈ S(F)^∗. The right multiplication of

G^∗ on G is given by

y₁∧ · · · ∧y_n y₁^∗∧ · · · ∧y_p^∗=

σ

ε(σ) det

1≤i, j≤p(y^∗_j(y_σ(i)))y_σ(p+1)∧ · · · ∧y_σ(n) for y₁, . . . , y_n ∈ G and y₁^∗, . . . , y_p^∗ ∈ G^∗, where σ runs through the set Sn,p

of permutations ofn elements which are increasing on the intervals [1, p] and

[p+1, n]. Thegeneralized Koszul complexeswe have in mind, are the complexes Cψ(t) : · · · −→_t+m+p

G⊗S_p(F)^∗ −→ · · ·^∂ψ −→^∂ψ _t+m

G⊗S₀(F)^∗

ν_ψ

−→_t

G⊗S₀(F)−→ · · ·^∂ψ −→^∂ψ ₀

G⊗S_t(F)−→0;

ν_ψ is the right multiplication byψ^∗(f₁^∗)∧ · · · ∧ψ^∗(f_m^∗)∈ G^∗.

Similarly, we can associate complexes with a map ϕ : H → G from a finite free R-module H into G, generalizing the dual version of the classical Koszul complex. Let h₁, . . . , h_l be a basis of H and h^∗₁, . . . , h^∗_l the dual basis of H^∗. Then ϕ, as an element of S(H^∗)⊗

G, has the presentation ϕ = _l

j=1h^∗_j⊗ϕ(h_j).Defined_ϕ to be the left multiplication byϕonD(H)⊗ G, i.e.,

d_ϕ(x^(k₁ ¹⁾. . . x^(k_p^p)⊗y) =

j

x^(k₁ ¹⁾. . . x^(k_j ^j−1). . . x^(k_p^p)⊗ϕ(x_j)∧y forx_i∈H and y∈

G, and onS(H^∗)⊗

G(in an obvious way). We obtain the family of complexes

Dϕ(t) : 0−→D_t(H)⊗₀

G−→ · · ·^dϕ −→^dϕ D₀(H)⊗_t G−→^νϕ

ν^ϕ

−→S₀(H^∗)⊗_t+l

G−→ · · ·^dϕ −→^dϕ S_p(H^∗)⊗_t+l+p

G−→ · · ·, whereν^ϕ is the left multiplication by ϕ(h₁)∧ · · · ∧ϕ(h_m)∈

G.

Now, let ψ◦ϕ = 0. Then there is a simple associativity formula (see Proposition 2.1 in [8]) concerning the right and left multiplications from above, which allows us to assemble the complexes Cψ(t) and Dϕ(t) to the Koszul bicomplexesK.,.(t):

... ... ... ...

· · ·H⊗^t+mG⊗F^∗−−−−−−→^t+m+1G⊗F^∗−−−−−−→^{±νϕ t+l+m+1}G⊗F^∗−−−−−−→H^∗⊗^t+l+m+2G⊗F^∗· · ·

∂ψ ∂ψ

· · ·H⊗^t+m−1G −−−−−−→^{dϕ t+m}G −−−−−−→^{±νϕ t+l+m}G −−−−−−→^dϕ H^∗⊗^t+l+m+1G· · ·

±νψ ±νψ ±νψ ±νψ

· · ·H⊗^t−1G −−−−−−→^{dϕ t}G −−−−−−→^{±νϕ t+l}G −−−−−−→^dϕ H^∗⊗^t+l+1G· · ·

∂ψ ∂ψ

· · ·H⊗^t−2G⊗F −−−−−−→ ^t−1G⊗F −−−−−−→^{±νϕ t+l−1}G⊗F −−−−−−→ H^∗⊗^t+lG⊗F· · ·

... ... ... ...

