Minimal free resolutions and asymptotic behavior of multigraded regularity

0.26 MB·English

by  Huy Tai Ha

MINIMAL FREE RESOLUTIONS AND ASYMPTOTIC BEHAVIOR OF
MULTIGRADED REGULARITY
7
0 HUY TA`I HA` AND BRENT STRUNK
0
2
Abstract. Let S be a standard Nk-graded polynomial ring over a field k, let I be a
n
a multigraded homogeneous ideal of S, and let M be a finitely generated Zk-graded S-
J module. We prove that the resolution regularity, a multigraded variant of Castelnuovo-
2 Mumford regularity, of InM is asymptotically a linear function. This shows that the
2 well known Z-graded phenomenon carries to the multigraded situation.

]
C
A
.
h 1. Introduction
t
a
InthispaperweinvestigatetheasymptoticbehaviorofamultigradedvariantofCastelnuovo-
m
Mumford regularity, the resolution regularity.
[
Throughout the paper, N will denote the set of non-negative integers. Let e ,...,e
1 k
2
be the standard unit vectors of Zk and let 0 = (0,...,0). Let k be a field and let S
v
2 be a standard Nk-graded polynomial ring over k. That is, S0 = k and S is generated
1 over S0 by elements of k Se . For l = 1,...,k, suppose Se is generated as a vector
3 l=1 l l
6 space over S0 by {xl,1,..L.,xl,Nl}. For simplicity, we shall use xl to represent the elements
0 {x ,...,x } and (x ) to denote the S-ideal generated by {x ,...,x }.
6 l,1 l,Nl l l,1 l,Nl
0
/ Definition 1.1. Let M = n∈Zk Mn be a finitely generated Zk-graded S-module. Let
h
L
t F : 0 → F → ··· → F → F → M → 0
a p 1 0
m
be a minimal Zk-graded free resolution of M over S, where F = S(−c[1],...,−c[k])
v: i j ij ij
i for i = 1,...,p. For each 1 ≤ l ≤ k, set c[l] = max {c[l]} for i = 0,..L.,p and let
X i j ij
r [l]
a res-reg (M) = max{c −i}.
l i
i
The multigraded resolution regularity of M is defined to be the vector
res-reg(M) = (res-reg (M),...,res-reg (M)) ∈ Zk.
1 k
Example 1.2. Let S = Q[x ,x ,y ,y ], where degx = e ∈ Z2 and degy = e ∈ Z2
0 1 0 1 i 1 j 2
for all 0 ≤ i,j ≤ 1. Then S is a standard N2-graded polynomial ring over Q. Consider
the ideal I = (x2,x y ,x y ,y2) as a Z2-graded S-module. Using any computational
0 0 1 1 0 0
algebra package (such as, CoCoA [5] or Macaulay2 [15]), we obtain the following minimal
2000 Mathematics Subject Classification. 13D02, 13D45, 14B15, 14F17.
Key words and phrases. Regularity, multigraded regularity,powers of ideal, asymptotic behavior.
The first author is partially supported by Louisiana Board of Regents Enhancement Grant.
1
2 HUYTA`I HA` AND BRENTSTRUNK
Z2-graded free resolution of I over S:
S(−1,−2)
S(−L2,−1) S(−2,0)
S(−3,−2)2
0 → S(−3,−3) → → S(−L3,−1) → S(−1L,−1)2 → I → 0.
S(−2L,−3)2
S(−2L,−2)2 S(0L,−2)
S(−L1,−3)
By definition, res-reg (I) = 2 and res-reg (I) = 2. Thus, res-reg(I) = (2,2) ∈ Z2.
1 2
Remark 1.3. For a vector n ∈ Zk we shall use n to denote its l-th coordinate. It can
l
be seen that TorS(M,k) = H (F⊗k). Thus, the resolution regularity of M can also be
i i
calculated as follows. For each 1 ≤ l ≤ k,
res-regl(M) = max{nl | ∃n ∈ Zk,i ≥ 0 so that TorSi (M,k)n+iel 6= 0}.
We shall often make use of this observation.
