Length and Multiplicities in Graded Commutative Algebra
aa r X i v : . [ m a t h . A T ] J u l LENGTH AND MULTIPLICITIES IN GRADED COMMUTATIVE ALGEBRA
MARK BLUMSTEIN
Contents
1. Introduction 12. A Review of Standard Definitions and Facts in the Graded Category 13. Graded Length 34. Graded Localization 65. Krull Dimension in grmod(A) 96. Graded ideals of definition and graded systems of parameters 137. Multiplicities for graded modules 148. Koszul complexes in grmod(A) and *Koszul multiplicities 199. Multiplicities and Degree for Positively Graded Rings 2210. Multiplicities and Euler-Poincar´e series 23References 251.
Introduction
This paper is a synthesis of the main ideas from the author’s matster’s thesis. The author wouldlike to thank Jeanne Duflot for her steady guidance and dedication as advisor. This paper came to bevia the study of the commutative algebra of equivariant cohomology rings H ∗ G ( X ) associated to a group G acting on a topological space X , which are of course naturally graded. This study was really begunabout 50 years ago by Quillen [11], [12] who described the (Krull) dimension of these graded rings andgave a decomposition of their spectra (in the sense of algebraic geometry). Many authors have followedwith their own studies of these rings from the point of view of commutative algebra; recent contributionsinclude the work of Symonds [16], Lynn [9] and Duflot [6].We were attempting to generalize the work of Lynn [9], resulting in the paper [4], and found that weneeded a careful exploration of various notions of multiplicity for graded rings which were nonstandard intwo ways: not positively graded (for example, they might be graded localizations) and/or not generatedby elements of degree 1. Since, as algebraic topologists, the degree of homogeneous elements in theserings can have geometric or representation-theoretic meaning to us, we were not comfortable with usingthe geometer’s trick of the Veronese embedding to get around the second problem.In this paper, there is nary a word about equivariant cohomology or algebraic topology. It is allabout graded commutative algebra and much of it is expository. The main results of interest to us arethe theorems about length, multiplicity and a second notion of “degree” (which is really another sort ofmultiplicity) for the Poincar`e series of a graded ring. When we were able to write results for rings froma larger collection than simply those of cohomology type, we tried to do so. We hope that workers infields other than algebraic topology find this exposition useful.2. A Review of Standard Definitions and Facts in the Graded Category
We consider only strictly commutative Z -graded rings and modules in this paper and use the standardnotation: if A is a graded ring, M is a graded A -module and n ∈ Z , M n is the set of homogeneous elementsof degree n (although “degree” will also have another meaning here); for every x ∈ M , x may be writtenuniquely as x = Σ n ∈ Z x n , where x n ∈ M n and x j = 0 for all but finitely many j , the element x n is thehomogeneous component of x of degree n . An element x ∈ M is a homogeneous element if and only if x has at most one nonzero homogeneous component. If d ∈ Z , we use the following convention for thesuspended A -module M ( d ): M ( d ) j . = M d + j , Date : July 16, 2020. or every j ∈ Z ; also, if M and N are graded A -modules, and ψ : M → N is an A -module homomorphism,then ψ is a graded homomorphism of degree d if for every integer n , ψ ( M n ) ⊆ N n + d . Definition 2.1.
Suppose A is a graded ring. The category grmod ( A ) has objects finitely generatedgraded A -modules. The morphisms of grmod ( A ) are the A -module homomorphisms which are graded ofdegree zero (i.e. degree-preserving).Recall that a submodule N of a graded A -module M is a graded submodule if and only if it is generatedover A by homogeneous elements; this is equivalent to the condition that for every element of N , all ofits homogeneous components are in N .Whether or not M and A are graded, the set of associated primes for M in A is denoted by Ass A ( M )and the support of an A -module M is the set Supp A ( M ) . = { p ∈ Spec ( A ) : M p = 0 } . For M finitelygenerated over A , p ∈ Supp A ( M ) if and only if Ann A ( M ) ⊆ p . A prime ideal of A that contains Ann A ( M ), and is minimal amongst all primes containing Ann A ( M ) is called a minimal prime for M . If M = A/ I for an ideal I of A , then a minimal prime for M is called a minimal prime in A over I . Notethat Ass A ( M ) ⊆ Supp A ( M ). Definition 2.2.
The graded support of M , ∗ Supp A ( M ), is the set of all graded prime ideals in thesupport of M . If I is a graded ideal in A , the graded variety of I , ∗ V ( I ), is the set of all graded primesin A containing I . (Recall that if J is any ideal in A , graded or not, V ( J ) is the set of prime ideals in A containing J .)We collect some standard results about Ann A ( M ) and Ass A ( M ) for the graded category below. Proposition 2.3.
Let A be a graded ring with M a graded A -module. i) Ann A ( M ) is a graded ideal in A and Ann A ( M ) = Ann A ( M ( d )) for every d ∈ Z . ii) If p ∈ Ass A ( M ) , then p is a graded ideal of A and is the annihilator of a homogeneous elementin A . iii) Therefore, if I is a graded ideal in A , all primes in Ass A ( A/ I ) are graded. iv) If p is a minimal prime for M , then p ∈ Ass A ( M ) ; thus, all minimal primes for M are graded. Finally, for an ideal I in a graded ring, graded or not, I ∗ is defined as the largest, graded idealcontained in I ; i.e. I ∗ is the ideal generated by all homogeneous elements of I ; if p is a prime ideal in A , p ∗ is also a prime ideal in A .2.1. Noetherian graded rings.
When we say that a graded A -module M is a Noetherian A -module,we mean that it is Noetherian in the usual sense, forgetting the grading.One can show [5] that the following conditions on A are equivalent: • A is Noetherian. • Every graded ideal in A is generated by a finite set of homogeneous elements. • A is Noetherian and A is a finitely generated A -algebra by a set of homogeneous elements.So, if M is a finitely generated graded A -module, and A is Noetherian, then M is Noetherian, and • every A -submodule N of M is finitely generated over A , and if N is graded, it is generated over A by a finite set of homogeneous elements; • for every j , M j is a Noetherian A -module and so every A -submodule of M j is finitely generated:if one has an ascending chain X ⊆ X ⊆ · · · of A -submodules of M j , then letting AX i be the(graded) A -submodule generated by X i , we must have AX i = AX i +1 for all i greater than orequal to some fixed N . By degree considerations, X i = AX i ∩ M j for every i , so X i = X i +1 for i ≥ N . Lemma 2.4. If A is a graded Noetherian ring and M ∈ grmod ( A ) , a. Supp A ( M ) = V ( Ann A ( M )) . = V ( M ) , so that ∗ Supp A ( M ) = ∗ V ( Ann A ( M )) . = ∗ V ( M ) . b. If I is a graded ideal in A , ∗ V ( M/ I M ) = ∗ V ( M ) ∩ ∗ V ( I ) = ∗ V ( Ann A ( M ) + I ) . c. If I and ˜ I are two graded ideals in A then their radicals are also graded, and ∗ V ( I ) = ∗ V (˜ I ) ifand only if √I = p ˜ I .Proof. The proof of a. can be found in [15]; also [15] tells us that V ( M/ I M ) = V ( M ) ∩ V ( I ) = V ( Ann A ( M ) + I ) and so b. follows from this. For c., the forward implication follows since all minimalprimes over I are graded, thus occur as minimal elements both in V ( I ) and ∗ V ( I ), and √I is theintersection of the (finite number of) minimal primes over I . (cid:3) he type of filtration described in the following lemma will be used several times in this paper; andwe provide a brief discussion of its proof. Lemma 2.5. If A is a Noetherian graded ring and M ∈ grmod ( A ) is nonzero, there exists a finitefiltration M • of M by graded submodules ( M = 0; M n = M ), integers d j and graded primes p j ∈ Spec ( A ) with graded isomorphisms of graded A -modules, M i +1 /M i ∼ = A/ p i +1 ( − d i +1 ) . Furthermore,given a finite list of graded primes ( p j | ≤ j ≤ n ) in Spec ( A ) (not necessarily distinct), and a gradedfiltration M • of M by graded submodules as above, we must have Ass A ( M ) ⊆ { p j | ≤ j ≤ n } ⊆ ∗ Supp A ( M ) and these three sets must have the same minimal elements, the set of which consists of the minimalprimes of M . Finally, if p is a minimal prime for M , forgetting all gradings and using the fact thatthe ordinary localization M p is a finitely generated Artinian A p -module, the number of times that A/ p ,possibly shifted, occurs as a graded A -module isomorphic to a subquotient of M • is always equal to thelength of M p as an A p -module and is thus independent of the choice of the graded filtration M • .Proof. We remind the reader of the proof of the first statement, adapted to the graded case: Using theNoetherian hypothesis, since M = 0, Ass A ( M ) = ∅ , so we may pick an element p ∈ Ass A ( M ). Then p is graded and there exists a homogeneous element m ∈ M such that p = ann A ( m ). Supposedeg( m ) = d , then A/ p ( − d ) is graded isomorphic to a graded A -submodule of M which we call M .If M = M , we are done. If not, we take the A -module M/M , notice that it is nonzero, and producean associated prime p ∈ Ass A ( M/M ). Since M/M is a graded A -module p is also graded. Suppose p = Ann A ( m ) where m / ∈ M is a homogeneous element in M and deg( m ) = d ; m is the coset of m in M/M . Thus there is a graded submodule M ⊆ M such that M /M is graded isomorphic to A/ p ( − d ). At some point there must be a smallest n ≥ M n = M , since otherwise the Noetherianhypothesis would be violated.For the last two statements, we refer to [15]. (cid:3) Graded Length
We’ve already started using the notation “ ∗ P ” for a modification of a property or definition P inthe ungraded category to obtain a property or definition in the graded category, and we continue it inthis section. From now on, unless stated otherwise, all modules and rings are graded, although we willsometimes redundantly restate this. Definition 3.1. If A is a graded ring, a graded ideal N is *maximal if and only if N 6 = A and N is amaximal element in the set of all proper graded ideals of A . Definition 3.2.
