Growth of Hilbert coefficients of Syzygy modules

0.28 MB·English

by  Tony J. Puthenpurakal

GROWTH OF HILBERT COEFFICIENTS OF SYZYGY
MODULES
5
1
TONYJ.PUTHENPURAKAL
0
2
n Abstract. Let(A,m)beacompleteintersectionringofdimensiondandletI
a beanm-primaryideal. LetM beamaximalCohen-MacaulayA-module. For
J i=0,1,···,d,leteI(M)denotetheith Hilbert-coefficientofM withrespect
i
9 toI. Weprovethat fori=0,1,2,thefunctionj7→eIi(SyzAj(M))isofquasi-
2 polynomialtypewithperiod2. LetGI(M)betheassociatedgradedmoduleof
M withrespecttoI. IfGI(A)isCohen-MacaulayanddimA≤2wealsoprove
] thatthefunctionsj7→depthGI(SyzA2j+i(M))areeventuallyconstantfori=
C 0,1. Let ξI(M)=liml→∞depthGIl(M). Finallyweprovethat ifdimA=2
A and GI(A) is Cohen-Macaulay then the functions j 7→ ξI(SyzA2j+i(M)) are
eventually constantfori=0,1.
.
h
t
a
m
[ 1. introduction

1
Let (A,m) be a Noetherian local ring of dimension d and let M be a finitely
v
generatedA-module ofdimensionr. LetI be anm-primaryideal. Letℓ(N)denote
3
0 the length of an A-module N. The function H(1)(M,n) = ℓ(M/In+1M) is called
I
4 the Hilbert-Samuel function of M with respect to I. It is well-known that there
7 existsapolynomialP (M,X)∈Q[X]ofdegreer suchthatP (M,n)=H(1)(M,n)
0 I I I
. for n ≫ 0. The polynomial PI(M,X) is called the Hilbert-Samuel polynomial of
1
M with respect to I. We write
0
5 r
X+r−i
1 P (M,X)= (−1)ieI(M) .
: I i r−i
v i=0 (cid:18) (cid:19)
X
i
X TheintegerseI(M)arecalledtheith-HilbertcoefficentofM withrespecttoI. The
i
r zeroth Hilbert coefficent eI0(M) is called the multiplicity of M with respect to I.
a For j ≥0 let SyzA(M)denote the jth syzygy ofM. In this paper we investigate
j
the function j 7→ eI(SyzA(M)) for i ≥ 0. It becomes quickly apparent that for
i j
reasonable answers we need that the minimal resolution of M should have some
structure. Minimal resolutions of modules over complete intersection rings have a
goodstructure. IfA=B/(f ,··· ,f )withf =f ,··· ,f aB-regularsequenceand
1 c 1 c
projdim M isfinitethenalsotheminimalresolutionofM hasanicestructure. The
B
definitiveclassofmoduleswitha goodstructuretheoryoftheir minimalresolution
is the class of modules with finite complete intersection dimension, see [2]. We are
able to prove our results for a more restrictive class of modules than modules of
finite CI-dimension.
Date:January30,2015.
1991 Mathematics Subject Classification. Primary13D40;Secondary13A30.
1
2 TONYJ.PUTHENPURAKAL
Definition 1.1. We say the A module M has finite GCI-dimension if there is a
flat local extension (B,n) of A such that
(1) mB =n.
(2) B =Q/(f ,··· ,f ),whereQislocalandf ,...,f isaQ-regularsequence.
1 c 1 c
(3) projdim M ⊗ B is finite.
Q A
We note that every finitely generated module over an abstract complete in-
tersection ring has finite GCI dimension. If A = R/(f ,··· ,f ) with f ,··· ,f a
1 c 1 c
R-regularsequenceandprojdim M isfinitethenalsoM hasfiniteGCI-dimension.
R
We also note that if M has finite GCI dimesnion then it has finite CI-dimension.
If M has finite CI-dimension then the function i 7→ ℓ(TorA(M,k)) is of quasi-
i
polynomial type with degree two. Set cx(M) = degree of this function +1. (See
2.7 for degree of a function of quasi-polynomial type).