The resolution regularity is a multigraded variant of the well known Castelnuovo-
Mumford regularity, developed via the theory of Hilbert functions and minimal free reso-
lutions by Aramova, Crona and DeNegre [1] (for Z2-gradings), and Sidman and Van Tuyl
[21] (for Zk-gradings in general). This complements the notion of multigraded regular-
ity, another variant of Castelnuovo-Mumford regularity, studied by Hoffman and Wang
[13] (for Z2-graded), and Maclagan and Smith [16] (for G-graded, where G is an Abelian
group). Roughly speaking, the resolution regularity of M captures the maximal coordi-
nates of minimal generating multidegrees of syzygy modules of M. In particular, it pro-
vides a crude bound for the generating degrees of M. Thus, resolution regularity can be
viewed asa refinement of Castelnuovo-Mumford regularity. Resolution regularity, further-
more, shares important similarities with the original definition of Castelnuovo-Mumford
regularity (cf. [12]).
Many authors [1, 11, 12, 13, 16, 17, 18, 21, 22] in recent years have tried to extend
our knowledge of Castelnuovo-Mumford regularity to the multigraded situation. On the
other hand, there has been a surge of interest in the asymptotic behavior of Castelnuovo-
Mumford regularity of powers of an ideal in an N-graded algebra (cf. [2, 4, 8, 6, 7, 10,
14, 23, 27]). It is known that if S is a standard N-graded algebra (over a Noetherian
ring A), I is a homogeneous ideal in S and M is a finitely generated Z-graded S-module,
then reg(InM) is asymptotically a linear function in n with slope ≤ d(I), where d(I) is
the maximal generating degree of I. The aim of this paper is to show that the Z-graded
phenomenon carries to the multigraded setting.
SupposeM isaZk-gradedS-module,minimallygeneratedindegreesd (M),...,d (M),
1 v
where d (M) = (d ,...,d ). Then, for each 1 ≤ l ≤ k, we define
i i,1 i,k
d[l](M) = max{d | i = 1,...,v}
i,l
to be the maximal l-th coordinate among the minimal generating degrees of M. Our first
main result is stated as follows.
ASYMPTOTIC BEHAVIOR OF MULTIGRADED REGULARITY 3
Theorem 1.4 (Theorem 4.1). Let S be a standard Nk-graded polynomial ring over a field
k and let M be a finitely generated Zk-graded S-module. Let I be a multigraded homoge-
neous ideal in S minimally generated in degrees d (I),...,d (I). Then, res-reg(InM) is
1 v
asymptotically a linear function with slope vector at most d[1](I),...,d[k](I) componen-
twise. (cid:0) (cid:1)
The slope vector of res-reg(InM), in fact, can be described explicitly via the theory
of reductions.
Definition 1.5. We say that a multigraded homogeneous ideal J ⊂ I is an M-reduction
of I if InM = JIn−1M for all n ≫ 0. For each l = 1,...,k, we define
ρ[l](I) = min{d[l](J) J is an M −reduction of I}
M
(cid:12)
(here, d[l](J) is defined similarly to d[l](I(cid:12))), and
beg[l](M) = min{nl ∃n ∈ Zk such that Mn 6= 0}.
(cid:12)
Theorem 1.6 (Theorem 4.5). Let S (cid:12)be a standard Nk-graded polynomial ring over a
field k and let M be a finitely generated Zk-graded S-module. Let I be a multigraded
homogeneous ideal in S. Then, there exists a vector b ≥ (beg[1](M),...,beg[k](M)) such
that for n ≫ 0,
res-reg(InM) = n ρ[1](I),...,ρ[k](I) +b.
M M
(cid:0) (cid:1)
It is well known that in the Z-graded case, Castelnuovo-Mumford regularity can be
defined either via the shifts of a minimal free resolution or by local cohomology modules
withrespecttothehomogeneousirrelevantideal(cf. [9]). Inthemultigradedsituation(i.e.
when k ≥ 2), the two approaches, using minimal free resolutions and local cohomology
with respect to the irrelevant ideal, to define variants of regularity do not agree and thus
give different invariants (see [12] for a discussion on the relationship between these two
multigraded variants of regularity). The incomparability between these approaches makes
it difficult to generalize Z-graded results to the multigraded setting. Another conceptual
difficulty is the simple fact that Zk is not a totally ordered set. Thus, it is not easy to
capture maximal coordinates of shifts in the minimal free resolution by a single vector.
