ALGEBRAIC AND ARITHMETIC RESIDUAL INTERSECTIONS

arXiv:1405.4586v1 [math.AC] 19 May 2014

ALGEBRAIC AND ARITHMETIC RESIDUAL INTERSECTIONS 2 ´ S. H. HASSANZADEH1 AND J. NAELITON

Abstract. Several long-standing questions (conjectures) about the structure of the residual intersections are answered (proved). More precisely for an s-residual intersection J = a : I in a Cohen-Macaulay ring R where I satisfies some sliding depth conditions on its Koszul homology modules (e.g. Strongly Cohen-Macaulay), it is shown that J is unmixed, depth(R/a) = dim(R)− s, Symk (I/a) is CM for all 1 ≤ k ≤ s − g + 2 and an upper bound for the type of Symk (I/a), 1 ≤ k ≤ s − g + 1, is presented. The concept of arithmetic s-residual intersection (a special kind of the algebraic s-residual which contains the geometric one) is introduced and it is shown that if J is an arithmetic sresidual and R is Gorenstein then the canonical module of R/J is Syms−g+1 (I/a). In quite general setting and for any residual intersection, a linear sharp upper bound for the Castelnuovo-Mumford regularity of Symk (I/a) is presented. Among others, it is also clarified why the Hilbert functions of R/a, Symk (I/a) for all 1 ≤ k ≤ s−g +1, and R/J (in the arithmetic case) are dependent just on the Koszul homology modules of I and the degrees of the generators of a.

1. Introduction The concept of residual intersection essentially goes back to Artin and Nagata [AN]. The notion is a vast generalization of linkage (or liaison) which is more ubiquitous, but also harder to understand, than linkage. Let X and Y be two irreducible closed subschemes of a Noetherian scheme Z with codimZ (X) ≤ codimZ (Y ) = s and Y 6⊂ X, then Y is called a residual intersection

of X if the number of equations needed to define X ∪ Y as a subscheme of Z is the smallest

possible, i.e. s. However to become clearer and also to include the reducible schemes and permit the existence of embedded components we state the following definition: Definition 1.1. Let R be a Noetherian ring, I an ideal of height g and s ≥ g an integer.

• An (algebraic) s-residual intersection of I is a proper ideal J of R such that ht(J) ≥ s and J = (a :R I) for some ideal a ⊂ I which is generated by s elements.

Date: May 20, 2014. 0

Mathematics Subject Classification 2000 (MSC2000). Primary 13C40, 13H15, 13D02, Secondary 14C17.

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Partially supported by a CNPq grant.

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Under a CNPq Doctoral scholarship. 1

´ S.H.HASSANZADEH AND J. NAELITON

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• A geometric s-residual intersection of I is an algebraic s-residual intersection J of I such that ht(I + J) ≥ s + 1.

The theory of residual intersections entered to the center of interest since the late 80’s. Its development was along other progresses in commutative algebra and algebraic geometry such as deformation theory[HU1] and families of approximation complexes [HSV],[KU]. The deep works of Huneke, Ulrich, Eisenbud, Chardin, Kustin, Miller among others, such as [HU], [KU],[BKM], [U] etc. show the importance and the intricacy of the problems involving the residual intersection, as well as in [BEM],[CU],[CPU],[E],[CEU], as some examples, one finds applications of residual intersections in the theories of resultants, rational singularities, multiplicities, projective dimensions, secant varieties, etc. One should add to the above lists the very recent works [CEU1],[EU],[HHU] which show the renewed interest to the subject. Many questions can be asked about the structure of residual intersections aiming to determine properties of J in terms of information of I and a. However undoubtedly the following are among the most important open questions which still wait for satisfactory answers: (1) When is R/J Cohen-Macaulay? (2) What is the structure of the canonical module of R/J, in the case where it makes sense? (3) What is the Hilbert series of R/J? (4) What is the minimal free resolution of R/J? It is quite convenient to suppose that R is a Cohen-Macaulay local (abbr. CM) ring [AN]. However in the attempts so far as we can tell the ideal I comes endowed with two conditions. One is the Artin-Nagata Gs condition which bounds the local number of generators of an ideal; more precisely, we say that an ideal I satisfies the condition Gs , if µ(Ip ) ≤ ht(p) for all prime ideal

p containing I such that ht(p) ≤ s − 1. The second condition is a request to have enough depth

for Koszul homology modules of I such as Strongly Cohen-Macaulay (SCM) and Sliding Depth

condition (SD)(see Definition 2.2 also for SDC condition). The point is that in the presence of both of these conditions the involved ideals are generated by some especial sequences such as regular sequences, proper sequences or d-sequences [U], [HSV]. Of course by mentioning this we don’t want to diminish the achievements obtained by now, in contrary these occurrence give a substantial role to the theory of residual intersection in the study of the arithmetic of blow-up algebras [V]. Some sort of depth assumptions were known to be necessary to proving the Cohen-Macaulayness of residual, [U], [H]. On the other hand removing the Gs condition remained as the main challenge in the theory, since it is essentially a technical hypothesis in the cases where the object of study is a single s-residual intersection and not a sequence of geometric i-residual intersections

ALGEBRAIC AND ARITHMETIC RESIDUAL INTERSECTIONS

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for all g ≤ i ≤ s. During the development of the theory, the following long-standing assertions were implicitly conjectured:

Conjecture. Let R be a Cohen-Macaulay(CM) local ring and I satisfy SCM or even SD. Then (1) R/J is Cohen-Macaulay, [HU]. (2) The canonical module of R/J is the (s − g + 1)th symmetric power of I/a, if R is Gorenstein.

(3) The Hilbert series of R/J depends only on I and the degrees of the generators of a, [CEU]. The fourth mentioned question in the above paragraph is far out of reach to be conjectured yet (but see [BKM] and [KU]). The first author could prove the Cohen-Macaulay-ness of R/J in [Ha] in the case where J = a : I is a geometric s-residual intersection. The simplification behind the geometric hypothesis is that this hypothesis does not allow irreducible components of J to contain I. However what has been proved in [Ha] is a little bit more general than the geometric residual. In fact conjecture (1) has been proved for a third variation of residual intersections whom we call arithmetic s-residual intersections: Definition 1.2. Let R be a Noetherian ring, I an ideal of codimension g and let s ≥ g be an integer. An arithmetic s-residual intersection J = (a : I) is an algebraic s-residual intersection

such that µRp ((I/a)p ) ≤ 1 for all prime ideal p with ht(p) ≤ s. ( µ denotes the minimum number of generators.)

Clearly any geometric s-residual intersection is arithmetic and any arithmetic one is algebraic. Also we notice that there exist as many arithmetic s-residual intersections as algebraic ones. Indeed for any algebraic s-residual intersection J = (a : (f1 , · · · , fr )) which is not geometric all

of the colon ideals (a : fi ) are arithmetic s-residual intersection and at least one of them is not geometric. The theorem proved in [Ha] is the following: Theorem 1.3. [Ha, Theorem 2.11]Suppose that (R, m) is a CM local ring and that J = a : I is an s-residual intersection of I. If I satisfies SDC1 at level min{s − g, r − g}, then there exists

an ideal K ⊆ J such that

(1) K is CM of height s; √ √ (2) K = J; (3) K = J, whenever the residual is arithmetic.


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The proof of this theorem makes use of a finite complex of not necessarily free modules. This approximation complex is a good substitution for free resolutions of residual intersections, which is sufficient to obtain results on the Castelnuovo-Mumford regularity and other numerical invariants of residual intersections. In the current paper we introduce a family of complexes {i Z•+ }∞ i=0 which approximate kth

symmetric powers of I/a, SymkR (I/a). The complex which was applied to prove Theorem 1.3 is just 0 Z•+ . These complexes work very well in any commutative Noetherian ring, however to

prove the acyclicity of i Z•+ , we appeal to the acyclicity lemma of Peskine-Szpiro; so that we impose the CM assumption on the ring and the sliding depth conditions on I.