Let G (A)= In/In+1 be the associated graded ring of A with respect to
I n≥0
I. Let G (M) = InM/In+1M be the associated graded module of M with
I L n≥0
respect to I. Our main result is
L
Theorem 1.2. Let (A,m) be a Cohen-Macaulay local ring of dimension d and let
M be a maximal Cohen-Macaulay A-module. Let I be an m-primary ideal. Assume
M has finite GCI dimension. Then for i = 0,1,2, the function j 7→ eI(SyzA(M))
i j
is of quasi-polynomial type with period two and degree ≤cx(M)−1.
Nextweconsidertheasymptoticbehaviorofdepthofassociatedgradedmodules
of syzygy modules. We prove
Theorem 1.3. Let (A,m) be a Cohen-Macaulay local ring of dimension ≤ 2 and
let M be a maximal Cohen-Macaulay A-module. Let I be an m-primary ideal with
G (A) Cohen-Macaulay. AssumeM has finite GCI dimension. Then the functions
I
j 7→depthG (SyzA(M)) and j 7→depthG (SyzA (M)) are constant for j ≫0.
I 2j I 2j+1
If M is a Cohen-Macaulay A-module and I is m-primary then it is known that
depthG (M)isconstantforalls≫0,see[6,2.2](alsosee[10,7.6]). Setξ (M)=
Is I
limdepthG (M). We prove the following:
Is
Theorem 1.4. Let (A,m) be a Cohen-Macaulay local ring of dimension 2 and let
M be a maximal Cohen-Macaulay A-module. Let I be an m-primary ideal with
G (A) Cohen-Macaulay. AssumeM has finite GCI dimension. Then the functions
I
j 7→ξ (SyzA(M)) and j 7→ξ (SyzA (M)) are constant for j ≫0.
I 2j I 2j+1
1.5. Dual Hilbert-Samuel function: Assume A has a canonical module ω. The
functionDI(M,n)=ℓ(Hom (M,ω/In+1ω)is calledthe dual Hilbert-Samuel func-
A
tion of M with respect to I. In [12] it is shown that there exist a polynomial
tI(M,z)∈Q[z]ofdegree d suchthat tI(M,n)=DI(M,n) for all n≫0. We write
d
X+r−i
tI(M,X)= (−1)icI(M) .
i r−i
i=0 (cid:18) (cid:19)
X
The integerscI(M) arecalledthe ith- dualHilbert coefficientof M withrespect to
i
I. The zeroth dual Hilbert coefficient cI(M) is equal to eI(M), see [12, 2.5]. We
0 0
prove:
Theorem 1.6. Let (A,m) be a Cohen-Macaulay local ring of dimension d, with a
canonical module ω and let I be an m-primary ideal. Let M be a maximal Cohen-
MacaulayA-module. AssumeM has finiteGCIdimension. Then for i=0,1,··· ,d
GROWTH OF HILBERT COEFFICIENTS 3
the function j 7→cI(SyzA(M)) is of quasi-polynomial type with period two. If A is
i j
a complete intersection then the degree of each of the above functions ≤cx(M)−1.
Although Theorem 1.6 looks more complicated than Theorem 1.2, its proof is
considerably simpler.
Let Hi(−) denote the ith local cohomology functor of G (A) with respect to
I
G (A) = In/In+1. Set
I + n>0
L reg(G (M))=max{i+j |Hi(G (M)) 6=0},
I I j
the regularity of G (M). Set
I
a (G (M))=max{j |Hi(G (M)) 6=0}.
i I I j
Assume that the residue field of A is infinite. Let J be a minimal reduction of
I. Say Ir+1 = JIr. Let M be a maximal Cohen-Macaulay A-module. Then it is
well-known that a (G (M))≤r−d; see [13, 3.2](also see [3, 18.3.12]). We prove
d I
Theorem 1.7. Let (A,m) be a Cohen-Macaulay local ring of dimension d ≥ 2
and let M be a maximal Cohen-Macaulay A-module. Let I be an m-primary ideal.