Our method in this paper is inspired by a recent work of Trung and Wang [27], where
the authors study Castelnuovo-Mumford regularity via filter-regular sequences (which
originated from [19]). To start, we first define a multigraded variant of the notion of
filter-regular sequences. We then need to establish the correspondence between resolu-
tion regularity and filter-regular sequences, which is no longer apparent after passing to
the multigraded situation. To achieve this, we develop previous work of R¨omer [18] and
Trung [26] to fuller generality to link resolution regularity, filter-regular sequences and lo-
cal cohomology with respect to different graded ideals, via the theory of Koszul homology.
Our main theorems are obtained by generalizing to the multigraded setting techniques of
[27] in investigating filter-regular sequences. Furthermore, as a byproduct of our work,
it seems possible to define multigraded regularity for Zk-graded modules over a standard
Nk-graded algebra over an arbitrary Noetherian ring A (not necessarily a field), even
though, in this case, a finite minimal free resolution as in Definition 1.1 may not exist.
This addresses a question Bernd Ulrich has asked us.
4 HUYTA`I HA` AND BRENTSTRUNK
Our paper is structured as follows. In Section 2, we define a multigraded variant of the
notionoffilter-regularsequences (Definition2.4),andgeneralizepropertiesoffilter-regular
sequences from the Z-graded case to the multigraded situation (Lemmas 2.6 and 2.7). In
Section 3, we establish the correspondence between resolution regularity, filter-regular
sequences and local cohomology with respect to different graded ideals (Theorem 3.5).
Theorem3.5laysthegroundworkfortherestofthepaper. InSection4,weproveourmain
theorems (Theorems 4.1 and 4.5). In the Appendix, making use of our work in Section 3,
we propose an alternative definition for a multigraded variant of Castelnuovo-Mumford
regularity when there might not exist finite minimal free resolutions (Definition 5.1).
Now, we shall fix some notations and terminology. If f ∈ M is a homogeneous element
of multidegree d = (d ,...,d ), then we define
1 k
deg (f) = d
l l
to be the l-th coordinate of the multidegree of f. Since coordinates of multidegrees
will be discussed frequently in the paper, we point out that the operator •[l] indicates
that the maximal l-th coordinate of involved multidegrees is being considered. For m =
(m ,...,m ) ∈ Zk and n ∈ Z, by writing (m,n) we refer to (m ,...,m ,n) ∈ Zk+1.
1 k 1 k
Acknowledgment: This project started when hurricane Katrina was about to hit New
Orleans, where the two authors were at the time. The collaboration was then made
possible with the help of many people from various places. We would like to thank the
University of Missouri-Columbia and the University of Arkansas for their hospitalities.
Especially, we would also like to express our thanks to Ian Aberbach, Jean Chan, Dale
Cutkosky, Mark Johnson, Hema Srinivasan and Bernd Ulrich. We thank an anonymous
referee for a careful reading of the paper.
2. Filter-regular sequences
In this section, we extend the notion of filter-regular sequences to the multigraded
situation. More precisely, we define M-filter-regular sequences with respect to a given
coordinate.
The first step in our proof of Theorem 1.4 is to express res-reg (InM) in terms of
l
invariants associated to M-filter-regular sequences with respect to the l-th coordinate. To
achieve this we show that, by a flat base extension, x can be taken to be an M-filter-
l
regular sequence with respect to the l-th coordinate for all 1 ≤ l ≤ k.
Definition 2.1. Let M = n∈Zk Mn be a finitely generated Zk-graded S-module. For
each 1 ≤ l ≤ k, define L
endl(M) = max{nl | ∃n ∈ Zk : Mn 6= 0}.
Remark 2.2. In general, end (M) may be infinite, and end (M) = −∞ if and only if
l l
M = 0.
Example 2.3. Consider S = k[x,y,z] as a standard N3-graded polynomial ring, where
deg(x) = (1,0,0), deg(y) = (0,1,0), and deg(z) = (0,0,1). Let M = S/I where I =
(x2y,xy2,xyz,y3,y2z,yz2). Then end (M) = ∞, end (M) = 2 and end (M) = ∞.
1 2 3
ASYMPTOTIC BEHAVIOR OF MULTIGRADED REGULARITY 5
Definition 2.4. Fix an integer 1 ≤ l ≤ k. A sequence f = hf ,...,f i of multigraded
1 s
homogeneous elements in S is said to be an M-filter-regular sequence with respect to the
l-th coordinate if
(i) (f ,...,f )M 6= M.