The output of these constructions and assumptions are the proofs of the above mentioned conjectures. Summing up, a flavor of our main results are the following Results. Let (R, m) be a CM local ring of dimension d, I an ideal with ht(I) = g > 0. Let s ≥ g, 1 ≤ k ≤ s − g + 2 and J = (a : I) any (algebraic) s-residual intersection. Suppose that I

is SCM . Then

• k Z•+ is acyclic, H0 (k Z•+ ) = Symk (I/a) and the latter is CM of dimension d − s; the acyclicity of k Z•+ implies conjecture(3). (Theorem 2.5 and Proposition 3.1).

• depth(R/a) = d − s; this equation is crucial in the theory (Corollary 2.8). • J is unmixed of codimension s (Corollary 2.8).

• ωR/J = Syms−g+1 (I/a), if the residual is arithmetic and R is Gorenstein; compare with conjecture(2) (Corollary 2.12)

In comparison, obtaining the structure of the canonical module in the absence of Gs condition is more astonishing than the other results. To achieve this we show that under the above mentioned hypotheses, Syms−g+1 (I/a) is a faithful maximal Cohen-Macaulay R/J-module of type 1. Indeed we prove even more we show in Theorem 2.10 that the following type inequality holds for any 1 ≤ k ≤ s − g + 1, k

rR (Sym (I/a)) ≤

r+s−g−k rR (R). r−1

As was already stated, the free resolution of R/J is not still predictable. Although in the case where I is a complete intersection a slight modification in the structure of the family {k Z•+ } makes

each member of this family a free resolution of its zeroth homology, alas, these free resolutions (in the complete intersection case) are so huge that except the immediate consequences such as projective dimension we could not derive more information. Nevertheless, a careful study of the graded structure of k Z•+ enables us to compute the Castelnuovo-Mumford regularity of H0 (k Z•+ ). In Proposition 3.2 it is shown that under the SD1 condition,


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reg(Symk (I/a)) ≤ reg(R) + dim(R0 ) + σ(a) − (s − g + 1 − k) indeg(I/a) − s. for k ≥ 1.

Also the graded structure of the canonical module of R/J is exploited in Proposition 3.3.

Finally by a combination of older and newer facts, we show that if I is SCM and evenly linked to a Gs ideal then H0 (0 Z•+ ) = R/J for any (algebraic) s-residual intersection J, Proposition 3.6. 2. Residual Intersection of SCM ideals In this section we introduce a family of complexes which approximate the residual intersection + and some of its related symmetric powers. We denote this family by {i Z•+ }∞ i=0 . The complex 0 Z•

was already defined in [Ha] and used to prove the CM-ness of arithmetic residual intersection of ideals with sliding depth. Throughout this section R is a Noetherian ring of dimension d, I = (f ) = (f1 , · · · , fr ) is an

ideal of grade g ≥ 1, although by adding one variable we may as well treat the case g = 0 but

for simplicity we keep the assumption g ≥ 1. a = (a1 , · · · , as ) is an ideal contained in I, s ≥ g,

J = a :R I, and S = R[T1 , · · · , Tr ] is a polynomial extension of R with indeterminates Ti ’s.

We denote the symmetric algebra of I over R by SI or in general the symmetric algebra of an R-module M by SymR (M ) and the kth symmetric power of M by SymkR (M ). We consider SI

as an S-algebra via the ring homomorphism S → SI sending Ti to fi as an element of (SI )1 = I P P then SI = S/L. Let ai = rj=1 cji fj , γi = rj=1 cji Tj , γ = (γ1 , · · · , γs ) and g := (T1 , · · · , Tr ).

For a sequence of elements x in a commutative ring A and an A-module M , we denote the

koszul complex by K• (x; M ), its cycles by Zi (x; M ) and homologies by Hi (x; M ). For a graded module M , indeg(M ) := inf{i : Mi 6= 0} and end(M ) := sup{i : Mi 6= 0}. Setting deg(Ti ) = 1 for all i, S is a standard graded ring over S0 = R.

To set one more convention, when we draw the picture of a double complex obtained from a N tensor product of two finite complexes (in the sense of [W, 2.7.1]), say A B; we always put A in the vertical direction and B in the horizontal one. We also label the module which is in the

up-right corner by (0, 0) and consider the labels for the rest, as the points in the third-quadrant. 2.1. k Z•+ complexes. The first object in the construction of the family of k Z•+ complexes is one

of the approximation complexes- the Z-complex [HSV]. We consider the approximation complex Z• (f ):

0 → Zr−1 ⊗R S(1 − r) → · · · → Z1 ⊗R S(−1) → Z0 ⊗R S → 0 where Zi = Zi (f ) is the ith cycle of the Koszul complex K• (f , R). The second object is the Koszul complex K• (γ, S):


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0 → Ks (γ1 , ..., γs )(−s) → · · · → K1 (γ1 , ..., γs )(−1) → K0 (γ1 , ..., γs ) → 0. Let D• = Tot(K• (γ, S) ⊗S Z• (f )). Then, assuming Zj = 0 for j < 0, min{i,r−1}

M

Di =

j=i−s

s [Zj ⊗R S](i−j ) (−i).

(2.1)

For a graded S-module M the kth graded component of M is denoted by M[k] . Let {(D• )[k] }k∈Z

be the kth graded strand of D• . We have (Di )[k] = 0, for all k < i, in particular Hk ((D• )[k] ) = Ker(Dk → Dk−1 )[k] .

(2.2)

ˇ The last object which is applied for the construction is the Cech complex of S with respect to the sequence g = (T1 , · · · , Tr ) denoted by Cg• = Cg (S): 0 → Cg0 (= S) → Cg1 → · · · → Cgr → 0. We now consider the bi-complex Cg• ⊗S D• which rises up to two spectral sequences for which

the second terms of the horizonal spectral are 2

−i,−j Ehor = Hgj (Hi (D• )),

(2.3)

and the first terms of the vertical spectral are ( 0 → Hgr (Dr+s−1 ) → · · · → Hgr (D1 ) → Hgr (D0 ) → 0 if j = r 1 −i,−j (2.4) Ever = 0 otherwise L L Since Hgr (Di ) = Hgr ( j [Zj ⊗ S](−i)) = j Zj ⊗ Hgr (S)(−i) and end(Hgr (S)) = −r, it follows

that end(Hgr (Di )) = i − r, thus Hgr (Di )[i−r+j] = 0, for all j ≥ 1. It then leads to define the

following sequence of complexes indexed by k ≥ 0: 0

/ H r (Dr+s−1 )[k] g

/ ...