The rows in the upper half arise from Dϕ(t+m+j) tensored with S_j(F)^∗, j = 0,1, . . ., while the rows below are built from Dϕ(t−j) tensored with S_j(F), j = 0,1, . . .; we abbreviate d_ϕ ⊗1_S(F∗) and d_ϕ ⊗1_S(F) to d_ϕ, and correspondinglyν^ϕ⊗1_S(F∗) and ν^ϕ⊗1_S(F) to ν^ϕ. The columns are obtained analogously: in western direction we have to tensorizeD_i(H) with Cψ(t−i), i = 0,1, . . ., while going east we must tensorize S_i(H^∗) with Cψ(t+l+i), i= 0,1, . . .; as before we shorten the complex maps to∂_ψ and ν_ψ. The signs ofν^ϕ and ν_ψ are determined by the associativity formula.

A detailed and more general treatment of the generalized Koszul com- plexes from above and of the Koszul bicomplexes just defined may be found in [7].

3. GRADE SENSITIVITY

The following contains some preparing material for the main results in the next section. In a sense it is a matter of generalizing part of the considerations of Section 5 in [8].

Our general assumption throughout the rest of this section will be that R isnoetherian, that H, G and F are free R-modules of finite ranks l, nand m, and that

H −−−−→^ϕ G −−−−→^ψ F

is a complex. Although much of what we will do holds formally for any l,n and m, the applications will refer to the case in which n≥m and n≥l. So we suppose that r = n−m ≥ 0, s = n−l ≥ 0. By I_ψ (I_ϕ) we denote the ideals inR generated by the maximal minors of a matrix representingψ (ϕ).

We setg= gradeI_ψ and h= gradeI_ϕ.

Since G is finitely generated, the generalized Koszul complexes Cψ(t) and Dϕ(t) have only a finite number of non-vanishing components. To iden- tify the homology, we fix their graduations as follows: position 0 is held by the leftmost non-zero module. AsCψ(t) and Dϕ(t) are isomorphic to well-known generalizations of the classical Koszul complex (see [8], Section 3), their ho- mology behaves grade sensitively in the sense of the following theorem (see Theorem 5.2 in [8]).

Theorem 3.1. Set C = Cokerψ and D = Cokerϕ^∗. Furthermore, let S₀(D) = R/I_ϕ, S₋₁(D) = _s+1

Cokerϕ, S₀(C) = R/I_ψ and S₋₁(C) = _r+1

Cokerψ^∗. Then the following hold.

(a) Hⁱ(Dϕ(t)) = 0for i < h; moreover, ift≤s+ 1andgradeI_k(ϕ)≥n− k+1for allkwithl≥k≥1, thenDϕ(t) is a free resolution ofS_s−t(D) (if −1≤t≤s+ 1 then it suffices to require that gradeI_ϕ ≥s+ 1.)

(b) Hⁱ(Cψ(t)) = 0 for i < g; moreover, if t ≥ −1 and gradeI_k(ψ) ≥ n−k+ 1 for all k with m ≥k≥1, then Cψ(t) is a free resolution of S_t(C) (if−1≤t≤r+1then it suffices to require thatgradeI_ψ ≥r+1.) Finally, if I_ϕ=R (I_ψ =R), then all sequences Dϕ(t) (Cψ(t)) are split exact.

By C.,.(t) we shall denote the bicomplex which is the lower part of the Koszul bicomplex K.,.(t) (the rows below the second row in the last diagram of the previous section). In other words,

C^0,0=C0,0(t) =









D_t(H)⊗₀

G⊗S₀(F) if 0≤t, S₀(H^∗)⊗_t+l

G⊗S₀(F) if −l≤t <0, S_−t−l(H^∗)⊗₀

G⊗S₀(F) ift <−l.

The row homology atC^p,q=Cp,q(t) is denoted by H_ϕ^p,q, the column homology byH_ψ^p,q. ThusH_ϕ^p,0 is thep-th homology module ofDϕ(t).

SetN^p = Ker (C^p,0→^∂ψ C^p,1). The canonical injections N^p →C^p,0 yield a complex morphism

0 −−−−→ N⁰ −−−−→ N¹ −−−−→ · · · N^p −−−−→^d¯ϕ N^p+1 · · ·

  

0 −−−−→ C^0,0 −−−−→ C^1,0 −−−−→ · · ·C^p,0 −−−−→^dϕ C^p+1,0· · ·,

where the maps ¯d_ϕ are induced by d_ϕ. The homology of the first row N(t) at N^p is denoted by ¯H^p. We shall now investigate this homology under the assumptionr ≤g.