1 s
(ii) For every 1 ≤ i ≤ s, we have
(f ,...,f )M : f
1 i−1 M i
end < ∞.
l
(f ,...,f )M
(cid:16) 1 i−1 (cid:17)
When f = hf ,...,f i is an M-filter-regular sequence with respect to the l-th coordi-
1 s
nate, we define
(f ,...,f )M : f
a[l](f) = max end 1 i−1 M i .
M 1≤i≤sn l(cid:16) (f1,...,fi−1)M (cid:17)o
Example 2.5. Consider S = k[x,y,z] and M = S/(x2y,xy2,xyz,y3,y2z,yz2) as in Ex-
ample 2.3. Then yM 6= M and end (0 : y) = 2. Thus, y is M-filter-regular with respect
2 M
to the second coordinate. Notice also that y is not M-filter-regular with respect to the
first or third coordinate, while hx,zi is an M-filter-regular sequence with respect to the
first, second, and third coordinates.
ThefollowinglemmagivesacharacterizationforM-filter-regularsequences withrespect
to the l-th coordinate via primes avoidance (see also [26, Lemma 2.1]).
Lemma 2.6. A sequence f = hf ,...,f i of elements in S is an M-filter-regular sequence
1 s
with respect to the l-th coordinate if and only if for each i = 1,...,s, f 6∈ P for any
i
associated prime P 6⊇ (x ) of M/(f ,...,f )M.
l 1 i−1
Proof. By replacing M with M/(f ,...,f )M, it suffices to prove the statement for
1 i−1
i = 1. For simplicity, we shall write f for f .
1
Observe that end (0 : f) < ∞ if and only if there exists an integer m such that
l M
(x )m(0 : f) = 0. In particular, this means that
l M
∞
0 : f ⊆ [0 : (x )j] = H0 (M).
M M l (xl)
j[=1
LetN = M/H0 (M). ObservefurtherthatAss(M)isthedisjointunionofAss(H0 (M))
(xl) (xl)
and Ass(N) (cf. [3, Exercise 2.1.12]).
Consider an arbitrary Q ∈ Ass(M) and assume that Q = ann (h) for some h ∈ M.
S
Then, Q ⊇ (x ) if and only if h(x )j = 0 for some j ≥ 1 (since Q is prime), that is,
l l
h ∈ H0 (M). This implies that Q ∈ Ass(H0 (M)). Thus, the set of associated primes of
(xl) (xl)
M which do notcontain(x )isAss(N). Hence, it remainstoshow that0 : f ⊆ H0 (M)
l M (xl)
if and only if f 6∈ P for any P ∈ Ass(N). Indeed, 0 : f ⊆ H0 (M) if and only if
M (xl)
0 : f = 0 in N, that is, f is a non-zerodivisor on N, and this is the case if and only if f
N
does not belong to any associated prime of N. The lemma is proved. (cid:3)
The next result allows us to assume that x is M-filter-regular with respect to the l-th
l
coordinate after a flat extension (see also [27, Lemma 1.2]).
6 HUYTA`I HA` AND BRENTSTRUNK
Lemma 2.7. For i = 1,...,N , let z = Nl u x , where U = (u ) is a square
l i j=1 ij l,j ij 1≤i,j≤Nl
matrix of indeterminates. Let k′ = k(U),PS′ = S ⊗k k′ and M′ = M ⊗k k′. Then,
(1) The sequence z = hz ,...,z i is an M′- filter-regular sequence with respect to the
1 Nl
l-th coordinate. Notice that (z ,...,z ) = (x )S′.
1 Nl l
(2) res-reg(M) = res-reg(M′) where res-reg(M′) is calculated over S′.
Proof. To prove (1) we first observe that since the elements z ,...,z are given by inde-
1 Nl
pendent sets of indeterminates, by induction it suffices to show that z is M′-filter-regular
1
with respect to the l-th coordinate. By Lemma 2.6, we need to show that z 6∈ P for
1
any associated prime P 6⊇ (x )S′ of M′. Indeed, if P 6⊇ (x )S′ is an associated prime of
l l
M′, then since S → S′ is a flat extension, we must have P = ℘S′ for some associated
prime ℘ 6⊇ (x )S of M. However, since (x )S = (x ,...,x )S 6⊆ ℘, we must have
l l l,1 l,Nl
z = u x +···+u x 6∈ ℘S′ = P.