Assume M has finite GCI dimension. Then the set
a (G (SyzA(M))
d−1 I i
( icx(M)−1 )
i≥1
is bounded.
Here is an overview of the contents of this paper. In section two we discuss
many preliminary facts that we need. In section three we introduce a technique
which is useful to prove our results. In section four(five) we discuss the case when
dimension of A is one(two) respectively. The proof of Theorem 1.2 is divided in
these two sections. The proof of Theorem 1.3 is in section five. In section six we
proveTheorem1.4. Theorem1.6isprovedinsectionseven. Finallyinsectioneight
we prove Theorem 1.7.
2. Preliminaries
In this section we collect a few preliminary results that we need.
2.1. The Hilbert function of M with respect to I is the function
H (M,n)=λ(InM/In+1M) for all n≥0.
I
Itiswellknownthattheformalpowerseries H (M,n)znrepresentsarational
n≥0 I
function of a special type:
P
h (M,z)
H (M,n)zn = I where r =dimM and h (M,z)∈Z[z].
I (1−z)r I
n≥0
X
It can be shown that eI(M)=(h (M,z))(i)(1)/i! for all0≤i≤r. It is convenient
i I
to set eI(M) = (h (M,z))(i)(1)/i! even when i ≥ r. The number eI(M) is also
i I 0
called the multiplicity of M with respect to I.
I: Superficial elements.
For definition and basic properties of superficial sequences see [8, p. 86-87]. The
followingresultiswell-knowntoexperts. Wegiveaproofduetolackofareference.
4 TONYJ.PUTHENPURAKAL
Lemma 2.2. Let (A,m) be a Cohen-Macaulay local ring of dimension d ≥ 1 and
let I be an m-primary ideal. Assume the residue field k =A/m is uncountable. Let
{M } be a sequence of maximal Cohen-Macaulay A-modules. Then there exists
n n≥1
x=x ,...,x ∈I such that x is an M -superficial sequence (with respect to I) for
1 d n
all n≥1.
Proof. We recall the construction of a superficial element for a finitely generated
module M of dimension r ≥ 1 (note M need not be Cohen-Macaulay). Let M be
the maximal ideal of G (A). Let
I
Ass∗(G (M))={P |P ∈Ass(G (M)) and P 6=M}.
I I
Set L = I/I2 and V = L/mL = I/mI. For P ∈ SpecG (A) let P = P ∩L and
I 1
P = (P +mI)/mI. If P 6= M then P 6= L. By Nakayama’s Lemma it follows
1 1 1
easily that P 6=V. If k is infinite then
1
S(M)=V \ P 6=∅.
1
P∈Ass∗[(GI(M))
Then u∈I such that u∈S(M) is an M-superficial element with respect to I.
Now assume that k is uncountable and let F = {M } be a sequence of
n n≥1
maximal Cohen-Macaulay A-modules. Set M =A. Then
0
S(F)=V \ P 6=∅.
1
n[≥0P∈Ass∗[(GI(Mn))
Ifu∈I suchthatu∈S(F)thenu∈S(M )foralln≥1. ThusuisM -superficial
n n
element with respect to I for all n ≥ 0. Set x = u. If d ≥ 2 then note that
1
{M /x M } is a sequence of maximal Cohen-Macaulay A/(x ) modules. So by
n 1 n 1
induction the result follows. (cid:3)
2.3. Associated graded module and Hilbert function mod a superficial
element:
Let x ∈ I be M-superficial. Set N = M/xM. Let r = dimM. The following is
well-known cf., [8].
(1) Set b (M,z) = ℓ (In+1M: x)/InM zn. Since x is M-superficial we
I i≥0 M
have b (M,z)∈Z[z].
I P (cid:0) (cid:1)
(2) h (M,z)=h (N,z)−(1−z)rb (M,z); cf., [8, Corollary 10 ].
I I I
(3) So we have
(a) e (M)=e (N) for i=0,...,r−1.
i i
(b) e (M)=e (N)−(−1)rb (M,1).
r r I
(4) The following are equivalent
(a) x∗ is G (M)-regular.