A *simple A -module is a nonzero graded A -module with no nonzero proper gradedsubmodules. A *composition series for a graded module M ∈ grmod ( A ) is a chain of graded A -submodulesof M , 0 = M ⊂ · · · ⊂ M n = M such that each successive quotient M i /M i − is isomorphic as a graded A -module to a *simple module. The *length of the *composition series 0 = M ⊂ · · · ⊂ M n = M isdefined to be n .The fundamental theorem about *composition series mirrors that in the ungraded case. The proof ofthe following is nearly identical to the ungraded case ([7],Theorem 2.13), with only minor adjustmentsmade to account for the grading, and we leave this effort to the reader. Theorem 3.3.
Suppose for M ∈ grmod ( A ) that a *composition series of length n for M exists. Then,every chain of graded submodules of M has length ≤ n , and can be refined to a *composition series oflength n . Every *composition series for M has length n . Definition 3.4. If M has a *composition series as an A -module, the *length of M ∈ grmod ( A ) is definedto be the length of a *composition series for M . We use the notation ∗ ℓ A ( M ) for this number; as usual,we say ∗ ℓ A ( M ) = ∞ if M does not have a *composition series.If we forget all gradings on A and M , ℓ A ( M ) denotes the usual length of M as an A -module.Some properties of ∗ ℓ A are as expected: • If 0 → M → N → P → grmod ( A ), then N has a *composition series ifand only if both M and P do; and in this case, ∗ ℓ A ( N ) = ∗ ℓ A ( M ) + ∗ ℓ A ( P ). • If d ∈ Z , then ∗ ℓ A ( M ( d )) = ∗ ℓ A ( M ). he only simple A -modules in the ungraded case are A -modules of the form A/ m , where m is amaximal ideal of A (recall all rings are commutative). Thus we are led to define graded fields; these arethe rings of *length zero as modules over themselves. Theorem 3.5. [7]
Let F be a graded ring. The following are equivalent: (1) Every nonzero homogeneous element in F is invertible. (2) F is a field and either F = F , or there exists a d > and an x ∈ F d such that F ∼ = F [ x, x − ] as a graded ring. In fact, in this last case, d > is the smallest positive degree with F d = 0 . (3) The only graded ideals in F are F and .A ring satisfying any of these three equivalent conditions is called a graded field . Example 3.6. If F is a graded field with a nonzero positive degree element, then F is *simple as amodule over itself, but it is not simple as such. To see this write F = F [ t, t − ], with deg( t ) = d >
0, and F a field. So F is certainly *simple, but if J is the ungraded ideal generated by t + 1, J is a nonzeroproper F -submodule of F , so F is not simple. Furthermore, F has a unique *maximal ideal, the zeroideal, but has as least as many ungraded nonzero maximal ideals as the nonzero elements of F . While F has a *composition series, it has no composition series.Similarly to the ungraded case, M is a *simple A -module if and only if there exists a *maximal ideal N of A , an integer d and a graded A -module isomorphism M ∼ = ( A/ N )( d ): if M is *simple, let x be anynonzero homogeneous element of M , say deg( x ) = − d . Then, the submodule of M generated by x isnonzero and graded, so must be all of M . The homomorphism A ( d ) → M of graded A -modules definedby a ax is thus surjective; its kernel is a graded ideal in A ( d ) of the form N ( d ) for some graded ideal N of A ; since M is *simple, N must be *maximal. The converse is left to the reader.Other facts parallel to the ungraded case include: 1) for every proper graded ideal I in A , there existsa *maximal ideal N containing I ; 2) if N is a proper graded ideal of A , then N is *maximal if and onlyif A/ N is a graded field. Thus, every *maximal ideal in A is a graded prime ideal. Furthermore, if N is*maximal in A , then N is a maximal ideal in A .The structure of finitely generated graded modules over graded fields mirrors that for the ungradedcategory: Lemma 3.7.
Suppose that M is a finitely generated graded module over a graded field F = F [ t, t − ] ,where t has positive degree d and F is a field. Then a) M is a free graded F -module, of finite rank, on a set of homogeneous generators. b) M is a finite-dimensional vector space over F of F -dimension less than or equal to the rankof M over F .Proof. Assume M = 0. Say M is finitely generated over F by homogeneous elements e , . . . , e r , where r ≥ M as an F -module. Then, M isfree on the e j s : certainly this set spans M over F . Suppose that there is a relation P j α j e j = 0 , with α j ∈ F . We may assume that all the α j s are homogeneous. If α r = 0, then it is invertible in F , so P r − j =1 α − r α j e j + e r = 0, implying that r is not minimal. Therefore α r = 0; and continuing the process, α j = 0 for every j .Set d j = deg e j . Now, note that X . = { t − d j /d e j | ≤ j ≤ r and d divides d j } is a basis for M over F ; of course, if d does not divide any d j , then M = 0. To see this, note that X is linearly independentover F , since the e j s are linearly independent over F . If x ∈ M , then x = P j α j e j , where α j is ahomogeneous element of F and deg α j + d j = 0 , ∀ j . Now, if α j = 0, d divides its degree, by definition of F . Thus, d divides d j for every j such that α j = 0. If d divides d j , then α j = β j t − d j /d , where β j ∈ F .Thus x is in the F -span of X . (cid:3) Definition 3.8. M ∈ grmod ( A ) is said to be a *Artinian module if M satisfies DCC on all chains ofgraded A -submodules of M .Unlike the Noetherian case, an A -module M can be *Artinian without being Artinian: an example isgiven by A = M , where A is a graded field with a nonzero positive degree element.Similarly to the ungraded case, we have Lemma 3.9.
Suppose that A is a graded Noetherian ring and M ∈ grmod ( A ) . Then the following areequivalent: a) M is *Artinian. ) ∗ ℓ A ( M ) < ∞ . c) ∗ V ( M ) consists of a finite number of *maximal ideals.Proof. The proof of the equivalence of a) and b) in the ungraded case, as in [2], adapts in a straightforwardway to the graded case. Note that the proof of “b) implies a)” does not require A to be Noetherian.To see how b) implies c), assume that M has a *composition series0 = M ⊂ M ⊂ · · · ⊂ M n − ⊂ M n = M ;the *simplicity of the subquotients means that there are *maximal graded ideals m i of A and integers d i such that M i /M i − ∼ = ( A/ m i )( d i ) as graded A -modules. Thus, m m · · · m n ⊆ Ann A ( M ). If p is aprime minimal over Ann A ( M ), then we have seen that p is graded. Since m · · · m n ⊆ p , we must have m i ⊆ p for at least one i . But m i is *maximal, so m i = p . Therefore ∗ V ( M ) ⊆ { m , . . . , m n } .For c) implies b), since p Ann A ( M ) is the intersection of the primes minimal over Ann A ( M ), andthere are a finite number of these, all graded, the hypothesis implies that this finite list of primesconsists entirely of *maximal ideals; say these ideals are m , . . . , m n . Thus, there is an N such that( m · · · m n ) N ⊆ Ann A ( M ) and there is a sequence ˜ m , . . . , ˜ m nN of *maximal ideals in A , not necessarilydistinct, whose product is contained in Ann A ( M ). Analogously to the ungraded case, one can thenconstruct a *composition series for M . (cid:3) If V is a graded vector space over an ordinary field k , where k is regarded as a graded ring concentratedin degree zero (this means that all nonzero elements have degree zero), thenvdim k ( V ) . = the dimension of V as a vector space over k. Lemma 3.10.