1 1,1 l,1 1,Nl l,Nl
Now, letFbeaminimalZk-gradedfreeresolutionofM overS. Toprove(2), weobserve
that since S → S′ is a flat extension, F ⊗ S′ is a minimal Zk-graded free resolution of
S
M′ over S′. Thus, by definition, res-reg(M) = res-reg(M′). (cid:3)
3. Koszul homology and a-invariant
The goal of this section is to relate the resolution regularity to invariants associated to
filter-regular sequences. Our techniques are based upon investigating Koszul complexes
andlocalcohomologymoduleswithrespect todifferent irrelevant ideals(givenbydifferent
sets of variables).
For u = (u ,...,u ) ∈ Zk such that u ≤ N for all l = 1,...,k, we shall denote by
1 k ≥0 l l
K(u)theKoszulcomplexwithrespect to(x ,...,x ,...,x ,...,x ). LetH (u;M)
1,1 1,u1 k,1 k,uk i
denotethehomologygroupH (K(u)⊗ M). Whenu = (N ,...,N ,r,0,...,0),forsim-
i S 1 s−1
plicity, we shall write K{s}(r) and H{s}(r;M) to denote K(u) and H (u;M), respectively.
i i
Lemma 3.1. For each s and 0 ≤ r ≤ N −1, let
s
H˜0{s}(r;M) = [0 :M/(x1,...,xs−1,xs,1,xs,2,...,xs,r)M xs,r+1].
Then we have the following exact sequence
··· → H{s}(r;M)(−e ) xs→,r+1 H{s}(r;M) → H{s}(r +1;M) → H{s}(r;M)(−e ) →
i s i i i−1 s
··· → H{s}(r;M)(−e ) xs→,r+1 H{s}(r;M) → H{s}(r +1;M) → H˜{s}(r;M)(−e ) → 0.
1 s 1 1 0 s
Proof. Consider the following exact sequence of Koszul complexes
(3.1) 0 → K{s}(r)(−e ) xs→,r+1 K{s}(r) → K{s}(r +1) → 0.
s
The conclusion is obtained by taking the long exact sequence of homology groups associ-
ated to (3.1); notice that the last term H˜{s}(r;M)(−e ) is given by the kernel of the map
0 s
H{s}(r;M)(−e ) xs→,r+1 H{s}(r;M). (cid:3)
0 s 0
The following theorem allows us to relate resolution regularity to invariants associated
to filter-regular sequences.
ASYMPTOTIC BEHAVIOR OF MULTIGRADED REGULARITY 7
Theorem 3.2. Let S be a standard Nk-graded polynomial ring over a field k and let M be
a finitely generated Zk-graded S-module. Suppose that x is an M-filter-regular sequence
l
with respect to the l-th coordinate. Then,
res-reg (M) = max{a[l](x ),d[l](M)}.
l M l
Proof. Without loss of generality, we shall prove the statement for l = 1.
For each 1 ≤ s ≤ k and 0 ≤ r ≤ N , let Q = N +···+N +r and define
s s,r 1 s−1
v[s](r) = max{n1 | ∃n ∈ Zk : Hi{s}(r;M)n+ie1 6= 0 for some 1 ≤ i ≤ Qs,r}
(we make the convention that v[1](0) = −∞). Observe that TorS0(M,k)d 6= 0 if and only
if d is a generating degree of M. Thus,
d[1](M) = max{n1 | ∃n ∈ Zk : TorS0(M,k)n 6= 0}.
Now, since K{k}(N ) gives a minimal Zk-graded free resolution of k over S, we have
k
H{k}(N ;M) = TorS(M,k) for all i ≥ 1. Hence,
i k i
res-reg (M) = max{v[k](N ),d[1](M)}.
1 k
It remains to show that
max{v[k](N ),d[1](M)} = max{a[1](x ),d[1](M)}.
k M 1
To this end, we proceed in the following steps.
(i) For any 1 ≤ r ≤ N ,
1
(x ,...,x )M : x
v[1](r) = max end 1,1 1,i−1 M 1,i .
1
1≤i≤rn (cid:16) (x1,1,...,x1,i−1)M (cid:17)o
(ii) For any s ≥ 2 and any 0 ≤ r ≤ N ,
s
max{v[s](r),d[1](M)} = max{a[1](x ),d[1](M)}.