I
(b) G (N)=G (M)/x∗G (M)
I I I
(c) b (M,z)=0
I
(d) e (M)=e (N).
r r
(5) (Sally descent) If depthG (N)>0 then x∗ is G (M)-regular.
I I
II:Base change.
2.4. Let φ: (A,m) → (A′,m′) be a flat local ring homomorphism with mA′ = m′.
Set I′ =IA′ and if N is an A-module set N′ =N ⊗A′. It can be easily seen that
(1) λA(N)=λA′(N′).
(2) projdimAN =projdimA′N′.
GROWTH OF HILBERT COEFFICIENTS 5
(3) HI(M,n)=HI′(M′,n) for all n≥0.
(4) dimM =dimM′ and grade(K,M)=grade(KA′,M′) for any ideal K of A.
(5) depthGI(M)=depthGI′(M′).
The specific base changes we do are the following:
(i)IfM isamaximalCohen-MacaulayA-modulewithfiniteGCIdimensionthen
by definition there exists a flat homomorphism (A,m) → (B,n) with mB = n and
B =Q/(f ,...,f ) for some local ring Q and a Q-regular sequence f ,...,f such
1 c 1 c
that projdim M ⊗ B <∞. In this case we set B =A′.
Q A
(ii)Ifk⊆k′ isanextensionoffieldsthenitiswell-knownthatthereexistsaflat
local ring homomorphism (A,m) → (A′,m′) with mA′ = m′ and A′/m′ = k′. We
use this construction when the residue field k of A is finite or countably infinite.
We take k′ to be any uncountable field containing k.
(iii) We can also take A′ to be the completion of A.
Remark 2.5. Let (A,m) be a Cohen-Macaulay local ring and let M be a maxi-
mal Cohen-Macaulay A-module with finite GCI dimension. After doing the base-
changes (i), (ii) and (iii) above we may assume that A = Q/(f ,...,f ) for some
1 c
complete Cohen-Macaulay local ring Q and a Q-regular sequence f ,...,f such
1 c
that projdim M <∞. Furthermore we may assume that the residue field of A is
Q
uncountable.
III:Quasi-polynomial functions of period 2.
Let us recall that a function f: Z →Z is said to be of quasi-polynomial type
≥0 ≥0
with period g ≥ 1 if there exists polynomials P ,P ,...,P ∈ Q[X] such that
0 1 g−1
f(mg+i)=P (m) for all m≫0 and i=0,··· ,g−1.
i
We need the following well-known result regarding quasi-polynomials of period
two.
Lemma 2.6. Let f: Z →Z . The following are equivalent:
≥0 ≥0
(i) f is of quasi-polynomial type with period 2.
(ii) f(n)zn = h(z) for some h(z)∈Z[z] and c≥0.
n≥0 (1−z2)c
Furthermore if P ,P ∈Q[X] are polynomials such that f(2m+i)=P (m) for all
P 0 1 i
m≫0 and i=0,1 then degP ≤c−1. (cid:3)
i
Convention: We set the degree of the zero-polynomialto be −∞.
Remark 2.7. Let f: Z → Z be of quasi-polynomial type with period 2. Let
≥0 ≥0
P ,P ∈ Q[X] be polynomials such that f(2m+i) = P (m) for all m ≫ 0 and
0 1 i
i=0,1. Then set degf =max{degP ,degP }.
0 1
As a consequence we get that
Corollary 2.8. Let f: Z → Z . If the function g(n) = f(n)+f(n−1) is of
≥0 ≥0
quasi-polynomial type with period 2 then so is f. Furthermore degf =degg.
Sketch of a proof. Set u(z) = f(n)zn. Then by our hypothesis and Lemma
n≥0
2.6 we have
P h(z)
(1+z)u(z)+f(0)z = ,
(1−z2)c
for some h(z)∈Z[z] and c≥0. An easy computation now shows that
r(z)
u(z)= ,
(1−z2)c
6 TONYJ.PUTHENPURAKAL
for some r(z)∈Z[z]. Thus f is of quasi-polynomial type with period two.