For this purpose we extend C.,.(t) by N(t) to the complex C.,.(t). So, C^p,−1 =C_p,−1(t) = N^p. We record some facts about the homology of C.,.(t).

To avoid new symbols, the column homology atC^p,q is again denoted byH_ψ^p,q (actually it differs from that ofC.,.(t) only atC^p,0). By construction,H_ψ^p,q= 0 forq=−1,0. Furthermore, we draw from Theorem 3.1 (a) that

(1) H_ϕ^p,q= 0 forp < h and q=−1.

Letr≤g. Then in case 0< p≤twe get from Theorem 3.1 (b) that (2) H_ψ^p,q=

0 forq <min(p−1, r), D_t−p(H)⊗S_p(C) forq =p,

and, if 0≤t < p, we obtain (3) H_ψ^p,q=

0 forq <min(p+l−2, r), S_p−t−1(H^∗)⊗S_p+l−1(C) forq =p+l−1,

whereC= Cokerψ as in Theorem 2.1.

For q ≥ −1 consider the qth row C_.,q of C_.,. and its image complex

∂_ψ(C_.,q) in C_.,q+1. Set E^p,q = H^p(∂_ψ(C_.,q−1)) for q ≥ 0. There are exact sequences

(4) H_ϕ^i−(j+1),j −→Ei−(j+1),j+1 −→E^i−j,j −→H_ϕ^i−j,j ifH_ψ^i−(j+1),j =H_ψ^i−j,j= 0, and because of (2) and (3) this holds if

0< i−(j+ 1)≤t and j <min

i−(j+ 1)−1, r or

0≤t < i−(j+ 1) and j <min

i−(j+ 1) +l−2, r .

Theorem 3.2. Let t≥0 be an integer. Assume that 1≤ r ≤g. Then, with the notation introduced above, H¯ⁱ = 0 for i = 0, . . . ,min(2, h−1). Set C= Cokerψ.

(a) For i odd, 3 ≤ i < min(h−1,2r,2t + 2), one has a natural exact sequence

0−→H¯ⁱ −→D_t−i−1

2 (H)⊗Si−1

2 (C)−→H_ψ^{i+12 ,i−12} −→H¯ⁱ⁺¹ −→0.

(b) Suppose that l >1.

(i) If 3≤2t+ 1< hthen H¯^2t+1=D₀(H)⊗S_t(C).

(ii) ¯Hⁱ= 0 for 2t+ 2≤i <min(h,2t+l+ 1).

(iii) If 2t+l+ 1< h then H¯^2t+l+1=H_ψ^t+1,t+l−1.

(c) Fori−l even,2t+l+ 2≤i <min(h−1,2r−l+ 2), one has a natural exact sequence

0−→H¯ⁱ−→Si−l

2 −t−1(H^∗)⊗Si+l

2 −1(C)−→H

i−l2 +1,^i+l₂ −1

ψ −→H¯ⁱ⁺¹−→0.

Proof. For the proof of the first statement we refer to [8], Theorem 5.3.

Leti be an odd integer, 3 ≤i <min(h−1,2r,2t+ 2). Using (4), we obtain the “southwest” isomorphisms

H¯ⁱ =E^i,0 ∼=E^i−1,1 ∼=· · · ∼=E^{i+12 ,i−12} and

H¯ⁱ⁺¹ =E^i+1,0 ∼=E^i,1 ∼=· · · ∼=E^{i+32 ,i−12} .