Let A be a graded Noetherian ring which is a finitely generated graded algebra over a field k ⊆ A , M ∈ grmod ( A ) , V a graded finite dimensional vector space over k , and say that vdim k ( V ) = d .If a ∈ A , m ⊗ v ∈ M ⊗ k V , then give M ⊗ k V an A -module structure by a · ( m ⊗ v ) . = ( a · m ) ⊗ v , andgrade M ⊗ k V in the usual way. Then ∗ ℓ A ( M ⊗ k V ) = ∗ ℓ A ( M ) · d. Proof.
Since V is finite dimensional over k , we may suspend V appropriately and assume, without lossof generality, that V j = 0 for j <
0; in this case, there exists an n such that j > n implies V j = 0.Define a graded filtration of M ⊗ k V by graded A -modules: F i . = M ⊗ k ( V ⊕ · · · ⊕ V n − i ) for 0 ≤ i ≤ n ,and F n +1 . = 0. Consider that F i / F i +1 ∼ = M ⊗ k V n − i , and the additive property of length allows ∗ ℓ A ( M ⊗ k V ) = P ni =0 ∗ ℓ A ( F i / F i +1 ) = P ni =0 ∗ ℓ A ( M ⊗ k V n − i ).By hypothesis, each graded component V j of V is a finite dimensional graded vector space concentratedin degree j . Thus, there is a graded isomorphism for each j , V j ∼ = k f ( j ) ( − j ) where f ( j ) is a functiongiving the vector space dimension of V j . Since M ⊗ k V j ∼ = M ⊗ k k f ( j ) ∼ = ⊕ f ( j )1 M ( − j ), we have that ∗ ℓ A ( M ⊗ A V j ) = ∗ ℓ A ( M ) · f ( j ). By hypothesis, P nj =0 f ( j ) = d , the total vector space dimension of V ,and finally ∗ ℓ A ( M ⊗ k V ) = P ni =0 ∗ ℓ A ( M ⊗ k V n − i ) = P ni =0 ∗ ℓ A ( M ) · f ( n − i ) = ∗ ℓ A ( M ) P ni =0 f ( n − i ) = ∗ ℓ A ( M ) · d . (cid:3) Positively or Negatively Graded Rings.
A graded ring S is positively (resp. negatively) graded ifand only if S i = 0 for i < i > S + (resp. S − ) of S is defined as ⊕ i> S i (resp. ⊕ i< S i ). Note that if M ∈ grmod ( S ), since S is positively (resp. negatively) graded, there existsan integer e such that M i = 0 for all i < e (resp. i > e ). Also, for a proper, graded ideal m of S , thefollowing are equivalent: • m is *maximal in S . • m = m ⊕ S + , (resp. m ⊕ S − ) and m (the degree zero elements of m ) is a maximal ideal in S . • S/ m is a graded field, concentrated in degree zero; i.e. S/ m is an ordinary field. • m is a maximal ideal in S .For positively or negatively graded rings, there is no difference between *length and length: Lemma 3.11.
Suppose that S is a positively or negatively graded Noetherian ring, and M ∈ grmod ( S ) is such that ∗ ℓ S ( M ) < ∞ . Then, ∗ ℓ S ( M ) = ℓ S ( M ) .Proof. Since ∗ V ( M ) consists of a finite number of *maximal ideals, there is a sequence of graded S -modules 0 = M ⊂ M ⊂ · · · ⊂ M n − ⊂ M n = M, maximal graded ideals m i of S and integers d i such that M i /M i − ∼ = ( S/ m i )( d i ) as graded S -modules.By the remark above, S/ m i is concentrated in degree 0 and each m i is a maximal ideal in S . So, forgettinggradings everywhere, the given *composition series is a composition series. (cid:3) Even in the cases where *length and length coincide, we’ll usually just talk about *length, emphasizingconstructions using graded modules only. For example,
Lemma 3.12.
Suppose S is a positively graded ring and X ∈ grmod ( S ) . a) If ∗ ℓ S ( X ) < ∞ , there exists an integer J such that if j > J , then X j = 0 . b) If S i is finitely generated as an S -module for every i , then X j is a finitely generated S -module,for every j . c) Suppose S is Artinian, S i is finitely generated as an S -module for every i , and there exists aninteger J such that if j > J , then X j = 0 . Then, ℓ S ( X j ) < ∞ for every j , and ∗ ℓ S ( X ) = ℓ S ( X ) < ∞ , where ℓ S ( X ) . = P j ℓ S ( X j ) is the (total) S -length of X .Proof. For every t ∈ Z define X ≥ t . = ⊕ s ≥ t X s . Since S is positively graded, X ≥ t is a graded S -submoduleof X . Since X is finitely generated over S , and S is positively graded, there exists a t ∈ Z such that X ≥ t = X . So we have a descending chain of graded S -submodules of X · · · ⊆ X ≥ t + k ⊆ X ≥ t + k − ⊆ · · · ⊆ X ≥ t +1 ⊆ X ≥ t = X. ( ∗ )For a), if ∗ ℓ S ( X ) < ∞ , X is *Artinian, so this chain stabilizes. By definition, this means that thereexists an J ≥ t such that X j = 0 for j > J .For b), let t be defined as in the first paragraph above; assume that X t = 0. Then, one can prove,by induction on j , that each X j is finitely generated over S as follows. If j = t , then since X is finitelygenerated as an S -module, say by x , . . . x N , if β t = { x i | deg( x i ) = t } , X t must be generated by β t as an S -module. Assume that j > t and X u is finitely generated over S for u < j . Then, X
0, then deg( x i ) is strictly less than j so that x i is in the S -span of β
Localizing in the graded category can be done in a few ways. We may localize as usual, forgetting thegraded structures, we may localize at sets consisting of homogeneous elements, or as in Grothendieck[8], consider the degree zero part of this last localized module. In this section we make the relevantdefinitions, and compare the different methods.
Definition 4.1.
Let T be a multiplicatively closed subset (MCS) consisting entirely of homogeneouselements of A . We’ll call this a “GMCS”. Since T is an MCS we may construct the localization T − M as usual. By definition, T − M is graded by: ( T − M ) i . = { mt ∈ T − M | m is homogeneous and deg m − deg t = i } .With this grading, T − M becomes a graded T − A -module. In the case where p ∈ Spec ( A ),and T is the set of homogeneous elements of A − p , we use the notation M [ p ] to denote the localization T − M , graded as above.For a GMCS T , we’ll assume from now on that 1 ∈ T and 0 / ∈ T .The following list of lemmas collect some facts about graded localizations; we leave the proofs to thereader. Lemma 4.2.
Let p ∈ Spec ( A ) . The set of homogeneous elements in A − p is equal to the set ofhomogeneous elements in A − p ∗ . Therefore, M [ p ] = M [ p ∗ ]6 emma 4.3. Let p and q be prime ideals of A , with q graded. Then, ( A/ q ) [ p ] = 0 if and only if q ⊆ p ∗ .If p is a minimal prime of A , then ( A/ q ) [ p ] = 0 if and only if q = p . Lemma 4.4. If M ∈ grmod ( A ) , and T is a GMCS in A , then a) T − M ∈ grmod ( T − A ) . b) If A is a Noetherian ring then T − A is a Noetherian ring and T − M ∈ grmod ( T − A ) . c) There is a one-one, inclusion-preserving correspondence between the prime ideals in A that aredisjoint from T , and the prime ideals in T − A given by p T − p ; moreover this correspondencerestricts to a one-one correspondence between the graded prime ideals in A disjoint from T andthe graded prime ideals in T − A , and further restricts to a one-one correspondence between theideals (all graded) in Ass A ( M ) that are disjoint from T , and the ideals (also all graded) in Ass T − A T − M . Lemma 4.5.