As f(n)≤g(n) we clearly get degf ≤degg. As g(2n)=f(2n)+f(2n−1) and
g(2n+1)=f(2n+1)+f(2n) we get that degg ≤degf. (cid:3)
The following result is also well-known:
Lemma 2.9. Let f: Z ×Z →Z be a function such that
≥0 ≥0 ≥0
h(z,w)
f(m,n)zmwn =
(1−z2)c(1−w)d
m,n≥0
X
Then for m,n≫0 we have
d−1
n+d−1−i
f(m,n)= (−1)ie (m) ,
i
d−1−i
i=0 (cid:18) (cid:19)
X
where the functions m 7→ e (m) for i = 0,...,d−1 are of quasi-polynomial type
i
with period 2 and degree ≤c−1. (cid:3)
IV: Eisenbud operators
Let Q be a Noetherian ring and let f = f ,...f be a regular sequence in Q. Set
1 c
A=Q/(f). Let M be a finitely generated A-module with projdim M finite.
Q
2.10. Let F:···F →···F →F →0 be a free resolution of M as a A-module.
n 1 0
Let t ,...t : F(+2)→F be the Eisenbud-operators; see [5, section 1.]. Then
1 c
(1) t are uniquely determined up to homotopy.
i
(2) t ,t commute up to homotopy.
i j
LetT =A[t ,...,t ]be a polynomialringoverAwithvariablest ,...,t ofdegree
1 c 1 c
2. Let D be an A-module. The operators t give well-defined maps
j
t : Exti (M,D)→Exti+2(M,D) for 1≤j ≤c and all i,
j A R
t : Tori+2(M,D)→Tori (M,D) for 1≤j ≤c and all i.
j A R
ThisturnsExt∗(M,D)= Exti (M,D)andTor∗(M,D)= Tori (M,D)
A i≥0 A A i≥0 A
intomodulesoverT (herewegiveanelementt∈Tori (M,D)degree−i). Further-
L A L
more these structure depend only on f, are naturalin both module arguments and
commute with the connecting maps induced by short exact sequences.
2.11. Gulliksen, [7, 3.1], proved that if projdim M is finite then Ext∗(M,D) is
Q A
a finitely generated T-module. If A is local and D = k, the residue field of A,
Avramov in [1, 3.10] proved a converse; i.e., if Ext∗(M,k) is a finitely generated
A
T-module then projdim M is finite. For a more general result, see [2, 4.2].
Q
Definition 2.12. (with notation as above:) Assume A is local with residue field
k. Set cxM =dim Ext∗(M,k), the complexity of M.
T A
We need the following result regarding the growth of lengths of certain Tor’s.
Recalla gradedmodule X overT =A[t ,...,t ] is saidto be *-ArtinianT-module
1 c
if every descending chain of graded submodules of X terminates.
Proposition 2.13. Let (Q,n) be a complete Noetherian ring and let f =f ,...f
1 c
be a regular sequence in Q. Set A = Q/(f) and m = n/(f). Let M be a finitely
generatedA-modulewithprojdim M finite. LetD beanon-zeroA-moduleoffinite
Q
length. Let Ext∗(M,D) be a finitely generated T = A[t ,...,t ]-module as above.
A 1 c
Then
GROWTH OF HILBERT COEFFICIENTS 7
(1) dim Ext∗(M,D)≤cx(M).
T A
(2) The function n7→ℓ(Extn(M,D)) is of quasi-polynomial type with period 2 and
A
degree ≤cx(M)−1.
(3) Tor∗(M,D) is a *-Artinian T-module. Here t∈TorA(M,D) has degree −n.
A n
(4) The function n7→ℓ(Torn(M,D)) is of quasi-polynomial type with period 2 and
A
degree ≤cx(M)−1.
Proof. (1) This follows from [2, Theorem 5.3].
(2) This easily follows from (1).
For (3), (4) we use the following result. Let E be the injective hull of k. Then
we have
(†) Hom (TorA(M,D),E)∼=Extn(M,Hom (D,E)).