We abbreviate ∂_ψ^p,q = (C^p,q −→^∂ψ C^p,q+1) and d^p,q_ϕ = (C^p,q −→^dϕ C^p+1,q). The diagram

0 0 0 0



  

−−−−→Im∂

i−12 ,ⁱ⁻³₂

ψ −−−−→ Im∂

i+12 ,ⁱ⁻³₂

ψ −−−−→ Im∂

i+32 ,ⁱ⁻³₂

ψ −−−−→ Im∂

i+52 ,ⁱ⁻³₂

 ψ

   

−−−−→Im∂_ψ^{i−12 ,i−32} −−−−→Ker∂_ψ^{i+12 ,i−12} −−−−→Ker∂_ψ^{i+32 ,i−12} −−−−→Ker∂_ψ^{i+52 ,i−12}



  

0 −−−−→ H_ψ^{i+12 ,i−12} −−−−→ 0 −−−−→ 0

 0

is induced byC.,.(t) and has exact columns. Its row homology at Im∂

i+12 ,ⁱ⁻³₂

ψ is

E^{i+12 ,i−12} = ¯Hⁱ, and at Ker∂_ψ^{i+12 ,i−12} it coincides withD_t−i−1

2 (H)⊗Si−1 2 (C) as shown by the diagram below, with exact columns and exact middle row (we writeD_t−i−1

2 forD_t−i−1

2 (H) and Si−1

2 forSi−1 2 (C)):

0 0 0 0



  

Im∂_ψ^{i−12 ,i−32} −−−−→ Ker∂_ψ^{i+12 ,i−12} −−−−→ Ker∂_ψ^{i+32 ,i−12} −−−−→ Ker∂_ψ^{i+52 ,i−12}



  

0−→ Cⁱ⁻¹2 ,ⁱ⁻¹₂ −−−−→ Cⁱ⁺¹2 ,ⁱ⁻¹₂ −−−−→ Cⁱ⁺³2 ,ⁱ⁻¹₂ −−−−→ Cⁱ⁺⁵2 ,ⁱ⁻¹₂



  

0−→D_t−i−1 2 ⊗Si−1

2 −−−−→ Im∂_ψ^{i+12 ,i−12} −−−−→^dϕ Im∂_ψ^{i+32 ,i−12} −−−−→ Im∂_ψ^{i+52 ,i−12}



  

0 0 0 0

In this diagram, the row homology at Im∂

i+12 ,ⁱ⁻¹₂

ψ vanishes since d

i+12 ,ⁱ⁺¹₂ ϕ

is injective. So, the row homology at Ker∂_ψ^{i+32 ,i−12} is zero. Altogether, we

obtain an exact sequence (∗) 0→H¯ⁱ→D_t−i−1

2 (H)⊗Si−1

2 (C)→H

i+12 ,ⁱ⁻¹₂

ψ →E^{i+32 ,i−12} →0.

Since

H¯ⁱ⁺¹∼=E^{i+32 ,i−12} , (a) has been proved.

For (b) we note thatl >1 implies g≥r > r−l+ 1, so we may use [8], Theorem 5.3.

The proof of (c) is similar to the proof of (a). We just mention that for alliunder consideration, we have the “southwest” isomorphisms

H¯ⁱ =E^i,0 ∼=E^i−1,1 ∼=· · · ∼=E^{i−l2 +1,i+l2 −1}, H¯ⁱ⁺¹ =E^i+1,0 ∼=E^i,1 ∼=· · · ∼=E^{i−l2 +2,i+l2 −1}.

If t is a negative integer, we obtain a similar result. We indicate it without the proof which is completely analogous to the proof of the previous theorem.

Theorem3.3. Let t <0 be an integer. Assume that 1≤r≤g, and use the notation from above.

(a) Suppose that t+l >0 (this implies l >1). Then (i) ¯Hⁱ= 0 for 0≤i <min(h,max(2, t+l));

(ii) if 2≤t+l < h then H¯^t+l=H_ψ^0,t+l−1;

(iii) ifi−t−l is odd, and t+l+ 1≤i < h−1, then one has a natural exact sequence

0→H¯ⁱ→Si−t−l−1

2 (H^∗)⊗Si+t+l−1

2 (C)→H

i−t−l+1 2 ,^i+t+l−1₂

ψ →H¯ⁱ⁺¹→0.

(b) Suppose that t+l≤0. Then H¯ⁱ = 0 for i= 0, . . . ,min(2, h−1). For i odd, 3≤i <min(h−1,2r), one has a natural exact sequence

0→H¯ⁱ →Si−1

2 −t−l(H^∗)⊗Si−1

2 (C)→H

i+12 ,ⁱ⁻¹₂

ψ →H¯ⁱ⁺¹→0.