/ H r (Dr+k+1 )[k] g

φk

/ H r (Dr+k )[k] g

/ 0

(2.5)

Since the vertical spectral (2.4) collapses at the second step, the horizonal spectral converges to the homologies of Hgr (D• ). Since all of the homomorphisms are homogeneous of degree 0, the convergence will be the case for any graded component. Therefore for any k ≥ 0 there exists a filtration · · · ⊆ F2k ⊆ F1k ⊆ Coker(φk ) such that

Coker(φk ) −k,0 ≃ (∞ Ehor )[k] . F1k

(2.6)


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−k,0 ⊆ Hg0 (Hk (D)), for all Observing that Hg−t (Hh (D)) = 0, for all (t, h) ∈ N × N0 , one has l Ehor

l ≥ 2; Hence we have the following chain of maps for which except the isomorphism in the middle

all are canonical, recall that the map on the right is given by (2.2): / Coker(φk )[k] / Coker(φk ) F1k ❤❤ ❤ ❤ ❤ ❤❤ ≃ ❤❤❤❤❤❤ ❤❤❤❤ ❤❤❤❤ ❤ ❤ ❤ s ❤1−1 ❤ / H 0 (Hk (D))[k] 1−1 / Hk (D)[k] g

Hgr (Dr+k )[k]

∞ (E−k,0 ) hor [k]

(2.7)

1−1

/ (Dk )[k]

We denote the composition of the above chain of R-homomorphisms by τk . Finally, we define the promised family of complexes as follow: For any integer k, k Z•+ is a

complex of length s consists of two parts: the right part which is (0 Ehor )[k] and the left part that is (1 Ever )[k] . These parts are joined via τk . More precisely: + k Z•

:

0

/ kZ + s

wherein + k Zi

φk

/ ···

=

(

/ kZ+ k+1

τk

(Di )[k] , r Hg (Di+r−1 )[k] ,

/ kZ+ k

/ ···

i ≤ min{k, s} i>k

/ kZ + 0

/ 0 (2.8)

(2.9)

Since Hgr (M ⊗R S) = M ⊗R Hgr (S) for all R-module M , k Z•+ looks like D• , while replacing S

with the free module Hgr (S)[k] and giving a shift by r − 1 for i > k.

The structure of k Z•+ is in two ways depending on the generating sets, namely it depends on

the generating set of I, f , and the expression of the generators of a in terms of the generators of I, cij . However, for k ≥ 1, we have H0 (k Z•+ ) = H0 (D• )[k] = (SI /(γ)SI )[k] = Symk (I/a). 2.2. Acyclicity and Cohen-Macaulayness. One more occasion where the properties of k Z•+

are independent of the generating sets of I and a is the following crucial case for the acyclicity of k Z•+ . Lemma 2.1. Let R be a Noetherian ring and a ⊆ I be two ideals of R. If I = a, then every complex k Z•+ defined in (2.8) is exact.

Proof. Since the approximation complex Z• (f ) (resp.

K• (γ)) is a differential graded alge-

bra, Hi (Z• ) (resp. Hi (K• (γ))) is a SI = S/L-module (resp. S/(γ)-module) for all i. The bi-complex K• (γ) ⊗S Z• , gives rise to the horizontal spectral sequence with second terms


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2 E−i,−j hor

= Hi (K• (γ; Hj (Z• ))). It follows that (L + (γ)) annihilates ∞ E−i,j hor

annihilates

2 E−i,−j . hor

which is a sub-quotient of

2 E−i,−j hor

and consequently

By the convergence of the spectral

sequence to the homologies of the total complex H• (D• ), it is straightforward to deduce that Hi (D• ) is (L + (γ))- torsion modules for all i, i.e. (L + (γ))N Hi+j (D• ) = 0 for all i, j and for some N . Considering the equation I = a in (SI )1 , we have L + (γ) = L + g. Therefore j j Hgj (Hi (D• )) = Hg+L (Hi (D• )) = H(γ)+L (Hi (D• )) =

(

Hi (D• ), j = 0 0.

(2.10)

j>0

Once more we study the spectral sequences arisen from Cg• ⊗ D• as it was already mentioned in

(2.8). Applying (2.10), we draw both horizontal and vertical spectral sequence simultanously as follow:

∞E

hor

:

···

0

0

Hk (D• )[k] ✈

···

H0 (D• )[k]

✈ ✈ ✈ 1E

ver

:

τk

···

0

✈ ✈

0

···

0

/ 0

0

···

0

✈ ✈ ✈ ✈ ✈

Hgr (Dr+s−1 )[k]

/ ...

φk

✈

/ H r (Dr+k )[k] g

Since Hj (D• ) is a sub-quotient of Dj , Hj (D)[k] = 0, for all j > k. Also by (2.10) there is only one non-zero row in the horizontal spectral. On the other hand g is a regular sequence on Dj ’s which in turn shows that the vertical spectral is just one line. Consequently Hj (k Z•+ ) := Hj ((Hgr (Hj+r−1 (D• )))[k] ) = Hj (D• )[k] = 0 whenever j > k. On the

other hand, if j < k, Hj (k Z•+ ) := Hj (D• )[k] = Hj (Hgr (Dr+j )[k] ) however the latter is zero since

end(Hgr (Dr+j ) ≤ j. This shows that k Z•+ is exact on the left and also on the right hand of τk .

It just remains to proof the exactness in the joint points k and k + 1. For this, note that by

(2.10), F1k = 0 in (2.6). Therefore the map τk defined in (2.7) is exactly the canonical map Can.

Hgr (Dr+k )[k] −−−→ Coker(φk )[k] which satisfies all we desired.

As already mentioned in the introduction, some sliding depth conditions are needed to prove the acyclicity of the k Z•+ complexes. Definition 2.2. Let (R, m) be a Noetherian local ring of dimension d and I = (f1 , ..., fr ) an ideal of grade g ≥ 1. Let k and t be two integers, we say that the ideal I satisfies SDk at level t


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if depth(Hi (f ; R)) ≥ min{d − g, d − r + i + k} for all i ≥ r − g − t (whenever t = r − g we simply

say that I satisfies SDk , also SD stands for SD0 ).

I is strongly Cohen-Macaulay, SCM, if Hi (f ; R) is CM for all i. Clearly SCM is equivalent to SDr−g . Similarly, we say that I satisfies the sliding depth condition on cycles, SDCk , at level t, if depth(Zi (f , R)) ≥ min{d − r + i + k, d − g + 2, d} for all r − g − t ≤ i ≤ r − g. Again if t = r − g we simply say that I satisfies SDCk .

Some of the basic properties and relations between SDk condition and SDCk are explained in the following proposition. Proposition 2.3. Let (R, m) be a CM local ring of dimension d and I = (f1 , ..., fr ) an ideal of grade g ≥ 1. Let k and t be two integers. Then (1) The properties SDCk and SDk at level t localizes; they only depend on I and not the generating set, if t = r − g.

(2) SDk implies SDCk+1 .

(3) SDk at level t < r − g implies SDCa at level t for a ≤ g + t + 2 − d.

(4) SDCk+1 at level t implies SDk at level t for any t, if g ≥ 2. This implication is also the case if g = 1 and k = 0.

(5) SD0 at level t ≥ 1 implies SDC0 at level t. Proof. (1) is essentially proved in [V]. (2) was already proved in [Ha, 2.4]. (3) follows after analyzing the spectral sequences derived from the tensor product of the truncated Koszul complex 0 −→ Zi (f ) −→ Ki −→ Ki−1 −→ · · · −→ K0 −→ 0. • (R). To prove (4), we consider the depth inequalities derived from the ˇ and the Cech complex, Cm

short exact sequences 0 −→ Zi+1 (f ) −→ Ki+1 −→ Bi (f ) −→ 0, and 0 −→ Bi (f ) −→ Zi (f ) −→

Hi (f ) −→ 0. To show (5) we use the fact that SDCk holds for any k at level −1 then a recursive induction applying depth(Zi ) ≥ min{depth(Hi ), d, depth(Zi+1 ) − 1} proves the assertion.

We are now ready to present our first main theorem concerning the acyclicity of the family {k Z•+ }. This theorem shows how the family {k Z•+ } behaves well with s-residual intersections. Theorem 2.4. Let (R, m) be a CM local ring of dimension d and J = (a : I) an s-residual intersection. Assume that I = (f1 , ..., fr ) and grade(I) = g ≥ 1. Let 0 ≤ k ≤ s − g + 2. Then

the complex k Z•+ is acyclic, if any of the following hypotheses holds: (1) 1 ≤ s ≤ 2 and s = k, or


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(2) r + k ≥ s + 1, k ≤ 2 and I satisfies SDC1 at level s − g − k, or

(3) r + k ≤ s and I satisfies SD, or

(4) r + k ≥ s + 1, k ≥ 3, I satisfies SDC1 at level s − g − k, and depth(Zi (f )) ≥ d − s + k, for 0 ≤ i ≤ k.