Let M ∈ grmod ( A ) , T a GMCS in A , and let d be any integer. Then there is a gradedisomorphism of graded T − A -modules T − ( M ( d )) ∼ = ( T − M )( d ) . Lemma 4.6.
Suppose that M ∈ grmod ( A ) . a) p ∈ Supp A ( M ) if and only if M [ p ∗ ] = 0 if and only if p ∗ ∈ ∗ V ( M ) . Therefore, ∗ V ( M ) = ∗ Supp A ( M ) = { q ∈ ∗ V ( A ) | M [ q ] = 0 } . b) If → M → N → P → is a short exact sequence in grmod A , then ∗ V ( N ) = ∗ V ( M ) ∪ ∗ V ( P ) . Proof.
Since V ( N ) = V ( M ) ∪ V ( P ), b) follows.For a ), it’s straightforward to see that the ungraded object M p = 0 implies that M [ p ∗ ] = 0. If M [ p ∗ ] = 0, and Ann A ( M ) is not contained in p ∗ , then since both are graded ideals, there exists ahomogeneous element r ∈ Ann A ( M ) such that r / ∈ p ∗ . But then, m/t = 0 /r = 0 for every m ∈ M andhomogeneous t / ∈ p ∗ . Finally, suppose that Ann A ( M ) ⊆ p ∗ , yet M p = 0. If x , . . . , x j are homogeneouselements of M generating M as an A -module, since x i / i , there exist s i / ∈ p such that s i x i = 0 for each i . We may assume that each s i is homogeneous, since x i is. Since s i / ∈ p , s i / ∈ p ∗ ,so that s = s s · · · s j / ∈ p ∗ and is homogeneous. Furthermore, sm = 0 for every m ∈ M , so s ∈ p ∗ , acontradiction. (cid:3) Lemma 4.7.
For p a graded prime in A , T a GMCS, ( T − p ) = T − p ∩ ( T − A ) , and if p ∩ T = ∅ then ( T − p ) is a prime ideal in ( T − A ) . Definition 4.8. [8] If p ∈ Spec ( A ), then we denote the degree 0 part of M [ p ] by M ( p ) .If M is an A -module, M ( p ) is an A ( p ) -module. Example 4.9. If A is a graded ring, p is a graded prime ideal in A , and T is the GMCS consisting of thehomogeneous elements of A − p , then T − p . = p [ p ] is a *maximal ideal in T − A . = A [ p ] and p ( p ) = ( p [ p ] ) is a maximal ideal in A ( p ) .Now, if M is a graded A -module and p is a graded prime ideal, we know that the standard localization M p isn’t usually graded as we allow inhomogeneous elements of A not in p to be inverted. If p is a minimalprime ideal for M , it must be graded, as we have seen, and from ordinary commutative algebra, we knowthat M p has finite length as an A p -module. But we can also consider the graded localization M [ p ] andthe comparison between length and *length: Theorem 4.10.
Suppose that A is a Noetherian graded ring. Let M ∈ grmod ( A ) , and p be a primeminimal over the graded ideal Ann A ( M ) . Then, a *composition series exists for the graded A [ p ] -module M [ p ] . Moreover, ∗ ℓ A [ p ] ( M [ p ] ) = ℓ A p ( M p ) . Proof.
We will produce a *composition series for M [ p ] , as an A [ p ] -module and calculate its length.Construct a graded filtration M • as in Lemma 2.5, and then localize this filtration using the gradedlocalization. We now have a filtration of M [ p ] by graded A [ p ] -submodules which looks like 0 = ( M ) [ p ] ⊆ ( M ) [ p ] ⊆ · · · ⊆ ( M ) [ p ] . By exactness of localization and the condition on successive quotients of M • wehave that ( M i +1 /M i ) [ p ] ∼ = ( A/ p i +1 ( − d i +1 )) [ p ] is a graded isomorphism of A [ p ] -modules, for appropriateintegers d i , where the graded primes p i are as in 2.5.There is a graded isomorphism (( A/ p i )( − d i )) [ p ] ∼ = ( A/ p i ) [ p ] ( − d i ), and( A/ p i ) [ p ] ( − d i ) = 0 if and only if p = p i by minimality of p ).In the case that p = p i , ( A/ p i ) [ p ] = 0 and we have ( M i ) [ p ] = ( M i − ) [ p ] . Now throw away all suchsubmodules ( M i ) [ p ] which are equal to the submodule ( M i − ) [ p ] to get a reduced filtration (( M j ) [ p ] ) of M [ p ] , where for each j , ( M j ) [ p ] ⊂ ( M j +1 ) [ p ] is a strict inclusion, ( M s ) [ p ] = M [ p ] for some s , and thezeroth term of the filtration is zero. The claim is that this reduced filtration forms a *composition seriesfor M [ p ] of *length equal to the number of times that A/ p , shifted, appeared as a successive quotient inthe original filtration M • .For each j the successive quotient ( M j +1 ) [ p ] / ( M j ) [ p ] is graded isomorphic to ( A/ p ) [ p ] ( − d j +1 ), asan A [ p ] -module. But ( A/ p ) [ p ] is a graded field, since A [ p ] has a unique graded prime ideal p [ p ] ; thus,( M j +1 ) [ p ] / ( M j ) [ p ] ∼ = ( A/ p ) [ p ] ( − d j +1 ) is a *simple A [ p ] -module for each j .Going back to the original filtration M • and forgetting the grading everywhere, recall that the numberof times that A/ p appears as a successive quotient in any finite filtration of M which has successivequotients isomorphic to A/ q for some prime q , graded or not, is always the same, and is equal to ℓ A p ( M p ). (cid:3) *Local rings.Definition 4.11. If A is a graded ring, then A is *local if and only if there is one and only one *maximalideal of A .Some examples of *local rings are immediate. For example, a graded field is always *local, with unique*maximal ideal 0. This shows that generally, a *maximal ideal of a graded ring A may not be a maximalideal of A . If p is a graded prime in A , then A [ p ] is a *local ring with unique *maximal ideal p [ p ] . Theend of this section gives a partial characterization of *local rings.Also, as one might expect, if A is a *local graded ring, with unique *maximal ideal N , then • For every proper ideal I (graded or not) of A , I ∗ ⊆ N . • Every homogeneous element of A − N is invertible: i.e., for every x ∈ A − N with deg x = d ,there exists a y ∈ A − N of degree − d such that xy = 1 ∈ A . • A/ N is a graded field; also, for every y ∈ N j and every x ∈ A − j , 1 − xy ∈ A is a unit in A . Lemma 4.12. (Graded Nakayama’s lemma) Suppose ( A, N ) is a *local ring and M is a finitely generatedgraded A -module with N a graded A -submodule of M . If q is a proper graded ideal in M , then N + q M = M implies that M = N .Proof. (Slight variation of proof of Nakayama’s lemma in [2].) We may assume N = 0 by passing to M/N . Say M = 0; choose a homogeneous generating set x , . . . , x r for M over A with a minimal number r ≥ q M = M ; then there are homogeneous elements α j ∈ q ⊆ N such that x r = α x + · · · + α r x r ; we must have deg α j +deg x j = deg x r for every j such that α j x j = 0. By minimality, α r x r = 0 and so deg α r = 0. Using the remarks above, 1 − α r is an invertibleelement of A . Thus, we may write x r as an A -linear combination of x , . . . , x r − , contradicting theminimality of r . (cid:3) Proposition 4.13. If A is *local and Noetherian with unique *maximal ideal N , and M is a nonzerofinitely generated graded A -module with N a minimal prime over Ann A ( M ) (equivalently, ∗ V ( M ) = {N } ), then M is a *Artinian A -module, and for each j ∈ Z , M j is an Artinian A -module. Furthermore,for each j ∈ Z , ℓ A M j ≤ ∗ ℓ A M. If, in addition, there is a homogeneous element of degree 1 (or,equivalently, -1) in A − N , ℓ A M j = ∗ ℓ A M for every j .Proof. M is *Artinian, since ∗ V ( M ) = {N } . In fact, in this case, M has a *composition series with theproperty that each successive quotient is annihilated by N and is also free of rank one over the gradedfield A/ N . Taking the degree j part of each module in this *composition series, we get a chain of A -submodules of M j and the dimension of each successive quotient over the field K . = ( A/ N ) . = A / N is either zero or 1. Thus, since N also annihilates each successive quotient in this “degree j” filtration,we see that we can make appropriate deletions in the “degree j” part of the *composition series for M to yield a composition series for M j over A of length less than or equal to ∗ ℓ A ( M ).For the last statement, supposing that there is a homogeneous element of degree 1 in A − N , thenthere are nonzero elements of every degree in the graded A -module ( A/ N )( d ), for every d ∈ Z ; to seethis, note that A/ N is a graded field, equal to K [ T, T − ], where T = 0 has least positive degree in A/ N , namely degree 1. So, each successive quotient in the *composition series for M is nonzero in every egree. After taking the “degree j” part of this *composition series, each quotient must be of rank 1 over K . Thus the equality holds. (cid:3) Corollary 4.14.