We supply Theorem 3.2 by some simple results concerning the homology ofN(t) at N^h.

Proposition 3.4. As in Theorem 3.2 assume that 1 ≤ r ≤ g and set C= Cokψ. Let µ= min(h,2r+ 1) and t≥ ^µ₂ −1.

(a) There is an exact sequence 0 →E^µ−1,1 → H¯^µ → H_ϕ^µ,0; in particular, if µ <3 we get an exact sequence 0→H¯^µ→H_ϕ^µ,0.

(b) Forµ≥3 odd, there is an exact sequence 0→E^µ−1,1 →D_t−µ−1

2 (H)⊗Sµ−1

2 (C)→H

µ+12 ,^µ−1₂

ψ .

(c) Forµ≥3 even, there is an exact sequence 0→H¯^µ−1 →D_t−µ−2

2 (H)⊗Sµ−2

2 (C)→H

µ2,^µ−2₂

ψ →H¯^µ→H_ϕ^µ,0.

Remark 3.5. One may easily deduce similar sequences in caset < ^µ₂ −1.

Proof. (a) This follows immediately from (4) and the fact that H_ψ^p,0 = H_ψ^p,1= 0 for all p.

(b) In order to cover the caseµ= 2r+ 1, we modify the first diagram in the proof of Theorem 3.2: the diagram

0 0 0 0



  

−→Im∂

µ−12 ,^µ−3₂

ψ −−−−→Im∂

µ+12 ,^µ−3₂

ψ −−−−→Im∂

µ+32 ,^µ−3₂

ψ −−−−→ Im∂

µ+52 ,^µ−3₂

 ψ

   

−→Im∂

µ−12 ,^µ−3₂

ψ −−−−→Ker∂

µ+12 ,^µ−1₂

ψ −−−−→Ker∂

µ+32 ,^µ−1₂

ψ −−−−→ Ker∂

µ+52 ,^µ−1₂

 ψ

   

0 −−−−→ H

µ+12 ,^µ−1₂

ψ −−−−→ H

µ+32 ,^µ−1₂

ψ −−−−→ H

µ+52 ,^µ−1₂

 ψ

  

0 0 0

has exact columns. Then, as in the proof of Theorem 3.2, we obtain an exact sequence

0→E^{µ+12 ,µ−12} →D_t−µ−1

2 (H)⊗Sµ−1 2 (C)→

→Ker

H

µ+12 ,^µ−1₂

ψ →H

µ+32 ,^µ−1₂ ψ

→E^{µ+32 ,µ−12} .

SinceE^{µ+12 ,µ−12} ∼=E^µ−1,1 in this case, we are done.

(c) Seti= µ−1 and use the sequence (∗) in the proof of Theorem 3.2 to get the exact sequence

0→H¯^µ−1 →D_t−µ−2

2 (H)⊗Sµ−2

2 (C)→H

µ2,^µ₂

ψ →E^{µ+22 ,µ−22} →0.

SinceE^µ+22 ,^µ−2₂ ∼=E^µ−1,1,we can glue this sequence and the sequence obtained under (a) to get the result.

4. LENGTH FORMULAS

In this section,R is a noetherian ring, andM an R-module which has a presentation

0 −−−−→ F −−−−→ G −−−−→^χ M −−−−→ 0,

where F, G are free modules of ranks m and n. Then, in particular, r = n−m≥0.

Let ¯λ:M → H be an R-homomorphism into a finite free R-moduleH of rankl≤n. Byλ:G → Hwe denote the corresponding lifted map. In case gradeI_χ has the greatest possible value r+ 1, Theorem 6.10 in [8] provides a comparably satisfactory description of the homology of the generalized Koszul complex Cλ¯(t):

0→_n

M⊗S_p(H)^∗ −→ · · ·^∂λ¯ −→^∂¯λ _t+l

M −→^νλ¯ _t

M −→ · · ·^∂¯λ −→^∂λ¯ S_t(H)→0, depending on the grade ofI_λ.