A n A A
(3) From (†) it follows that the Matlis dual of Tor∗(M,D) is
A
Ext∗(M,Hom (D,E)). ByGulliksen’sresultwehavethatExt∗(M,Hom (D,E))
A A A A
isafinitelygeneratedgradedT-module. SobyMatlis-duality[4,3.6.17]wegetthat
Tor∗(M,D) is a *-Artinian T-module.
A
(4) This follows from (2) and (†). (cid:3)
As an immediate consequence we obtain:
Corollary 2.14. Let (A,m) be a Cohen-Macaulay local ring of dimension d and
let I be an m-primary ideal. Let M be a maximal Cohen-Macaulay A-module. If
M has finite GCI dimension over A then the function i 7→ eI(SyzA(M)) is of
0 i
quasi-polynomial type with period 2 and degree =cx(M)−1.
Proof. ByRemark2.5wemayassumeA=Q/(f)whereQiscompletewithinfinite
residue field, f = f ,··· ,f is a Q-regular sequence and projdim M is finite. Let
1 c Q
F be a minimal resolution of M. Then rankF = ℓ(TorA(M,k)). So by 2.13 we
i i
get that the function i 7→ rankF is quasi-polynomial of period two. Set M =
i i
SyzA(M). The exact sequence 0 → M → F → M → 0 yields eI(M ) +
i i+1 i i 0 i
eI(M )=(rankF )eI(A). The result now follows from 2.8. (cid:3)
0 i+1 i 0
We also prove:
Proposition 2.15. Let (A,m) be a Cohen-Macaulay local ring of dimension d
and let I be an m-primary ideal. Let M be a maximal Cohen-Macaulay A-module.
Assme M has finite GCI dimension over A. Fix n≥0. Then the function
InSyzi (M)
i7→ℓ A
In+1Syzi (M)
(cid:18) A (cid:19)
is of quasi-polynomial type with period two and degree ≤cx(M)−1.
Proof. ByRemark2.5wemayassumeA=Q/(f)whereQiscompletewithinfinite
residuefield,f =f ,··· ,f isaQ-regularsequenceandprojdim M isfinite. LetF
1 c Q
be a minimal resolutionof M. Set M =SyzA(M). Fix n≥0. The exactsequence
i i
0→M →F →M →0 yields an exact sequence
i+1 i i
0→TorA (M,A/In+1)→M /In+1M →F /In+1F →M /In+1M →0.
i+1 i+1 i+1 i i i i
Similarly we have an exact sequence
0→TorA (M,A/In)→M /InM →F /InF →M /InM →0.
i+1 i+1 i+1 i i i i
8 TONYJ.PUTHENPURAKAL
Computing lengths we have
InM InM In
ℓ i +ℓ i+1 =(rankF )ℓ +ℓ TorA (M,A/In+1)
In+1M In+1M i In+1 i+1
(cid:18) i(cid:19) (cid:18) i+1(cid:19) (cid:18) (cid:19)
−ℓ TorA (M,A/In) .(cid:0) (cid:1)
i+1
By 2.13 the terms in the right hand sid(cid:0)e are of quasi-poly(cid:1)nomial type with period
2 and degree ≤cx(M)−1. The result follows from Corollary 2.8. (cid:3)
As a consequence of the above Proposition we get:
Corollary 2.16. Let (A,m) be an Artin local ring and let I be an m-primary ideal.
LetM be afinitely generatedA-module of finiteGCI-dimension. Then for all j ≥0
the function i7→eI(SyzA(M)) is of quasi-polynomial type with period 2 and degree
j i
≤cx(M)−1.
Proof. We note that Ir =0 for some r ≥1. The result now follows from Corollary
2.15. (cid:3)
2.17. Eisenbud operators modulo a regular element.
Let x be A⊕M-regular. Let y ∈ Q be a pre-image of x in Q. As permutation of
regularsequencesisalsoaregularsequencewealsogetthaty isQ-regular. Wenow
note that projdim (M/xM) is finite. Let F be a minimal resolution of M over
Q/(y)
A. Then F/xF is a minimal resolution of M/xM over A/(x). Let t ,...,t be the
1 c
Eisenbud operators over F. By the construction of Eisenbud operators it follows
that t ,...,t induce operators t∗,...,t∗ over F/xF. Let N be an A/(x)-module.