Proof. Fix 0 ≤ k ≤ s − g + 2. To prove the acyclicity, we avail ourselves of ”lemme d’aciclicit´e”

of Peskine-Szpiro; so that it is enough to verify the following two conditions: (A) k Z•+ is acyclic on the punctured spectrum. (B) depth(k Zi+ ) ≥ i for all 0 ≤ i ≤ s.

However instead of condition (B), for later use, we verify the following stronger condition: (B ′ ) depth(k Zi+ )p ≥ i for all p with ht(p) ≥ s.

The sketch of the proof is as follows. We first show that (B ′ ) holds, then by induction on

ht(p) we verify condition (A)- the basis of the induction is all the cases where ht(p) ≤ s − 1 due to Lemma 2.1.

Consider the k Z•+ complex: 0

/ H r (Dr+s−1 )[k] g

/ ···

φk

/ H r (Dr+k )[k] g

τk

/ (Dk )[k]

/ ···

/ (D0 )[k]

/ 0

(2.11)

This complex is divided into two parts. The right side of τk and the left one. The components on the right part are (Dj )[k] which are direct sums of some copies of Zi (f ) for max{0, j − s} ≤

i ≤ min{r − 1, j}, see (2.1).

In the case where k ≤ 2 the condition (B ′ ) is automatically fulfilled on the right part of k Z•+ ,

since R is CM and Zi ’s are second syzygies.

Under condition (1) the left part does not exist so that (B ′ ) is satisfied in this case. Now suppose that condition (2) happens. We verify (B ′ ) on the left of τk . For any j = k, ..., s − 1 the first cycle which appears in Dr+j is Zr−s+j . Therefore (B ′ ) holds if we show that

depth(Zr−s+j )p ≥ j + 1 for all j ≥ k. For the latter, appealing the CM-ness of R in conjunction

with [V, Section 3.3], it is enough to show that depth(Zr−s+j ) ≥ d − s + j + 1. However by standing SDC1 assumption, we know that for r − s + k ≤ i ≤ r − g, depth(Zi ) ≥ d − r + i + 1.

As for i > r − g one always has depth(Zi ) = d − r + i + 1. It then follows that depth(Zr−s+j ) ≥

d − r + (r − s + j) + 1 = d − s + j + 1 as desired. Notice that by condition (2), k ≤ 2, hence (B ′ )

holds on the right, always.

Now suppose that we are in the situations of condition (3). We verify (B ′ ) for the left part of τk . In this case Dr+k , ..., Ds all started from Z0 also for i = 0, ..., r−1, Hgr (Ds+i )[k] (=

+ k Zs−r+i+1 )

starts from Zi . Hence to verify (B ′ ), it is enough to show that depth(Zi ) ≥ d−s+(s−r +i+1) =


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d − r + i + 1 for all i. This is just the SDC1 condition at level r − g which is equivalent to SD by Proposition 2.3. As for the right part of k Z•+ , we shall prove that depth((Dj )[k] )p ≥ j

for j = 0, 1, ..., k. Note that since k ≤ s, Dj contains all Z0 , Z1 , ..., Zj . Therefore it will be

enough to see depth(Zi ) ≥ d − s + k, for 0 ≤ i ≤ k. However, by the SD assumption we have

depth(Zi ) ≥ d − r + i + 1 which completes the proof since by assumption −r ≥ k − s. Therefore

(B ′ ) holds under (3).

In the case (4), the verification of (B ′ ) for the left part is the same as that in the case (2). As for the right part we just saw in the above paragraph that depth(Zi ) must be at least d − s + k which the standing assumption in (4).

Therefore the condition (B ′ ) of acyclicity criterion holds in any case. To verify (A) we induct on ht(p). If ht(p) ≤ s − 1, then Ip = ap . Therefore by Lemma 2.1

(k Z•+ )p is acyclic. Now suppose that ht(p) ≥ s and that for any q with ht(q) ≤ ht(p) − 1, (k Z•+ )q

is acyclic, in particular, for any q ⊂ p, (k Z•+ )q is acyclic. We recall that the stronger condition

(B ′ ) holds under any of the standing assumptions which says that depth(k Zi+ )p ≥ i; so that the

acyclicity criterion for (k Z•+ )p proves that (k Z•+ )p is acyclic as desired.

Before proceeding to the rest of the paper, we may ask for giving more attention to the conditions (1) − (4) in the above theorem in particular cases. For instance the cases where k = 0, 1 and s − g + 1 are the most important ones. For k = 0 we refer to [Ha]. Also one

interesting situation is where the residual intersection is on the nose that is when s = g + 1. In these extreme cases we may get the acyclicity of k Z•+ for free, without any sliding depth condition

on I; for example if s = g + 1 then

+ s−g+1 Z•

is always acyclic and resolves Syms−g+1 (I/a). Even

in the case of linkage i.e. s = g we obtain non-trivial information since in this case condition (2) is automatically fulfilled for k = 1, 2. Having a finite acyclic complex whose components have sufficient depth, one can estimate the depth of the zeroth homology. Theorem 2.5. Let (R, m) be a CM local ring of dimension d, I = (f1 , ..., fr ) an ideal with ht(I) = g ≥ 1. Let s ≥ g and 0 ≤ k ≤ s − g + 2. Suppose that one the following hypotheses holds: (i) r + k ≤ s and I satisfies SD condition.

(ii) r + k ≥ s + 1, I satisfies SDC1 at level s − g − k and depth(Zi (f )) ≥ min{d, d − s + k} for 0 ≤ i ≤ k.

Then for any s-residual intersection J = (a : I), the complex k Z•+ is acyclic. Furthermore, H0 (0 Z + ) and Symk (I/a) for 1 ≤ k ≤ s − g + 2, are CM of dimension d − s.


12

Proof. Under any of the conditions (i) or (ii), the complex k Z•+ is acyclic by Theorem 2.4.

Moreover these conditions imply that depth(Zi (f )) ≥ d − s + i. Consider the double complex

Cm• ⊗R (k Z•+ ) and put it in the third quadrant with Cmd ⊗R (k Z0+ ) in the (0, 0) coordinate. By

the acyclicity of k Z•+ we have: ∞

−i,−j Ehor =

(

Hgd−j (H0 (k Z•+ )), i = 0 0,

i 6= 0

(2.12)

On the other hand the sliding depth hypotheses imply that depth(k Zi+ ) ≥ d−s+i for all i (see

−i,−j the proof of Theorem 2.4). Thus 2 Ever = 0 for any coordinate under the line x + y = −d + s.

Thus the convergence of the spectral sequences implies Hgi (H0 (k Z•+ )) = 0 for i ≤ d − s − 1. This

shows that depth(H0 (k Z•+ )) ≥ d − s. On the other hand, recall that H0 (k Z•+ ) = Symk (I/a) for

k ≥ 1, and that J = (a : I) is of codimension at least s, therefore dim(Symk (I/a)) ≤ d − s and

hence it is CM of dimension d − s.

At this moment it is worth mentioning a couple of remarks about the conditions and levels in Theorem 2.5. Remark 2.6. All of depth conditions in Theorem 2.5 are satisfied, if any of the following stronger conditions holds: (iii) k ≤ s − r + 2 and I satisfies SD.

(iv) depth(Hi (f )) ≥ min{d − s + k − 2, d − g}, for 0 ≤ i ≤ k − 1 and I satisfies SD. (v) I is strongly Cohen-Macaulay.