In the following we deal with another extremal case. We assume that gradeI_χ = dimR. ThenR/I_χhas finite length(R/I_χ). (Generally, the length of anR-moduleN is denoted by(N).)

First, we dualize the sequence F → G^χ → H^λ . Then we setF =F^∗, G= G^∗, H =H^∗ and ψ=χ^∗, ϕ=λ^∗, to attain the setup which we studied in the previous sections.

We recall some notation and facts from Section 6 in [8]. As there, we consider the upper part of the Koszul bicomplexK.,.(t) which we rewrite as

... ... ...



 

· · · −−−−→ B_t^0,−1 −−−−→ · · · −−−−→^dϕ B_t^t,−1 −−−−→^±νϕ B_t^t+1,−1 −−−−→ · · ·^dϕ



^∂ψ ^∂ψ

· · · −−−−→ B_t^0,0 −−−−→ · · · −−−−→ B_t^t,0 −−−−→^±νϕ B_t^t+1,0 −−−−→ · · ·, where

B_t^0,0 =









D_t(H)⊗_m

G⊗S₀(F)^∗ if 0≤t, S₀(H^∗)⊗_t+l+m

G⊗S₀(F)^∗ if−l≤t <0, S_−t−l(H^∗)⊗_m

G⊗S₀(F)^∗ ift <−l.

SetM^p = Coker(B_t^p,−1 →^∂ψ B_t^p,0). The canonical surjection B_t^p,0 → M^p yields a complex morphism

· · · −→B_t^−1,0 −−−−→ B_t^0,0 −−−−→ · · · −−−−→^dϕ B_t^p,0 −−−−→ B^p+1,0_t −→ · · ·



  

· · · −→M⁻¹ −−−−→ M⁰ −−−−→ · · · −−−−→^d¯ϕ M^p −−−−→ M^p+1−→ · · ·, where the maps ¯d_ϕ are induced by d_ϕ. The lower row is denoted by M(t).

By [8], Proposition 6.8, there is an isomorphism M(ρ −t) → Cλ¯(t) whose inverse composed with the induced complex mapM(ρ−t)→ N(ρ−t) yields a complex mapµ:C¯λ(t)→ N(ρ−t).

Theorem 4.1. Let M, ¯λ, and λ be as above. Set ρ = r −l, g = gradeI_χ, h = gradeI_λ. Suppose that g = dimR = r. Equip C¯λ(t) with the graduation induced by the complex morphismµ:C¯λ(t)→ N(ρ−t). Then the homology modulesHⁱ of C_¯^.

λ(t) have finite length for i≤min(h−1,2r).

Set C= Cokerχ^∗, S₀(C) =R/I_χ, and assume that h >0.

(a) Let l= 1. Then for all t∈Z and iodd, 0< i <min(h−1,2r), (Hⁱ)−(Hⁱ⁺¹) =(Si−1

2 (C))−(Si+1 2 (C)).

(b) Let l >1. We distinguish four cases.

(i) For all t≤ ^ρ₂ and i odd, 0< i < h−1, (Hⁱ)−(Hⁱ⁺¹) =

r−t−ⁱ⁺¹₂ l−1

(Si−1

2 (C))−

r−t−ⁱ⁺³₂ l−1

(Si+1

2 (C)).

(ii) Suppose that ^ρ₂ < t≤ρ. If i is odd, 0< i <min(h−1,2(ρ−t)), then

(Hⁱ)−(Hⁱ⁺¹) =

r−t−ⁱ⁺¹₂ l−1

(Si−1

2 (C))−

r−t−ⁱ⁺³₂ l−1

(Si+1

2 (C)).

If i−l is even,2(ρ−t) +l+ 2≤i < h−1, then (Hⁱ)−(Hⁱ⁺¹) =

_i+l

2 −ρ+t−2 l−1

(Si+l

2 −1(C))−

− _i+l

2 −ρ+t−1 l−1

(Si+l

2 (C)).