Suppose that A is a Noetherian graded ring. Let M ∈ grmod ( A ) , and p be a primeminimal over Ann A ( M ) , necessarily graded. Then, M [ p ] is an *Artinian A [ p ] -module and M ( p ) is anArtinian A ( p ) -module. Also, ℓ A ( p ) ( M ( p ) ) ≤ ∗ ℓ A [ p ] ( M [ p ] ) = ℓ A p ( M p ) . In addition, if there is a homogeneous element of degree 1 (or -1) in A − p , then ℓ A ( p ) ( M ( p ) ) = ∗ ℓ A [ p ] ( M [ p ] ) = ℓ A p ( M p ) . In ending this section, we point out that *local graded rings are often graded localizations of positively(or negatively) graded rings at graded prime ideals.Suppose that ( A, N ) is a *local ring. Then, there exists a homogeneous element of strictly positivedegree in A − N if and only if there exists a homogeneous element of strictly negative degree in A − N : if s ∈ A − N is homogeneous of degree d > A − N is invertible,there exists a t ∈ A − N that is homogeneous and st = 1 ∈ A . Necessarily, the degree of t is − d . Sincethe argument is reversible, we have the conclusion.Thus, we have alternatives: • There exist homogeneous elements of A − N in at least one strictly positive degree and at leastone strictly negative degree. The analysis of this alternative is given below and we see that A isa graded localization of a positively graded ring at a graded prime ideal. • A is a positively or negatively graded ring: in this case, A is the localization of itself (a positivelyor negatively graded ring) at the *maximal ideal N since we’ve seen that N is the uniquemaximal ideal in A and A e = N e for all e = 0. • A has nonzero elements of both positive and negative degree, N d = A d for all d = 0 and N isthe unique maximal ideal of A . In this case, since N is an ideal, we must have A d A − d ⊆ N ,for all d = 0. As an example, consider A = k [ s, t ] / ( st ) where k is a field (all elements of degree0), the degree of s is one and the degree of t is -1. In this case, one might not be able to obtain A as a graded localization of a positively (or negatively) graded object at a graded prime ideal.But we don’t fully analyze this case here.Anyway, in the case of the first of the alternatives, let S ( A ) = ⊕ d ≥ A d be the “positive part” of A , graded with the natural grading; this too is a graded ring, and it is certainlypositively graded. Considering the graded abelian subgroup S ( N ) = ⊕ d ≥ N d of S ( A ), we see that it isa graded prime ideal in S ( A ). We claim that S ( A ) [ S ( N )] is isomorphic as a graded ring to A , with N corresponding to S ( N ) [ S ( N )] , under the well-defined injective homomorphism of graded rings defined by a/b ab − , if a ∈ S ( A ) d and b ∈ A e − N e for d, e ≥
0. To see that the homomorphism is surjective,suppose that x is a homogeneous element of degree j in A . If j ≥ x/ x , and x/ ∈ S ( A ) [ S ( N )] . If j <
0, the assumption of the first alternative says that there is a homogeneous element t ∈ A − N withdeg( t ) = k >
0. Then, there is a positive integer l such that lk + j > t l x/t l ∈ S ( A ) [ S ( N )] and t l x/t l x . 5. Krull Dimension in grmod(A)
Remark 5.1.
From now on, we assume that A is a Noetherian graded ring, unless explicitly statedotherwise.The height of a prime ideal p of A , graded or not, is defined as usual: ht( p ) is the longest length n (which always exists, using the Noetherian hypothesis) of a chain of primes p ⊂ · · · ⊂ p n = p ; thus wedefine the graded height ∗ ht( p ) of a graded prime ideal p in the ring A , as the longest length m (whichalways exists, using the Noetherian hypothesis) of a chain of graded primes p ⊂ · · · ⊂ p m = p . Forevery graded prime p , ht( p ) ≥ ∗ ht( p ).Forgetting the grading on A and M , one defines the Krull dimension of a graded A -module M asusual; here this is denoted by dim A ( M ). As usual, dim( A ) . = dim A ( A ) . The graded Krull dimension of a graded A -module M , ∗ dim A ( M ), is the greatest D such that there exists a strictly increasing chain p ⊂ . . . ⊂ p D f graded prime ideals in A such that Ann A ( M ) ⊆ p . If no such greatest D exists, M has infinite gradedKrull dimension. For the zero module, we define ∗ dim(0) = −∞ . By definition, ∗ dim( A ) . = ∗ dim A ( A ) . For any graded A -module M , • ∗ dim A ( M ) ≤ dim A ( M ).Also, since Ann A ( M ) = Ann A ( M ( n )), for every n ∈ Z , • dim A ( M ) = dim A ( M ( n )) , for every n ∈ Z . • ∗ dim A ( M ) = ∗ dim A ( M ( n )) , for every n ∈ Z . Example 5.2.
Let F be a graded field of the form F ∼ = F [ t, t − ], where deg( t ) >
0. The only gradedprime in F is 0, so that ∗ dim( F ) = 0. On the other hand, dim( F ) = 1 . More generally,
Example 5.3. If A is has only one graded prime ideal N , then A is a *local, *Artinian ring with ∗ dim( A ) = 0, and A is an Artinian ring of Krull dimension zero with unique nilpotent maximal ideal N .Most of the proofs for the following lemma may be found in [5]. Lemma 5.4.
Suppose that p ∈ Spec( A ). ( p may or may not be graded.) We know that p has finiteheight; say ht( p ) = d . i) If q ∈ Spec ( A ) and p ∗ ⊆ q ⊆ p then either q = p or q = p ∗ . ii) There exists a chain of primes q ⊂ · · · ⊂ q d = p , such that q , · · · , q d − are all graded. iii) If p is graded then there exists a chain of graded prime ideals such that p ⊂ p ⊂ · · · ⊂ p d = p ,so that ht( p ) = ∗ ht( p ) . iv) If p is not graded ( p ∗ is a proper subset of p ) , then ht( p ) = ht( p ∗ ) + 1 = ∗ ht( p ∗ ) + 1 . Corollary 5.5. If A is a graded Noetherian ring, then ∗ dim( A ) ≤ dim( A ) ≤ ∗ dim( A ) + 1; therefore if M ∈ grmod ( A ) , ∗ dim A ( M ) ≤ dim A ( M ) ≤ ∗ dim A ( M ) + 1 . Proof. If A has finite Krull dimension, the first inequality is always true; also, there is a maximal ideal m of A , not necessarily graded, such that ht( m ) = dim( A ). But then dim( A ) = ht( m ) ≤ ∗ ht( m ∗ ) + 1 ≤ ∗ dim( A ) + 1 . If A does not have finite Krull dimension, then for every positive integer e there is a primeideal p of height larger than e . But then ∗ ht( p ∗ ) is larger than e −
1, so ∗ dim( A ) is infinite as well. (cid:3) Krull dimension for modules over positively graded rings.
Remark 5.6.
The results in this section also hold for negatively graded rings, after changing definitionsappropriately.