Further, under each of these conditions, the acyclicity of k Z•+ is independent of generating sets

of I and a.

Remark 2.7. We see in the situation of Theorem 2.5 and its remarks that conditions (i) and (v) (resp.) imply that depth(R/I) ≥ d − s + k and depth(R/I) ≥ d − g ≥ d − s, respectively.

The other conditions (ii), (iii) and (iv) merely yield depth(R/I) ≥ d − s + k − 2. Therefore to have a better consequence of the theorem it may not be too much to ask depth(R/I) ≥ d − s.

The next corollary demonstrates an important invariant of (algebraic) residual intersections which is depth(R/a). This consequently shows that all (algebraic) residual intersection of ideals with sliding depth condition are unmixed.


13

Corollary 2.8. Let (R, m) be a CM local ring of dimension d, I = (f1 , ..., fr ) an ideal of ht(I) = g ≥ 1. Suppose that I satisfies any of the conditions (i) − (v) in Theorem 2.5 and its remark,

and that depth(R/I) ≥ d − s (for instance I may be SCM). Then for any (algebraic) s-residual

intersection J = (a : I), we have

(1) depth(R/a) = d − s and thus a is minimally generated generated by s elements.

(2) J is unmixed of codimension s. S (3) Ass(R/a) ⊆ Ass(R/I) Ass(R/J). Moreover if I is unmixed then the equality holds and the union is disjoint.

(4) If the residual is arithmetic then Symi (I/a) is a faithful R/J-module for all i.

Proof. (1). Applying Theorem 2.5 for k = 1, we have depth(I/a) = d − s. Therefore the result

follows from the exact sequence 0 → I/a → R/a → R/I → 0 and the fact that depth(R/I) ≥ d−s by hypothesis.

(2). J has codimension at least s, thus for any p ∈ Ass(R/J), ht(p) ≥ s. On the other hand,

Ass(R/J) ⊆ Ass(R/a) and any prime ideal in the latter has codimension less than s by (1) (see [BH, 1.2.13]) which yields the unmixed-ness of J.

(3). Let p ∈ Ass(R/a), then ht(p) ≤ s, by (1). If ht(p) < s or p 6⊃ J, then ap = Ip and thus

p ∈ Ass(R/I) otherwise ht(p) = s and p ⊇ J which clearly means p ∈ Ass(R/J). Verification of the last statement is straightforward.

(4) To see this, we refer to the proof of Theorem 1.3 in [Ha, Theorem 2.11]. In loc.cit. it 2 0,0 is shown that in the spectral sequence defined in (2.3), K := (∞ E0,0 hor )[0] ⊆ J ⊆ ( Ehor )[0] .

Moreover in the case where the residual is arithmetic all of these inclusion are equality. We S∞ 0 i i+1 (I)). Thus once just notice that (2 E0,0 i=0 (γ · Sym (I) :R Sym hor )[0] = Hg (H0 (D• ))[0] =

the equality in the above line happens, J = γ · Symi (I) :R Symi+1 (I) for all i. Therefore Symi (I/a) = Symi+1 (I)/γ · Symi (I) is a faithful R/J-module.

In the case where the residual intersection is geometric we have stronger corollaries. Corollary 2.9. Let (R, m) a CM local ring of dimension d, I = (f1 , ..., fr ) an ideal of ht(I) = g ≥ 1. Suppose that I satisfies any of the conditions (i) − (v) in Theorem 2.5 and its remark,

and that depth(R/I) ≥ d − s (for instance I may be SCM). Then for any geometric s-residual intersection J = (a : I) and for any 0 ≤ k ≤ s − g + 2, we have: (1) R/J is CM. (2) I k /aI k−1 ≃ Symk (I/a) and it is a faithful maximal Cohen-Macaulay R/J-module.

(3) aI k−1 = I k ∩ J.


14

(4) I k + J is a CM ideal of codimension s + 1 and ht(I k + J/J) = 1. Proof. (1). This is an especial case of Theorem 1.3. Symk (I) γSymk−1 (I)

(2). Consider the natural map ϕ :

→

Ik aI k−1

that sends Ti to fi . ϕ is onto. Let k

Sym (I) K = Ker(ϕ). We show that Ass(K) = ∅, hence K = 0. Ass(K) ⊆ Ass( γSym k−1 (I) ) and by k

Sym (I) Theorem 2.5, depth( γSym k−1 (I) ) = d − s, hence for any associated prime p in the latter ht(p) ≤ s.

However, since J is a geometric residual, for any p of codimension ≤ s, either Ip = ap or Ip = (1).

In the former case Symk (I

p)

Symk (Ip ) γp Symk−1 (Ip )

= 0, for k ≥ 1, which implies that p 6∈ Ass(K). In the latter case,

= Rp , hence ϕp is an isomorphism, which means Kp = 0 thus a priori p 6∈ Ass(K).

Therefore Ass(K) = ∅, hence K = 0. The CM ness of Symk (I/a) was already shown in Theorem

2.5. To see that I k /aI k−1 is faithful, just notice that J ⊆ aI k−1 : I k and that the equality holds locally in all associated primes of J which are of codimension s. (3). Consider the canonical map ψ :

Ik aI k−1

→

I k +J J .

By (2), I k /aI k−1 is CM of dimension d − s

which in conjunction with the fact that the residual is geometric shows that ψ is an isomorphism for all 0 ≤ k ≤ s−g +2. Notice that

I k +J J

≃

Ik . J∩I k

Thus the canonical map imply J ∩I k = aI k−1 .

(4). By (2) and (3), (I k + J)/J is CM of dimension d − s also by (1), R/J is CM of dimension

d − s. Thus the exact sequence 0 → (I k + J)/J → R/J → R/(I k + J) → 0 implies depth(R/I k + J) ≥ d − s − 1. On the other hand, since the residual is geometric, ht(I k + J) ≥ s + 1 which

in turn completes the proof. To see that ht(I k + J/J) = 1, it just enough to notice that R/J is CM. We now proceed to determine the structure of the canonical module of R/J. This achievement is via a study of the type of the appropriate candidate for the canonical module. Recall that in a Notherian local ring (R, m) the type of a finitely generated module M is the dimension of depth(M )

R/m-vector space ExtR

(R/m, M ) and it is denoted by rR (M ) or just r(M ).

Theorem 2.10. Let (R, m) be a CM local ring of dimension d, I = (f1 , ..., fr ) an ideal of ht(I) = g ≥ 2. Let J = (a : I) be an s-residual intersection of I and 1 ≤ k ≤ s − g + 1. Suppose

that I is SCM. Then

k

rR (Sym (I/a)) ≤

r+s−g−k rR (R). r−1

Proof. Keep notations introduced at the beginning of the section in effect. First we deal with the case where g = 2. Consider the complex k Z•+ 0

/ H r (Dr+s−1 )[k] g

/ ···

φk

/ H r (Dr+k )[k] g

τk

/ (Dk )[k]

/ ···

/ (D0 )[k]

/ 0

(2.13)


15

+ We study the depth of the components of this complex. Zr−1 = R hence depth(k Zr+s−1 ) = d.

For k ≤ j ≤ s − 2, the first cycle which appears in Dr+j is Zr−s+j and it has depth equal to

d = d − g + 2, by SCM condition, which is at least d − s + (j + 1) + 1. As for the right part of

the complex we have depth(Zi ) = d − g + 2 ≥ d − s + k + 1 for all 0 ≤ i ≤ k, since by assumption k ≤ s − g + 1.

Summing up we have depth(k Zi+ ) ≥ d − s + i + 1 for all i = 0, · · · , s − 1.