1 c 1 c
Set E = TorA(M,N) = TorA/(x)(M/xM,N). Then the action of t on E is same
∗ ∗ i
as that of t∗.
i
2.18. Now assume that Q be a Noetherian ring and let f = f ,...f be a regular
1 c
sequence in Q. Set A = Q/(f). Let M be a finitely generated A-module with
projdim M finite. Let F be a minimal resolution of M. Let t ,··· ,t : F(+2) →
Q 1 c
F be the Eisenbud operators. Let I be an m-primary ideal of A and let R =
A[Iu] be the Rees algebra of A with respect to I. Let N = N be an R-
n≥0 n
module (not necessarily finitely generated). We claim that TorA(M,N) =
Li≥0 i
TorA(M,N ) has a bigraded R[t ,...,t ]-module structure. Here t have
i,n≥0 i n 1 c L i
degree(0,2)andifa∈In thendegaun =(n,0). Toseethisletv =aus ∈R . The
s
L v
map N −→N yields the following commutative diagram
n n+s
F⊗N // F⊗N (−2)
n n
tj
v v
(cid:15)(cid:15) (cid:15)(cid:15)
F⊗N //F⊗N (−2).
n+s n+s
tj
Takinghomologywegetthe requiredresult. Ananalogousargumentyieldsthat
Exti (M,N ) is a bigraded R[t ,...,t ]-module.
i,n≥0 A n 1 c
L
3. LI(M)
i
In this section we extend and simplify a technique from [9] and [10]. Let A be a
Noetherian ring, I an m-primary ideal and M a finitely generated A-module.
GROWTH OF HILBERT COEFFICIENTS 9
3.1. Set LI(M)= M/In+1M. Let R=A[Iu] be the Rees-algebra of I. Let
0 n≥0
S =A[u]. Then R is a subring of S. Set M[u]=M ⊗ S an S-module and so an
A
L
R-module. Let R(M) = InM be the Rees-module of M with respect to I.
n≥0
We have the following exact sequence of R-modules
L
0→R(M)→M[u]→LI(M)(−1)→0.
0
Thus LI(M)(−1) (and so LI(M)) is a R-module. We note that LI(M) is not a
0 0 0
finitely generated R-module. Also note that LI(M)=M ⊗ LI(A).
0 A 0
3.2. For i≥1 set
LI(M)=TorA(M,LI(A))= TorA(M,A/In+1).
i i 0 i
n≥0
M
We assertthat LI(M) is a finitely generatedR-module for i≥1. It is sufficient to
i
prove it for i = 1. We tensor the exact sequence 0 → R → S → LI(A)(−1) → 0
0
with M to obtain a sequence of R-modules
0→LI(M)(−1)→R⊗ M →M[u]→LI(M)(−1)→0.
1 A 0
Thus LI(M)(−1) is a R-submodule of R⊗ M. The latter module is a finitely
1 A
generated R-module. It follows that LI(M) is a finitely generated R-module.
1
3.3. Now assume that A is Cohen-Macaulay and M is maximal Cohen-Macaulay.
Set N = SyzA(M) and F = Aµ(M) (here µ(M) is the cardinality of a minimal
1
generator set of M). We tensor the exact sequence
0→N →F →M →0,
with LI(A) to obtain an exact sequence of R-modules
0
0→LI(M)→LI(N)→LI(F)→LI(M)→0.
1 0 0 0
As eI(F) = eI(M)+eI(N) we get that the function n → ℓ(TorA(M,A/In+1)) is
0 0 0 1
of polynomial type and degree ≤d−1. Thus dimLI(M)≤d.
1
We need the following result.
Proposition 3.4. Let (A,m) be Cohen-Macaulay local ring of dimension d ≥ 1
with infinite residue field and let I be an m-primary ideal. Let M be a MCM A-
module. Set M = SyzA(M). Let x be A⊕M ⊕M -superficial with respect to I.