If2(ρ−t)+l+1< hthen(H^{2(ρ−t)+l+1}) =(S_r−t(C)). Moreover, Hⁱ =

S_ρ−t(C) if i= 2(ρ−t) + 1< h,

0 if 2(ρ−t) + 2≤i <min(h,2(ρ−t) +l+ 1).

(iii) Suppose that ρ < t < r. If i+r−t is odd, r−t+ 1≤i < h−1, then

(Hⁱ)−(Hⁱ⁺¹) =

_i−r+t−3

2 +l

l−1

(Si+r−t−1 2 (C))−

−

_i−r+t−1

2 +l

l−1

(Si+r−t+1 2 (C)).

If r −t < h then (H^r−t) = (S_r−t(C)). Moreover, Hⁱ = 0 if 0≤i <min(h, r−t).

(iv) Suppose that r≤t and iodd, 0< i < h−1. Then

(Hⁱ)−(Hⁱ⁺¹) =

t−ρ+ⁱ⁻³₂ l−1

(Si−1

2 (C))−

t−ρ+ ⁱ⁻¹₂ l−1

(Si+1

2 (C)).

Remark 4.2. Remark that in the above formulas we use the fact that h can reach its maximal value only if it is even. More precisely, if l ≥ 2 and g= h=r, then l= 2 and r, h, g are even (see Corollary 6.2 in [8]). In this case,iodd,i < h−1, meansi≤r−3.

We further notice that forh <∞ the formulas under (b) cover the case where l = 1. We specified them for the readers convenience since we shall apply thel= 1 case in Section 4.

Proof. Ifr = 0 thenM = 0 sinceχis injective. So, we may assume that r≥1.

The graduation induced byN(ρ−t) onCλ¯(t) is completely determined by

C_¯λ⁰(t) =







 _r

M⊗S_ρ−t(H)^∗ ift≤ρ, _t

M⊗S₀(H) ifρ < t < r, _r

M⊗S_t−r(H) ifr≤t.

Forq > r the support of_q

M is contained in the variety ofI_χ. Consequently, Cⁱ_¯

λ(t) has finite length ifi <0, which in turn implies thatHⁱ has finite length.

In particular, there remains nothing to prove ifh= 0. Leth >0. Thenρ≥0 by Proposition 5.1. in [8].

By [8], Proposition 6.9, µ: Cλ¯(t) → N(ρ−t) induces the commutative diagram

0 0 0



  C_¯⁻¹

λ

∂_¯λ⁻¹

−−−−→ C_λ¯^{0 ∂}

¯0

−−−−→λ C_λ¹_¯ −−−−→ C_λ¯² −−−−→ C_¯λ³

µ0



^µ1 ^µ2 ^µ3 0 −−−−→ N^{0 d¯}

0ϕ

−−−−→ N¹ −−−−→ N² −−−−→ N³



   0 −−−−→ Cokerµ₀ −−−−→^α Cokerµ₁ −−−−→ 0 −−−−→ 0





0 0

with exact columns.

For arbitrary t and arbitrary i, the maps µ_i are isomorphisms at all prime ideals which do not containI_χ. Consequently, Kerµ_i and Cokerµ_i have finite length. In particular, Kerµ₀ equals the torsion submodule ofC⁰_¯

λ since N⁰ is free. On the other hand, C_¯¹

λ is a torsion free module. So, the torsion submodule ofC_¯⁰

λ is contained in Ker∂_¯⁰

λ. If we denote by ¯µ₀ and ¯∂_¯⁰

λ the maps induced byµ₀ and∂_λ⁰_¯ onC_λ⁰_¯/Kerµ₀, then we get the commutative diagram

0 0 0 0



   C⁰_¯

λ/Kerµ₀ ^∂¯

¯0

−−−−→λ C¹_¯

λ −−−−→ C²_¯

λ −−−−→ C_¯³

λ

¯ µ0



^µ1 ^µ2 ^µ3 0 −−−−→ N^{0 d¯}

0ϕ

−−−−→ N¹ −−−−→ N² −−−−→ N³



   0 −−−−→ Cokerµ₀ −−−−→^α Cokerµ₁ −−−−→ 0 −−−−→ 0





0 0

with exact columns.