(2.14)

Now, let F• → R/m be a free resolution of R/m. We put the double complex HomR (F• , k Z•+ )

in the third quadrant with HomR (F0 , k Z0+ ) in the center. The two arisen spectral sequences are: ( ExtjR (R/m, Symk (I/a)), i = 0 ∞ −i,−j Ehor = 0, i 6= 0 and

1 E−i,−j ver

= ExtjR (R/m,

+ k Zi ).

The latter vanishes for all (i, j) where i − j ≤ d − s and

i 6= s, according to (2.14). The convergence of both complexes to the homology module of the

k total complex implies that Extd−s R (R/m, Sym (I/a))(=

submodule of

ExtdR (R/m, k Zs+ ).

∞ E−i,−j ) hor

Therefore rR (Symk (I/a)) ≤

∞ E−s,−d . The latter ver d dimR/m (ExtR (R/m, k Zs+ )).

≃

is a

According the the structure of D• , Dr+s−1 = S(−r + 1 − s). Hence + k Zs

= Hgr (S)(−r+1−s+k) .

To calculate the dimension of the above Ext module we just need to know how many copies of R appear in the inverse polynomial structure of the above local cohomology module. The answer is simply the positive solutions of the numerical equation α1 + · · · + αr = r − 1 + s − k which is r−k+s−2 . Therefore rR (Symk (I/a)) ≤ r+s−2−k r(R) in the case where g = 2. r−1 r−1 Now, let ht(I) = g ≥ 2. We choose a regular sequence a of length g − 2 inside a which

is a part of a minimal generating set of a. I/a is still SCM by [H, Coroolary 1.5], moreover

J/a = a/a : I/a. Recall that µ(a) = s, Corollary 2.8, therefore J/a is an (s − g + 2)-residual

intersection of I/a which is of height 2. Hence we return to the case g = 2 so that r + (s − g + 2) − 2 − k r+s−g−k rR (Symk (I/a)) = rR/a (Symk (I/a)) ≤ rR/a (R/a) = rR (R). r−1 r−1

(see [BH, Exercise 1.2.26]).

Remark 2.11. In the above corollary the inequality holds as well for k = 0 saying that r+s−g + rR (H0 (0 Z• )) ≤ rR (R). r−1


16

Everything is now ready to exploit the structure of the canonical module Corollary 2.12. Suppose that (R, m) is a Gorenstein local ring of dimension d. Let I be a SCM ideal with ht(I) = g and J = (a : I) be an arithmetic s-residual intersection of I. Then ωR/J ≃ Syms−g+1 (I/a). It is isomorphic to I s−g+1 + J/J in the geometric case; Furthermore R/J is generically Gorenstein and, R/J is Gorenstein if and only if I/a is principal.

Proof. Without losing the generality, we may suppose that g ≥ 2, otherwise we may add a new variable x to R and consider ideals a + (x) and I + (x) in the local ring R[x]m+(x) wherein we

have J + (x) = (a + (x)) : (I + (x)). R/J is CM by Theorem 1.3. According to Theorem 2.5, Corollary 2.8 and Theorem 2.10 Syms−g+1 (I/a) is a maximal CM, faithful R/J-module and of type 1, respectively. Therefore it is the canonical module of R/J by [BH, 3.3.13]. Next fact follows from Corollary 2.9. Next, notice that for any prime ideal p ⊇ J of codimension s. r((R/J)p ) = µ(ωR/J ) =

µ(Syms−g+1 ((I/a)p )) which is 1 since the residual is arithmetic by the assumption. It then

implies that R/J is generically Gorenstein. Finally let t = µ(I/a) then r(R/J) = µ(ωR/J ) = s−g+t which is one if and only if t = 1 . t−1 Since a part of our results works well with arithmetic s-residual intersections, it is worth to

mention some general properties of which. Proposition 2.13. Let R be a Noetherian ring. a = (a1 , · · · , as ) ⊂ I ideals of R. Then V (i) J = a : I is an arithmetic s-residual intersection if and only if ht(0 : j+1 (I/a)) ≥ s + j for j = 0, 1.

(ii) If 2 is unit in R then I + J ⊆ (0 : arithmetic one.

V2

(I/a)); in particular any geometric s-residual is an

(iii) The followings are equivalent (1) For all i ≤ s, an i-residual intersection of I exists.

(2) For any prime ideal p ⊃ I such that ht(p) ≤ s − 1, µ(Ip ) ≤ ht(p) + 1.

(3) There exist a1 , · · · , as ∈ I such that for i ≤ s − 1, Ji = (a1 , · · · , ai ) : I is an

arithmetic i-residual intersection and J = (a1 , · · · , as ) : I is an (algebraic) s-residual intersection.

Proof. (i). J = a : I is an arithmetic s-residual if and only if ht(Fittj (I/a)) ≥ s + j for j = 0, 1 V [Eis, Proposition 20.6]. Moreover Fittj (I/a) and the annihilator of j+1 (I/a) have the same

radical [Eis, Exercise 20.10] which yields the assertion. (ii) is straightforward. (iii) goes along the same line as the proof of [U, Lemma 1.4].


17

3. Hilbert Function, Castelnuovo-Mumford regularity and Projective Dimension As it was already stated in the introduction, though the family {k Z•+ } does not consist of free

complexes, it can approximate R/J and Symk (I/a) very closely. Several numerical invariant or functions such as Castelnuovo-Mumford regularity, Projective dimension and Hilbert function can be estimated via this family. In [CEU], Chardin, Eisenbud and Ulrich restated an old question ([S]) asking for which open sets of ideals a the Hilbert function of R/a depends only on the degrees of the generators of a. In [CEU], the authors consider the following two conditions:

(A1) If the Hilbert function of R/a is constant on the open set of ideals a generated by s forms of the given degrees such that codim(a : I) ≥ s; and

(A2) if the Hilbert function of R/(a : I) is constant on this set. It is shown in [CEU, Theorem 2.1] that ideals with some slight depth conditions in conjunction with Gs−1 or Gs satisfy these two conditions. In this direction we have the following result. Proposition 3.1. Let R be a CM graded ring over an Artinian local ring R0 . Let J = (a : I) be an s-residual intersection. Let 0 ≤ k ≤ s − g + 2 and suppose that I satisfies any of the following

hypotheses:

(1) 1 ≤ s ≤ 2 and s = k, or

(2) r + k ≥ s + 1, k ≤ 2 and I satisfies SDC1 at level s − g − k, or

(3) r + k ≤ s and I satisfies SD, or

(4) r + k ≥ s + 1, k ≥ 3, I satisfies SDC1 at level s − g − k, and depth(Zi (f )) ≥ d − s + k, for 0 ≤ i ≤ k.

Then the Hilbert function of H0 (0 Z•+ ), if k = 0, and of Symk (I/a), if 1 ≤ k ≤ s − g + 2, depends

only on the degrees of the generators of a and the Koszul homologies of I. In particular if I

satisfies any of the above conditions for k = 1 then the Hilbert function of R/a satisfies (A1), if any of the above conditions holds for k = 0 and the residual is arithmetic then (A2) holds as well. Proof. By Theorem 2.4, the standing assumptions on I imply the acyclicity of k Z•+ . Hence

the Hilbert function of H0 (0 Z•+ ) is derived from the Hilbert functions of the components of the

complex k Z•+ which, according to (2.1) and (2.9), are just some direct sums of the Koszul’s cycles

of I shifted by the twists appearing in the koszul complex K• (γ, S). Since the Hilbert functions

of Koszul’s cycles are inductively calculated in terms ofthose of the Koszul’s homologies, the Hilbert function of H0 (0 Z•+ ) depends on the Koszul’s homologies of I and just the degrees of the

generators of a. In the case where k = 1 we know the Hilbert function of I/a = Sym1 (I/a) so


18

that we know the Hilbert function of R/a by 0 → I/a → R/a → R/I → 0. Also in the arithmetic

case we have H0 (0 Z•+ ) = R/J.