M 1
(x ,...,x )M : x
1,1 1,i−1 M 1,i
For simplicity of notation, we shall denote end by s for
1 i
(x ,...,x )M
(cid:16) 1,1 1,i−1 (cid:17)
i = 1,...,N . Then a[1](x ) = max{s ,...,s }.
1 M 1 1 N1
We shall prove (i) by induction on r. By Lemma 3.1 we have
0 → H{1}(1;M) → H˜{1}(0;M)(−e ) → 0.
1 0 1
Thus,
v[1](1) = max{n1 | ∃n ∈ Zk : H1{1}(1;M)n+e1 6= 0}
= max{n1 | ∃n ∈ Zk : H˜0{1}(0;M)n 6= 0}
= end (0 : x ) = s .
1 M 1,1 1
This proves the statement for r = 1.
Suppose that r > 1. By Lemma 3.1, we have
··· → H{1}(r;M) → H˜{1}(r−1;M)(−e ) → 0.
1 0 1
This implies that v[1](r) ≥ s . Observe that if v[1](r−1) = −∞ then v[1](r) ≥ v[1](r−1),
r
which implies that v[1](r) ≥ max{v[1](r−1),s }. Assume that v[1](r−1) > −∞. That is,
r
8 HUYTA`I HA` AND BRENTSTRUNK
there exist an integer i and n ∈ Zk with n1 = v[1](r−1) such that H{1}(r−1;M)n+ie1 6= 0
and H{1}(r −1;M) = 0. By Lemma 3.1, we have
n+(i+1)e1
{1} {1}
··· → Hi+1(r;M)n+(i+1)e1 → Hi (r−1;M)n+ie1 → 0.
This implies that H{1}(r;M) 6= 0. Thus, v[1](r) ≥ v[1](r −1), and therefore, we
i+1 n+(i+1)e1
also have v[1](r) ≥ max{v[1](r −1),s }.
r
We will prove the other direction, that is, v[1](r) ≤ max{v[1](r −1),s }. Consider an
r
arbitrary n = (n ,...,n ) ∈ Zk with n > max{v[1](r−1),s }. For i ≥ 2, by Lemma 3.1,
1 k 1 r
we have
{1} {1} {1}
··· → Hi (r −1;M)n+ie1 → Hi (r;M)n+ie1 → Hi−1(r −1;M)n+(i−1)e1 → ...
Since n1 > v[1](r −1), we have Hi{−1}1(r −1;M)n+(i−1)e1 = Hi{1}(r −1;M)n+ie1 = 0. This
{1}
implies that Hi (r;M)n+ie1 = 0. For i = 1, by Lemma 3.1, we have
(3.2) ··· → H1{1}(r −1;M)n+e1 → H1{1}(r;M)n+e1 → H˜0{1}(r−1;M)n → 0.
Since n1 > v[1](r −1), we have H1{1}(r −1;M)n+e1 = 0. Since n1 > sr, we have H˜0{1}(r −
1;M)n = 0. Thus, (3.2) implies that H1{1}(r;M)n+e1 = 0. Hence, n1 > v[1](r). This is
true for any n > max{v[1](r −1),s }, so we must have v[1](r) ≤ max{v[1](r −1),s }.
1 r r
We have shown that v[1](r) = max{v[1](r−1),s }. By induction,
r
v[1](r −1) = max{s ,...,s }.
1 r−1
Thus, v[1](r) = max{s ,...,s } and (i) is proved.
1 r
We shall prove (ii) by using double induction on s and on r. By part (i), we have
v[2](0) = v[1](N ) = max{s ,...,s } = a[1](x ). Therefore, thestatement istruefors = 2
1 1 N1 M 1
and r = 0. Assume that either s > 2 or r > 0. It can be seen that v[s](0) = v[s−1](N ).
s−1
Thus, we can assume that r > 0 (and s ≥ 2).
Let N = M/(x1,...,xs−1,xs,1,xs,2,...,xs,r−1)M. Observe first that Nm 6= 0 for some
m if and only if m equals the first coordinate of a generating degree of M (i.e., m =
1 1
d (M) for some i). This implies that
i,1
(3.3) end (0 : x ) ≤ end (N) = d[1](M).