Definition 5.7. If S is a positively graded ring, P roj ( S ) . = { p ∈ Spec( S ) | p is graded and S + p } . Note that if p ∈ P roj ( S ), then the set of homogeneous elements of S − p has at least one nonzeroelement of strictly positive degree. We’ve noted that N is a *maximal ideal in S if and only if N = N ⊕ S + , with N a maximal ideal in S . Thus, P roj ( S ) contains no *maximal ideals.For positively graded rings, there is no difference between ∗ dim and dim: Lemma 5.8.
Let the ring S be a positively graded ring of finite Krull dimension and M ∈ grmod ( S ) .Then, i) dim( S ) = ∗ dim( S ) ; therefore, ii) dim S ( M ) = ∗ dim S ( M ) . For graded localizations of positively graded rings, the following is well-known:
Theorem 5.9.
Suppose that S is a positively graded ring of finite Krull dimension, and p is a gradedprime ideal of S . Then, if S + ⊆ p , dim( S [ p ] ) = ∗ dim( S [ p ] ) , and if S + p , dim( S [ p ] ) = ∗ dim( S [ p ] ) + 1 . roof. Since S [ p ] is a localization of S , it is Noetherian. Ignoring the grading and recalling the standardorder-preserving correspondence between the set of all primes of S disjoint from T and the prime idealsof of T − S , for any MCS or GMCS T in S . So ∞ > dim( S ) ≥ dim( S [ p ] ) . We have already seen, then, that ∗ dim( S [ p ] ) ≤ dim( S [ p ] ) ≤ ∗ dim( S [ p ] ) + 1.Now let T be the GMCS consiting of all homogeneous elements of S not in p .In the case where S + ⊆ p , we must have p = ( p ∩ S ) ⊕ S + . For any element t ∈ T , this forces deg t = 0.Thus, S [ p ] is a positively graded ring of finite Krull dimension, so dim( S [ p ] ) = ∗ dim( S [ p ] ) . Now, S [ p ] / p [ p ] = ( S/ p ) [ p ] is a graded field, and it does have a positive degree element since S + p :Choose any homogeneous t ∈ S + , t / ∈ p . Then t ∈ T , and has positive degree, thus ( t + p ) / S/ p ) [ p ] . Forgetting the grading, this domain has dimension 1. Thus, theremust exist a prime q , necessarily ungraded, of S [ p ] such that p [ p ] ⊂ q . Therefore dim( S [ p ] ) ≥ ht( p [ p ] ) + 1 = ∗ ht( p [ p ] ) + 1 = ∗ dim( S [ p ] ) + 1 , yielding the conclusion. (cid:3) The following lemma establishes a relationship between primes in the localized ring and primes in thedegree 0 part of the localization, the ideas are implicit in [8].
Lemma 5.10.
Suppose that S is a positively graded ring, and T is any GMCS that contains at leastone element of positive degree. If q is a prime ideal in ( T − S ) , then there exists a unique graded prime p ∈ P roj ( S ) , disjoint from T , such that q = ( T − p ) .Proof. Uniqueness is left to the reader. To establish existence, let q ∈ Spec ( T − S ) . Define for i ≥ p i . = { x ∈ S i | ∃ j > , t ∈ T j s.t. x j t i ∈ q } , so that, since q is prime, p = { r ∈ S | r ∈ q } . Define p . = ⊕ i ≥ p i , we will show that p satisfies the required conditions.First, each p i is an abelian group with respect to +. For if x, y ∈ p i , i ≥
0, then there exists a k , k > s ∈ T k , t ∈ T k such that x k s i and y k t i are in q . Then, ( x + y ) k + k = P α + β = k + k c ( α,β ) x α y β , for thebinomial coefficient c ( α,β ) ∈ S . Now, either α ≥ k or β ≥ k . If α ≥ k , then x α y β s i t i = x k s i · x α − k y β t i . Thisis a product of an element in q with an element in ( T − S ) , so it must be in q . A similar computationhandles the case that β ≥ k . Therefore, ( x + y ) k k ( st ) i ∈ q , and so x + y ∈ p i .To show that that p is an ideal in S , one needs only to show that S i p j ⊆ p i + j for every i, j . Suppose s ∈ S i and x ∈ p j . There exists k > , t ∈ T k with x k t j ∈ q . Then, ( sx ) k t j + i = s k t i · x k t j , the product of anelement in ( T − S ) with an element in q , and therefore sx ∈ p i + j so that p is a graded ideal in S .Furthermore, p ∩ T = ∅ : if not, choose a t ∈ p i ∩ T . So, there exists a k > s ∈ T k such that t k s i ∈ q . However, the product s i t k · t k s i must also be in q , which contradicts that 1 q . Since T has atleast one nonzero element of positive degree, and p ∩ T = ∅ , S + p .To verify that p is prime, suppose that f ∈ S n , g ∈ S m , and f g ∈ p n + m . There exists a k > t ∈ T k such that ( fg ) k t m + n ∈ q . Now, ( fg ) k t m + n = f k t n · g k t m ∈ q , and by primality of q , together with the definitionof p , either f ∈ p n or g ∈ p m .We have established that p ∈ P roj ( S ), and it only remains to show that q = ( T − p ) . Suppose that ξ ∈ q , so ξ may be written as xt , with x ∈ S i , t ∈ T i . If i > x i /t i = ξ i ∈ q , so x ∈ p i by definition,and ξ = xt ∈ ( T − p ) . If i = 0, then t ξ = x ∈ q , so x ∈ p and ξ = xt ∈ ( T − p ) .On the other hand, suppose that xt ∈ ( T − p ) , x ∈ p i , t ∈ T i . By definition, there exists a k >
0, andan s ∈ T k such that x k s i ∈ q . Then, s i t k · x k s i ∈ q since s i t k ∈ ( T − S ) and x k s i ∈ q . Of course, s i t k · x k s i = (cid:0) xt (cid:1) k ,and by primality of q , xt ∈ q . (cid:3) Thus we have
Theorem 5.11.
Suppose S is a positively graded ring, and T is a GMCS in S containing at least oneelement of positive degree. Then, there exists a one-to-one inclusion-preserving correspondence { p ∈ Proj ( S ) | p ∩ T = ∅} ↔ { q ∈ Spec ( T − S ) } ; his correspondence takes p to ( T − p ) . Using the correspondence of the above theorem, we have, as expected,
Corollary 5.12.
Suppose S is a positively graded ing of finite Krull dimension. Let p ∈ P roj ( S ) . Then S ( p ) is a local Noetherian ring of finite Krull dimension and dim( S ( p ) ) = ∗ ht( p ) = ∗ ht( p [ p ] ) = ∗ dim( S [ p ] ) = dim( S [ p ] ) − . Poincar´e series and dimension for positively graded rings.
The ring Z [[ t ]][ t − ] is denoted by Z (( t ));thus an element of Z (( t )) is a formal Laurent series f ( t ) with integer coeffiicients; there always exists an n ∈ Z with t n f ( t ) ∈ Z [[ t ]] . In order to define the Poincar´e series for a graded abelian group M , we assume, in addition, that 1)each M j is a finitely generated module over a ordinary commutative Artinian ring S and 2) M j = 0 for j <<
0. Whenever we write down a Poincar´e series for a graded abelian group M , we will make theseassumptions.For example, if S is a positively graded Noetherian ring and M ∈ grmod ( S ), then as long as S isArtinian, 1) and 2) hold. Definition 5.13.
Suppose that M is a graded abelian group and S is an Artinian ring satisfying 1)and 2) above. Then the Poincar´e series of M is the formal Laurent series with integer coefficients P M ( t ) = X i ∈ Z ℓ S ( M i ) t i , where ℓ S ( M i ) is the length of the finitely generated module M i over the Artinian ring S .Sometimes the Poincar´e series is called the Hilbert series, or the Hilbert-Poincar´e series. Theorem 5.14. (The Hilbert-Serre Theorem) [2]
Let S be a positively graded Noetherian ring with S Artinian, M ∈ grmod ( S ) . Suppose that S is generated as a S -algebra by elements x , . . . , x n of positivedegrees d , . . . , d n . Then, P M ( t ) = q ( t ) Q ni =1 (1 − t d i ) , where q ( t ) ∈ Z [ t, t − ] .Furthermore, if M has no elements of negative degree, q ( t ) ∈ Z [ t ] . From now on, we assume that S is a positively graded Noetherian ring of finite (Krull) dimension,with S Artinian.Some facts to note about Poincar´e series: • Let ˆ S be another positively graded ring. Assume also that the graded abelian group M is in grmod ( S ), S = ˆ S is Artinian and M is also a graded ˆ S -module (but not necessarily finitely gen-erated as such). Then whether we consider M as an S -module or as a ˆ S -module, its Poincar´e se-ries does not change. For example, let y , . . . , y s ∈ S + be homogeneous. Define ˆ S = S h y , . . . , y s i to be the graded subring of S generated by S and y , . . . , y s . Now, whether we consider M asan S -module, or as an ˆ S -module, its Poincar´e series is the same. • If M has a Poincar´e series with respect to S , then so does M ( n ), for every n ∈ Z , and P M ( n ) ( t ) = t − n P M ( t ) . • If 0 → P → M → N → grmod ( S ), then P M ( t ) = P P ( t ) + P N ( t ) . • If M, N ∈ grmod ( S ), then P M ⊗ S N ( t ) = P M ( t ) P N ( t ) , if M ⊗ S N is given the usual grading.We end this section with a brief discussion of the connection of the Poincar´e series with (Krull)dimension. Definition 5.15.