1 1 1
Let R=A[Iu] be the Rees algebra of A with respect to I. Set X =xu∈R . If N
1
is a R-module, let H (X,N) denote the first Koszul homology of N with respect to
1
X. Then
(1) H (X,LI(M))= (In+1M: x)/InM.
1 0 n≥0
(2) X is LI(M) regular if and only if x∗ is G (M)-regular.
0 L I
(3) If x∗ is G (A)-regular then H (X,LI(M))∼=H (X,LI(M )).
I 1 1 1 0 1
Proof. (1)The mapM/InM −X→M/In+1M isgivenby m+InM 7→xm+In+1M.
The result follows.
(2) This follows from (1).
(3) We have an exact sequence 0 → LI(M) → LI(M ) → LI(F ) −→π LI(M) →
1 0 1 0 0 0
0. Let E = kerπ. As x∗ is G (A)-regular we get that H (X,LI(A)) = 0. So
I 1 0
H (X,E)=0. The result follows. (cid:3)
1
10 TONYJ.PUTHENPURAKAL
II: Ratliff-Rush filtration.
Let (A,m) be a Noetherian local ring and let I be an m-primary ideal in A. Let
M be a finitely generated A-module. The Ratliff-Rush submodule IM of M with
respect to I is defined by
g
IM = (Ik+1M: Ik).
k≥0
[
If depthM >0 then I]kM =IgkM for k ≫0.
3.5. Now assume d = dimA > 0 and A is Cohen-Macaulay. Also assume M is
MCM A-module. Let R be the Rees algebra of A with respect to I and let M be
its unique maximal graded ideal. Then by [10, 4.7] we have
I^n+1M
H0 (LI(M))= .
M 0 In+1M
n≥0
M
We need the following result:
Proposition 3.6. [with assumptions as above] If depthG (A)>0 then
I
H0 (LI(M))∼=H0 (LI(SyzA(M))).
M 1 M 0 1
Proof. Set M =SyzA(M). The exact sequence 0→M →F →M →0 yields an
1 1 1
exact sequence
0→LI(M)→LI(M )→LI(F)→LI(M)→0.
1 0 1 0 0
Therefore we have an exact sequence
0→H0 (LI(M))→H0 (LI(M ))→H0 (LI(F))
M 1 M 0 1 M 0
Now as depthG (A) > 0 we get that In = In for all n ≥ 1. So H0 (LI(A)) = 0.
I M 0
Thus H0 (LI(F))=0. The result follows. (cid:3)
M 0
f
3.7. Now assume that A = Q/(f) where f = f ,...,f is a Q-regular sequence
1 c
and M is a finitely generated A-module with projdim M finite. Let I be an m-
Q
primary ideal. Set L(M) = LI(M) = TorA(M,LI(A)). Give L(M)
i≥0 i i≥0 i 0
a structure of a bigraded S = R[t ,··· ,t ]-module as discussed in 2.18. Then
1 c
L L
H0 (L(M)) = H0 (LI(M)) is a bigraded S-module. To see this note that
M i≥0 M i
H0 (L(M))=H0 (L(M)).
M LMS
4. dimension one
InthissectionweassumedimA=1. WeprovethatifM isMCMA-modulewith
finite GCI dimension then the function i 7→ eI(SyzA(M)) is of quasi-polynomial
1 i
typewithperiodtwo. IfG (A)isCohen-Macaulaythenweshowthatthefunctions
I
j 7→depthG (SyzA(M)) and j 7→depthG (SyzA (M)) is constant for j ≫0.
I 2j I 2j+1
The following Lemma is crucial.
Lemma4.1. Let(A,m)beaCohen-Macaulay localringofdimension oneandwith
an infinite residue field. Let I be an m-primary ideal and let x be A-superficial with
respect to I. Assume Ir+1 =xIr. Let M be a MCM A-module. Then
(1) deghI(M,z)≤r
(2) The postulation number of the Hilbert-Samuel function of M with respect to I
is ≤r−2.