1 N s,r 1
Consider an arbitrary n = (n ,...,n ) ∈ Zk with n > max{v[s](r − 1),d[1](M)}. By
1 k 1
Lemma 3.1, we have
{s} {s} {s}
(3.4) Hi (r −1;M)n+ie1 → Hi (r;M)n+ie1 → Hi−1(r −1;M)n+ie1−es
{s} {s} ˜{s}
(3.5) H1 (r −1;M)n+e1 → H1 (r;M)n+e1 → H0 (r −1;M)n+e1−es → 0.
Since n > max{v[s](r −1),d[1](M)}, it follows from (3.3) and (3.5) that
1
{s}
H1 (r;M)n+e1 = 0.
Since n1 > v[s](r −1), we have Hi{s}(r −1;M)n+ie1 = Hi{−s}1(r −1;M)n+ie1−es = 0. Thus,
{s}
(3.4) implies that Hi (r;M)n+ie1 = 0 for all i ≥ 2. Hence,
(3.6) v[s](r) ≤ max{v[s](r −1),d[1](M)}.
ASYMPTOTIC BEHAVIOR OF MULTIGRADED REGULARITY 9
Observe now that if v[s](r −1) ≤ d[1](M) then (3.6) implies that
max{v[s](r),d[1](M)} = d[1](M) = max{v[s](r −1),d[1](M)}.
On the other hand, if v[s](r −1) > d[1](M) then there exist an integer i ≥ 1 and n ∈ Zk
withn1 = v[s](r−1)suchthatHi{s}(r−1;M)n+ie1 6= 0. Inthiscase, ifHi{s}(r;M)m+ie1 = 0
for any m ∈ Zk with m ≥ v[s](r−1), then by Lemma 3.1 we have
1
0 → Hi{s}(r −1;M)m+ie1−es x→s,r Hi{s}(r−1;M)m+ie1 → 0.
mM1=n1 mM1=n1
This implies that
{s} {s}
Hi (r−1;M)m+ie1 = xs,r Hi (r−1;M)m+ie1 ,
mM1=n1 (cid:16)mM1=n1 (cid:17)
which is a contradiction by Nakayama’s lemma. Therefore, there exists, in this case,
m0 = (m01,...,m0k) ∈ Zk with m01 = v[s](r −1) such that Hi{s}(r;M)m0+ie1 6= 0. That
is, v[s](r) ≥ v[s](r −1) > d[1](M). We, hence, have max{v[s](r),d[1](M)} = max{v[s](r −
1),d[1](M)}. The conclusion now follows by induction. (cid:3)
Example 3.3. Consider S = k[x,y,z] and M = S/(x2y,xy2,xyz,y3,y2z,yz2) as in Ex-
ample 2.3. Notice that dimM = 2. It can be seen that a[1](hx,zi) = 1, a[2](hx,zi) = 2,
M M
and a[3](hx,zi) = 1 while d[1](M) = d[2](M) = d[3](M) = 0. Theorem 3.2 implies that
M
res-reg(M) = (1,2,1).
The following lemma (see [26, Lemma 2.3]) exhibits the behavior of modding out by a
filter-regular element with respect a given coordinate.
Lemma 3.4. Let g ∈ Se be an M-filter-regular element with respect to the l-th coordinate
l
and let (x′) be the image of the ideal (x ) in S/gS. Then for all i ≥ 0,
l l
end Hi+1(M) +1 ≤ end Hi (M/gM)
l (xl) l (x′l)
(cid:0) (cid:1) (cid:0) (cid:1)
≤ max end Hi (M) ,end Hi+1(M) +1 .
l (xl) l (xl)
(cid:8) (cid:0) (cid:1) (cid:0) (cid:1) (cid:9)
Proof. It follows from the proof of Lemma 2.6 that (0 : g) ⊆ ∞ [0 : (x )n]. That is,
M n=0 M l
(0 :M g) is annihilated by some power of (xl). This implies thatS
(3.7) Hi (0 : g) = 0 ∀i ≥ 1.
(xl) M
Consider the exact sequence
0 → (0 : g) → M → M/(0 : g) → 0.
M M
(3.7) implies that Hi (M) = Hi (M/(0 : g)) for all i ≥ 1. Now, consider the long
(xl) (xl) M
exact sequence of cohomology groups associated to the exact sequence
g
0 → M/(0 : g)(−e) → M → M/gM → 0
M l
we get
(3.8) H(ixl)(M)n → H(ix′l)(M/gM)n → H(ix+l1)(M)n−el →g H(ix+l1)(M)n
for any i ≥ 0 and n ∈ Zk.