The next important numerical invariant associated to an algebraic or geometric object is the Castelnuovo-Mumford regularity. Here we present an upper bound for the regularity of H0 (0 Z•+ )

and Symk (I/a) (if 1 ≤ k ≤ s − g + 1). These upper bound generalize the previous results in the case where k ≥ 1 and improve them by removing the Gs condition when the bound is attained. L Assume that R = n≥0 Rn is a positively graded *local Noetherian ring of dimension d over

a Noetherian local ring (R0 , m0 ) set m = m0 + R+ . Suppose that I and a are homogeneous ideals

of R generated by homogeneous elements f1 , · · · , fr and a1 , · · · , as , respectively. Let deg ft = it

for all 1 ≤ t ≤ r with i1 ≥ · · · ≥ ir and deg at = lt for 1 ≤ t ≤ s. For a graded ideal b, the sum of the degrees of a minimal generating set of b is denoted by σ(b).

Proposition 3.2. With the same notations just introduced above, let (R, m) be a CM ∗ local ring, I an ideal with ht(I) = g ≥ 2, s ≥ g and 0 ≤ k ≤ s − g + 1. Suppose that J = (a : I) is an s-residual intersection of I and that one the following hypotheses holds: (i) r + k ≤ s and I satisfies SD1 condition.

(ii) r + k ≥ s + 1, I satisfies SDC2 at level s − g − k and depth(Zi (f )) ≥ d − s + k + 1 for 0 ≤ i ≤ k.

Then if k ≥ 1,

reg(Symk (I/a)) ≤ reg(R) + dim(R0 ) + σ(a) − (s − g + 1 − k) indeg(I/a) − s.

as well reg(H0 (0 Z•+ )) ≤ reg(R) + dim(R0 ) + σ(a) − (s − g + 1) indeg(I/a) − s.

Proof. The proof of this fact is essentially the same as that of [Ha, Theorem 3.6] wherein the case of k = 0 and H0 (0 Z•+ ) is verified. Here, we just need to substitute the complex C• in loc.cit. with k Z•+ in the current case and adopt the shifts.

The inequality in the above proposition becomes more interesting in particular cases, for example if R is a polynomial ring over a field and k = 1 this inequality reads as reg(I/a) + (s − g) indeg(I/a) ≤ σ(a) − s. Hence we have a numerical obstruction for J = (a : I) to be an s-residual intersection. At the other end of the spectrum we have k = s − g + 1, this case corresponds to the canonical

module of R/J in the arithmetic case; so that to compute the regularity of R/J it is crucial to find the graded structure of the canonical module. Proposition 3.3. Suppose that (R, m) is a positively graded Gorenstein *local ring, over a Noetherian local ring (R0 , m0 ), with canonical module ωR = R(b) for some integer b. Let I be a


19

homogeneous SCM ideal with ht(I) = g, a ⊂ I homogeneous and J = (a : I) be an arithmetic s-residual intersection of I. Then ωR/J = Syms−g+1 (I/a)(b + σ(a)).

Proof. We may and do assume that g ≥ 2. First suppose that g = 2. We keep the notations

defined at the beginning of the section 2 to define k Z•+ complex. Considering R as a subalgebra of S = R[T1 , · · · , Tr ], we write the degrees of an element x of R as the 2-tuple (deg x, 0)with the

second entry zero. So that deg ft = (it , 0) for all 1 ≤ t ≤ r, deg at = (lt , 0) for all 1 ≤ t ≤ s,

deg Tt = (it , 1) for all 1 ≤ t ≤ s, and thus deg γt = (lt , 1) for all 1 ≤ t ≤ s. With these notations

the Z-complex has the following shape N N N Z• : 0 → Zr−1 R S(0, −r + 1) → · · · → Z1 R S(0, −1) → Z0 R S → 0.

Consequently,

Zr−1 = R(−

Pr

t=1 it , 0)

N

R S(0, −r

+ 1),

and, taking into account that a1 , · · · , as is a minimal generating set of a by Corollary refcunmixed, P Dr+s−1 = S(− rt=1 it − σ(a), −s − r + 1)).

It then follows that Hgr (S)(−

r X t=1

+ (s−1) Zs

= Hgr (Dr+s−1 )[∗,s−1] is isomorphic to

it − σ(a), 0)[∗,−r] ≃ (RT1−1 · · · Tr−1 (−

Notice that deg(Tj−1 ) = (ij , −1)).

r X t=1

it ))(−σ(a)) ≃ R(−σ(a)).

(3.1)

We next turn to the proof of Theorem 2.10 and consider the spectral sequences therein

for k = s − g + 1 = s − 1. We have

to

ExtdR (R/m, R(b))(−b

1 E−s,−d ver

= ExtdR (R/m,

+ (s−1) Zs )

− σ(a)) by (3.1). The latter is in turn homogeneously isomorphic to

s−g+1 (I/a)) ≃ R/m(−b − σ(a)). The inclusion Extd−s R (R/m, Sym

ExtdR (R/m, k Zs+ )

which is isomorphic

∞ E0,−(d−s) hor

֒→

1 E−s,−d ver

=

is indeed an isomorphism, since the latter is a vector space of dimension one.

In the graded case all of the homomorphisms in the proof of Theorem 2.10 are homogeneous, s−g+1 (I/a)) ≃ R/m(−b − σ(a)). therefore the above isomorphism implies that Extd−s R (R/m, Sym

Since we already know that in the local case ωR/J = Syms−2+1 (I/a), Corollary 2.12, hence in the graded case we have ωR/J = Syms−2+1 (I/a)(b + σ(a)). Now suppose that g ≥ 2 and let a = a1 , · · · , ag−2 be a regular sequence in a which is a part

of its minimal generating set. Then ωR/a =

R a (b

+ σ(a)) and moreover we go back to the case

where g = 2. We then have ωR/J = Sym

(s−g+2)−2+1

I/a a/a

((b + σ(a)) + σ(a/a)) = Syms−g+1 (I/a)(b + σ(a)).


20

Having the graded canonical module in hand we can exploit better the Hilbert function and the Castelnuovo-Mumford regularity. Recall that in the case where (R, m) is a positively graded ∗ local

ring over a Noetherian ring R0 which is not necessarily a field, the invariant regm (M ) :=

max{end(Hmi (M ))+i} is considered as the regularity with respect to m. It shares many properties with the traditional regularity with respect to R+ (e.g. [J]). In a CM ring R one always has reg(M ) ≤ regm (M ) ≤ reg(M ) + dim(R0 ) [Ha, Proposition 3.4]. Corollary 3.4. Suppose that (R, m) is a positively graded Gorenstein *local ring over a Noetherian local ring (R0 , m0 ) with canonical module ωR = R(b) for some integer b. Let I be a homogeneous SCM ideal with ht(I) = g, a ⊂ I homogeneous and J = (a : I) be an arithmetic s-residual intersection of I. Then regm (R/J) = regm (R) + σ(a) − indeg(Syms−g+1 (I/a)) − s.

Proof. By theorem 1.3, R/J is CM of dimension d−s. Hence using the local duality regm (R/J) = end(Hmd−s (R/J)) + d − s = − indeg(ωR/J ) + d − s. The result now follows from the above proposition which states that ωR/J = Syms−g+1 (I/a)(b + σ(a)).