Let M be in grmod ( S ), M = 0. • If M ∈ grmod ( S ), d ( M ) is the least j such that there exist positive integers f , . . . , f j with( j Y i =1 (1 − t f i )) P M ( t ) ∈ Z [ t, t − ] . By definition, d ( M ) = 0 if and only if P M ( t ) is in Z [ t, t − ] . Note that the Hilbert-Serre theoremshows that d ( M ) < ∞ ; also, d ( M ) is the order of the pole at t = 1 for P M ( t ) . s ( M ) is the least s such that there exist homogeneous elements y , . . . , y s ∈ S + with M finitelygenerated over S h y , . . . , y s i ⊆ S . By definition, s ( M ) = 0 if and only if M is a finitelygenerated graded S -module. Note that for a finite set of homogeneous generators for S + , thenumber of elements in that set is an upper bound for s ( M ). • d (0) = s (0) = −∞ . Note that if n ∈ Z , then d ( M ( n )) = d ( M ), since P M ( n ) ( t ) = t − n P M ( t ). Also, s ( M ( n )) = s ( M )by definition. The following theorem and proposition could be considered “folklore”, but the paper ofSmoke cited is, as far as we know, the first appearance of these statements in the literature. Theorem 5.16.
Smoke’s Dimension Theorem (Theorem 5.5 of [14] )Suppose that S is a positively graded ring of finite Krull dimension, with S Artinian. Let M ∈ grmod ( S ) .If d ( M ) , s ( M ) are defined as above, we have d ( M ) = s ( M ) = ∗ dim S ( M ) < ∞ .Under the hypotheses of the theorem, we’ve already seen that ∗ dim S ( M ) = dim S ( M ) , so all of thesenumbers equal dim S ( M ) as well. Graded ideals of definition and graded systems of parameters
Returning to the more general case a graded ring A , not necessarily positively graded, we defineanalogously to Serre, a graded ideal of definition and a graded system of parameters. Definition 6.1.
Let A be a graded ring and M ∈ grmod ( A ). A proper, graded ideal I of A such that ∗ ℓ A ( M/ I M ) < ∞ is called a graded ideal of definition for M (a GIOD for M ).(This is a little different from Serre’s definition [15] of an ideal of definition in the ungraded case.)Lemmas 2.4 and 3.9 say that I is a graded ideal of definition for M if and only if all graded primescontaining I + Ann A ( M ) are *maximal. Definition 6.2.
Let A be a Noetherian ring of finite Krull dimension, and also assume that A is eithera positively graded ring or a *local ring with unique *maximal ideal N . Define m to be the gradedideal A + in the first case, and the ideal N in the second. Suppose M = 0 is in grmod ( A ). A sequence y , . . . , y D of homogeneous elements of m , such that • the graded A -module M/ ( y , . . . , y D ) M has finite *length over A and • D = ∗ dim A ( M )is called a graded system of parameters (GSOP) for the A -module M .Note that by definition, a GSOP (or a GIOD) for M is also a GSOP (resp. GIOD) for M ( n ), forevery n ∈ Z (and vice versa).In the positively graded case, an alternative characterization of some GIODs (and thus some GSOPs)is given by: Lemma 6.3.
Suppose that S is a positively graded Noetherian ring of finite Krull dimension, with S Artinian, and y , . . . , y u are homogeneous elements of S + . Let M ∈ grmod ( S ) . Then, M/ ( y , . . . , y u ) M has finite *length over S if and only if M is a finitely generated S h y , . . . , y u i -module.Proof. Recall that S h y , . . . , y u i is the subring of S generated by S and y , . . . , y u . Let t ∈ Z be chosensuch that M j = 0 for j < t . Suppose that X . = M/ ( y , . . . , y u ) M has finite *length over S . We’veseen that there exists an integer t ≤ t such that X j = 0 if j > t . Using Lemma ?? , M j is finitelygenerated over S , so for every j such that t ≤ j ≤ t we may choose a finite set E j of generators,possibly empty, for M j over S . Then, we prove that M is generated by the finite set E . = ∪ t j = t E j over S h y , . . . , y u i ; to do this we show, using induction on deg( z ), that a homogeneous element z of M is in the submodule of M generated by E over S h y , . . . , y u i . To start the induction, note that ifdeg( z ) ≤ t , the claim is certainly true. Let s > t and suppose that the inductive hypothesis holdsfor every homogeneous w of degree strictly less than s . Let z be a homogeneous element of M ofdegree s . Since s > t , ( M/ ( y , . . . , y u ) M )) s = 0, so s ∈ ( y , . . . , y u ) M. Write z = P uα =1 y α m α . Sincedeg( y α ) + deg( m α ) = s for every α such that y α m α = 0, and deg( y α ) > α , we must havedeg( m α ) < s for every α with y α m α = 0. Thus by induction, m α is a linear combination of elements of E , with coefficients in S h y , . . . , y u i . Clearly, then, so is z . Note that this part of the proof never usedthat S is Artinian. onversely, suppose M is generated by a finite set E of nonzero homogeneous elements as a graded S h y , . . . , y u i -module. Set I . = ( y , . . . , y u ). Let t = max { deg( e ) | e ∈ E } . Then, for j > t , ( M/ I M ) j =0: If x ∈ M j , j > t , write x = P e ∈ E f e e , where f e ∈ S h y , . . . , y u i is homogeneous. If deg( f e ) = 0, and f e e = 0, then f e e ∈ I M . Therefore, x is equivalent to P f e =0 , deg( f e )=0 f e e mod I M . However, for everysummand in this last sum, we must have deg( e ) = deg( x ) > t if f e e = 0, a contradiction. Thus, x isequivalent to 0 mod I M . Lemma 3.12 tells us that since S is Artinian, ∗ ℓ S ( M/ I M ) < ∞ . (cid:3) The following proposition is another part of the “folklore” knowledge, but the citation is the first thatwe know of in the literature.
Proposition 6.4. (Theorem 6.2 of [14] ) Suppose that S is a positively graded Noetherian ring of finiteKrull dimension, with S Artinian, and M = 0 is in grmod ( S ) . Let D ( M ) . = d ( M ) = s ( M ) = ∗ dim S ( M ) = dim S ( M ) , so Theorem 5.16 and Lemma 6.3 tell us that a GSOP exists for M . Moreover, D ( M ) is the length of any GSOP and if y , . . . , y D ( M ) ∈ S + is a GSOP for M , y , . . . , y D ( M ) arealgebraically independent over S . Multiplicities for graded modules
In this section, we define the *Samuel multiplicity and *Koszul multiplicity for modules in grmod ( A ).All of this work is done analogously to Serre [15], and since we only give brief discussions/proofs here, ifthe reader does not have in mind the development of multiplicities in [15], it’s advised to have a copy of[15] at hand. Another treatment of multiplicity in the graded case is given in [13].The *Samuel multiplicity is explored using the tools of the graded category which we have developedthus far: *length, *dimension, graded localization, etc. The *Koszul multiplicity is defined using toolsfrom homological algebra. In each case, to adapt the theory from the ungraded case, we have theadded complication of our objects being bi-graded - the internal grading that the module inherits from grmod ( A ), and an external grading coming from either the associated graded module in the case of Samuelmultiplicities, or the complex grading for Koszul multiplicities. Keeping track of the bi-grading, allmorphisms respect both gradings, and as one might expect, the bi-grading does not cause any problems.We show, as in the ungraded case, the two multiplicities (*Koszul and *Samuel) agree.Finally, we show that the graded multiplicity theory agrees with the ungraded theory by simplyforgetting the grading, when we work over positively graded rings. This is to be expected, for we haveshown that *length and length agree in the positively graded case.In the following, we will consider filtrations of A -modules; as previously, we will use upper indicesfor filtrations, whether working in a graded or an ungraded category. We’ll use notations like M • oroften F ( M ) for filtrations of M by A -modules. Filtrations will be indexed in different ways, accordingto convention. Definition 7.1.