10 HUYTA`I HA` AND BRENTSTRUNK
Bythedefinitionofthefunctionend ,thereexistsm ∈ Zk withm = end Hi+1(M) +1
l l l (xl)
suchthatH(ix+l1)(M)m = 0andH(ix+l1)(M)m−el 6= 0. (3.8)thenimpliesthatH(ix(cid:0)′l)(M/gM)(cid:1)m 6=
0. Thus,
end Hi (M/gM) ≥ end Hi+1(M) +1.
l (x′l) l (xl)
(cid:0) (cid:1) (cid:0) (cid:1)
On the other hand, for any n ∈ Zk such that
n > max end Hi (M) ,end Hi+1(M) +1 ,
l l (xl) l (xl)
(cid:8) (cid:0) (cid:1) (cid:0) (cid:1) (cid:9)
we have H(ixl)(M)n = H(ix+l1)(M)n−el = 0. (3.8) now implies that
end Hi (M/gM) ≤ max end Hi (M) ,end Hi+1(M) +1 .
l (x′l) l (xl) l (xl)
(cid:0) (cid:1) (cid:8) (cid:0) (cid:1) (cid:0) (cid:1) (cid:9) (cid:3)
The lemma is proved.
The next theorem lays the groundwork for the rest of the paper, allowing us to prove
our main theorems in Section 4.
Theorem 3.5. Let S be a standard Nk-graded polynomial ring over a field k and let M be
a finitely generated Zk-graded S-module. Suppose that x is an M-filter-regular sequence
l
with respect to the l-th coordinate. Then,
(1) a[l](x ) = max end Hi (M) +i 0 ≤ i ≤ N −1 .
M l n l (xl) l o
(cid:0)(x ,...,(cid:1)x )M(cid:12) : (x )
(2) a[l](x ) = max end l,1 l,i (cid:12) M l 0 ≤ i ≤ N −1 .
M l l (x ,...,x )M l
n (cid:16) l,1 l,i (cid:17) (cid:12) o
(x ,...,x )M : (x(cid:12))
l,1 l,i M (cid:12)l
(3) res-reg (M) = max end 0 ≤ i ≤ N .
l l (x ,...,x )M l
n (cid:16) l,1 l,i (cid:17) (cid:12) o
(cid:12)
Proof. We prove (1) using induction on N . Since x is an M(cid:12) -filter-regular element with
l l,1
respect to the l-th coordinate, as in the proof of Lemma 2.6, we have (0 : x ) ⊆
M l,1
∞ [0 : (x )n] = H0 (M). Thus, a[l](x ) ≤ end H0 (M) . On the other hand,
n=0 M l (xl) M l,1 l (xl)
tShere exists n ∈ Zk with nl = endl H(0xl)(M) such th(cid:0)at H(0xl)(M(cid:1))n 6= 0. Furthermore,
xl,1H(0xl)(M)n ⊆ H(0xl)(M)n+el = 0, w(cid:0)hich impl(cid:1)ies that H(0xl)(M)n ⊆ (0 :M xl,1)n. There-
fore, a[l](x ) ≥ end H0 (M) . Hence, a[l](x ) = end H0 (M) and the statement
M l,1 l (xl) M l,1 l (xl)
is true for N = 1. (cid:0) (cid:1) (cid:0) (cid:1)
l
Assume that N > 1. Let x′ denote the sequence of images of x ,...,x in S/x S
l l l,2 l,Nl l,1
and let N = M/x M. Then x′ gives an N-filter-regular sequence with respect to the
l,1 l
l-th coordinate. By induction and using Lemma 3.4, we have
max{end Hi (M) +i | i = 1,...,N −1}
l (xl) l
≤ max{en(cid:0)d Hi−1(N(cid:1) ) +i−1 | i = 1,...,N −1} = a[l](x′)
l (x′) l N l
l
(cid:0) (cid:1)
≤ max{end Hj (M) +j | j = 0,...,N −1}.
l (xl) l
(cid:0) (cid:1)
This, together with the fact that a[l](x ) = max{a[l](x ),a[l](x′)}, implies that
M l M l,1 N l
a[l](x ) = max end Hi (M) +i | i = 0,...,N −1 .
M l l (xl) l
(cid:8) (cid:0) (cid:1) (cid:9)
(1) is proved.