Remark 3.5. As we mentioned above, regm (R) ≤ reg(R) + dim(R0 ) and also one always has

indeg(Syms−g+1 (I/a)) ≥ (s−g+1) indeg(I/a). Therefore the equation in this corollary yields the

inequality in Proposition 3.2. Furthermore here we have the necessary and sufficient conditions

for which the equality in Proposition 3.2 holds. This fact is new and it was already known under some stronger condition including the Gs condition, see [Ha, Proposition 3.15]. Our next remark is about the projective dimension of H0 (k Z•+ ) for k ≤ s−g+2. The fact is that

the complexes {k Z•+ } are finite complexes with known components. Hence any finite-dimension-

condition on the Koszul homology modules of I imposes the same finite dimension property on H0 (k Z•+ ), in particular cases such as arithmetic residual it is also applied to R/J. A very

interesting case is where I is generated by a regular sequence where all of the Koszul homology

modules are zero and hence of finite projective dimension! which implies that P d(H0 (k Z•+ )) < ∞; So that if (R, m) is CM then P d(Symk (I/a)) = s for all 1 ≤ k ≤ s − g + 2 by Theorem 2.5.

Even better, in the case where I is complete intersection, the approximation complex Z• can be

replaced by a free complex (we replace each cycle Zi with the tail of the Koszul complex). Thus roughly speaking in case of complete intersection all of the complexes {k Z•+ } may be constructed

to be free. However even for geometric (or arithmetic) residuals these free complexes are not the minimal free resolutions, one can verify this fact by comparing the rank of the free modules here, for k = 0, with those in [BKM] (in the latter the residual is geometric).


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Finally one may ask whether under the Gs condition our methods can contribute more information about the structure of residual intersections than what is known so far. The next result shows that in the presence of the Gs condition the complex 0 Z•+ approximates R/J for any (algebraic) s-residual intersection J.

Proposition 3.6. Let (R, m) be a Gorenstein local ring and suppose that I is SCM evenly linked to an ideal which satisfies Gs . If J = a : I is any s-residual intersection of I then R/J = H0 (0 Z•+ ). In particular all of the

above facts about the regularity, projective dimension and the Hilbert function hold for R/J.

Proof. Again without losing the generality we suppose that ht(I) = g ≥ 2. It is shown in

[HU, Theorem 5.3] that under the above assumptions on I there exists a deformation (R′ , I ′ ) of (R, I) such that I ′ is SCM and Gs . Let I = (f1 , · · · , fr ), a = (a1 , · · · , as ) and suppose Pr ′ ′ ′ that ai = j=1 cij fj . Let sij be a lifting of cij to R and fi be a lifting of fi to R . Set

X = (xij ) to be an s × r matrix of invariants, α := (α1 , · · · , αs ) = X · (f1′ , · · · , fr′ )t . Let ˜ = R′ [xij ](m+(x −s )) and J˜ := (α1 , · · · , αs ) : ˜ I ′ R. ˜ By [HU, Lemma 3.2] J˜ is a geometric R ij ij R ˜ ˜ We construct the complex 0 Z•+ ((α1 , · · · , αs ); (f ′ , · · · , fs′ )) in R s-residual intersection of I ′ R. 1 ′ + ′ ˜ K ˜ = H0 (0 Z• ((α1 , · · · , αs ); (f , · · · , fs ))). and set R/ 1

˜ to R which sends xij to cij . By Theorem 1.3, Assume that π is the projection from R ˜ = J. ˜ By essentially the same proof as [Ha, Proposition 2.13], R/π(K) ˜ = H0 (0 Z•+ ) and by K ˜ = R/π(J) ˜ = R/J as desired. [HU, Theorem 4.7], π(J˜) = J. Therefore H0 (0 Z + ) = R/π(K) •

Futher Progress. In an upcoming work we will study a very general and common situation where colon ideals (or to say so the union of schemes) may fit in. More precisely we study J = a : I such that ht(J) ≥ ht(a) + 1(!). References [AN] M. Artin and M. Nagata, Residual intersection in Cohen-Macaulay rings, J.Math. Kyoto Univ. (1972), 307–323. [BKM] W. Bruns, A. Kustin, and M. Miller,The resolution of the generic residual intersection of a complete intersection, J. Algebra 128 (1990), 214–239. [BH] W. Bruns, J. Herzog, Cohen–Macaulay Rings, revised version, Cambridge University Press, Cambridge, 1998. [BEM] L. Bus´e, M. Elkadi, B. Mourrain, Resultant over the residual of a complete intersection. Effective methods in algebraic geometry (Bath, 2000). J. Pure Appl. Algebra 164 (2001), no. 1-2, 35–57. [BKM] W. Bruns, A. Kustin, and M. Miller,The resolution of the generic residual intersection of a complete intersection, J. Algebra 128 (1990), 214–239. [CPU] A.Corso, C.Polini, B.Ulrich, The structure of the core of ideals, Math. Ann. 321 (2001), no. 1, 89–105. [CU] M. Chardin and B. Ulrich, Liaison and Castelnuovo-Mumford regularity, American Journal of Mathematics 124 (2002), 1103–1124.


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[CEU] M. Chardin, D.Eisenbud and B. Ulrich, Hilbert Functions, Residual Intersections, and Residually S2 Ideals, Composito Mathematica 125 (2001), 193–219. [CEU1] M. Chardin, D.Eisenbud and B. Ulrich, Hilbert Series of Residual Intersections, arXiv:1304.0044 . [E] B.Engheta, On the projective dimension and the unmixed part of three cubics, J. Algebra 316 (2007), no. 2, 715–734. [Eis] D. Eisenbud, Commutative Algebra with a view toward algebraic geometry, Graduate Texts in Math. 150, Springer, New York, 1995. [EU] D.Eisenbud and B. Ulrich, Residual intersections and duality, arXiv:1309.2050. [HSV] J. Herzog, A. Simis, and W. Vasconcelos, Koszul homology and blowing-up rings, Commutative Algebra (Trento, 1981), Lecture Notes in Pure and Appl. Math., vol. 84, Marcel Dekker, NY,(1983), pp. 79–169. [HHU] R. Hartshorne, C. Huneke,B. Ulrich, Bernd Residual intersections of licci ideals are glicci, Michigan Math. J. 61 (2012), no. 4, 675–701. [Ha] S. H. Hassanzadeh, Cohen-Macaulay residual intersections and their Castelnuovo-Mumford regularity, Trans. Amer. Math. Soc. 364(2012), 6371–6394. [H] C. Huneke, Strongly Cohen-Macaulay schemes and residual intersections, Trans. Amer. Math. Soc. 277(1983), 739–763. [HU] C. Huneke, B. Ulrich, Residual Intersection, J. reine angew. Math.390(1988), 1–20. [HU1] C. Huneke, B. Ulrich, The Structure of Linkage, Ann. of Math.126(1987), 277–334. [J] Jahangiri, Catelnuovo-Mumford refularity and cohomological dimension, arXiv:1304.2517. [KU] A. Kustin and B. Ulrich, A family of complexes associated to an almost alternating map, with applications to residual intersections, Mem. Amer. Math. Soc. 461(1992). [M] H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986. [S] R. Stanley, Weyl groups, the Hard Lefschetz theorem and the Sperner property, SIAMJ. Algebra Discrete Math. 1 (1980), 168–184. [U] B. Ulrich, Artin-Nagata properties and reduction of ideals, Contemp. Math. 159(1994), 373–400. [V] W. V. Vasconcelos, Arithmetic of Blowup Algebras, 195, Cambridge University Press, 1994. [W] C. A. Weibel, An introduction to homological algebra, 38, Cambridge University Press, 1994. 1,2

´ tica Av. Jornalista Anibal Universidade Federal de Pernambuco Departamento de Matema ´ ria 50740-560, Recife, Pernambuco, Brazil. Fernandes, sn, Cidade Universita E-mail address: [email protected], and [email protected]