Suppose that I is an ideal in A . A filtration F ( M ) with F i +1 ( M ) ⊆ F i ( M ) for every i ≥
0, is called I -bonne if IF n ( M ) ⊆ F n +1 ( M ), for every n ≥
0, and with equality for n >> Example 7.2. If I is an ideal in A , the I -adic filtration · · · ⊆ I j +1 M ⊆ I j M ⊆ · · · ⊆ I M ⊆ M is I -bonne.If A is a graded ring, and M a graded A -module, a filtration F ( M ) is graded if and only if all thesubmodules F j ( M ) are graded submodules; if I is a graded ideal, the definition of an I -bonne gradedfiltration remains the same as in the ungraded case.7.1. The Ungraded Case.
We begin by outlining the procedure for defining the Hilbert and Samuelpolynomials in the ungraded case (see [15] for full discussion/proofs).Suppose that H is a positively graded ring with H Artinian, and that H is generated as an H -algebra by a finite number of homogeneous elements x , . . . , x u in H . Such a ring H is then called a“standard” graded ring. For any finitely generated, positively graded H -module M , M n is a finitelygenerated H -module for every n . Since H is Artinian, the Hilbert function, n ℓ H ( M n ), is definedfor all integers n ≥
0. Using induction on the number of generators for H as an H -algebra, and theadditivity of length over exact sequences, one may prove that the Hilbert function is polynomial-like; inother words there is a unique polynomial f with rational coefficients such that f ( n ) = ℓ H ( M n ) for all n sufficiently large. The polynomial describing the function n ℓ H ( M n ) is called the Hilbert polynomialof M (over H ).Recall the delta notation from the theory of polynomial-like functions: if f is a function with aninteger domain, then ∆ f is the function defined by ∆ f ( n ) . = f ( n + 1) − f ( n ). Then, we know that f is olynomial-like if and only if ∆ f is polynomial-like. We may iterate the operator “∆” on integer domainfunctions, obtaining operators ∆ r , for r ≥ A is an ungraded Noetherian ring, M an ungradedfinitely generated A -module, and I is an ideal of A such that M/ I M has finite length over A ; this lastis true if and only if V ( I + Ann A ( M )) consists of a finite number of maximal ideals in A .Summarizing the discussion in [15], given an ideal I with ℓ A ( M/ I M ) < ∞ and an I -bonne filtration F ( M ), ℓ A ( M/ F n ( M )), is well-defined. Now, V ( M/ I M ) = V ( Ann A ( M ) + I ) consists of a finite numberof maximal ideals; without loss of generality we may assume that Ann A ( M ) = 0 and V ( M/ I M ) = V ( I )consists of a finite number of maximal ideals, so that A/ I is an Artinian ring. The positively gradedassociated graded module gr ( M ) = ⊕ n ≥ F n ( M ) / F n +1 ( M ) is finitely generated over the positivelygraded associated graded ring gr ( A ) = ⊕ n ≥ I n / I n +1 . Furthermore gr ( A ) is generated over gr ( A ) = A/ I , an Artinian ring, by elements of degree one, and the Hilbert polynomial for gr ( M ) as a gr ( A )-module exists.Then, n ℓ A ( M/ F n +1 ( M )) − ℓ A ( M/ F n ( M )) = ℓ A ( F n ( M ) / F n +1 ( M )) is polynomial-like, and thegeneral theory of polynomial-like functions tells us that the Samuel function n ℓ A ( M/ F n ( M )) is alsopolynomial-like. The polynomial describing this function is called the Samuel polynomial p ( M, F , n ) ofthe A -module M with respect to the filtration F and the ideal I .7.2. The Graded Case.
We make new, similar definitions in the graded category, now assuming A isa graded Noetherian ring and M ∈ grmod ( A ). We do not assume that A is positively graded, nor thatit is generated by elements of degree 1.To define the *Hilbert polynomial, start with certain bigraded objects: Suppose that H is a bigradedring such that H i,j = 0 for i < H , ∗ . = ⊕ j ∈ Z H ,j is a graded ring that is *Artinian and H is generatedas an bigraded algebra over the graded ring H , ∗ by a finite number of elements in H , ∗ . = ⊕ j ∈ Z H ,j . M is taken to be a bigraded H -module such that M i,j = 0 for i < M is generated as an H -moduleby a finite number of bi-homogeneous elements. Then, for each k ≥ M k, ∗ . = ⊕ j ∈ Z M k,j is a finitelygenerated graded H , ∗ -module, so ∗ ℓ H , ∗ ( M k, ∗ ) is well-defined for every k ≥
0. Furthermore, the function k
7→ ∗ ℓ H , ∗ ( M k, ∗ ) is polynomial like. To see this, following the argument in [15] for the ungraded case, useinduction on the number of bihomogeneous generators (taken from H , ∗ ) for H as an H , ∗ -algebra, andadditivity of ∗ ℓ over exact sequences of graded modules. The exact sequence used in Theorem II.B.3.2of [15] becomes an exact sequence of graded modules, with middle map multiplication by a generator ofbidegree (1 , d ): 0 → N n, ∗ → M n, ∗ ( − d ) → M n +1 , ∗ → R n +1 , ∗ → H is generated as a bigraded algebra over H , ∗ by r elements of bidegree(1 , − ), then the *Hilbert polynomial has degree less than or equal to r − A is a graded ring, I is a graded ideal in A and F ( M ) is a graded I -bonne filtration of M .Note that if I is a graded ideal in A , F ( M ) is a graded I -bonne filtration of M , and d ∈ Z is afixed integer, we may shift degrees by d throughout the filtration yielding an I -bonne filtration F ( d )of M ( d ): F ( d ) n ( M ( d )) . = ( F n ( M ))( d ). To see that this filtration is also I -bonne, just compute that I ( F ( d ) n ( M ( d ))) = ( IF n ( M ))( d ) as follows. Suppose that x ∈ ( IF n ( M ))( d ) j = ( IF n ( M )) d + j , so that x = P t α t m t , where α t ∈ I , m t ∈ F n ( M ) are all homogeneous and deg( α t ) + deg( m t ) = d + j whenever α t m t = 0. Thus, deg( m t ) = d + ( j − deg( α t )) for all such t , so that m t ∈ ( F ( d ) n )( M ( d )) j − deg( α t ) , α t m t ∈ I ( F ( d ) n ( M ( d ))) j for every t and x ∈ I ( F ( d ) n ( M ( d ))) j . The converse is similarly proved. Inparticular, the d -suspension of the I -adic filtration on M is the I -adic filtration on M ( d ).Given a GIOD I for M , and a graded I -bonne filtration F ( M ), ∗ ℓ A ( M/ F n ( M )) < ∞ . Passing with-out loss of generality to the case Ann A ( M ) = 0 as in the ungraded case, we see that A/ I is a *Artinianring and that the associated bigraded module gr ( M ) = ⊕ n ≥ F n ( M ) / F n +1 ( M ), where gr ( M ) n,j . = F n ( M ) j / F n +1 ( M ) j , is finitely generated over the associated bigraded ring gr ( A ) = ⊕ n ≥ I n / I n +1 (where gr ( A ) n,j . = ( I n ) j / ( I n +1 ) j ). Note that gr ( A ) is generated by elements of bidegree (1 , − ), asan algebra over the *Artinian graded ring A/ I and thus the *Hilbert polynomial for gr ( M ) as a gr ( A )-module exists. efinition 7.3. Suppose that I is a GIOD for M ∈ grmod A and F is an I -bonne filtration of M . The*Samuel function with respect to F and I is defined on the nonnegative integers by n
7→ ∗ ℓ A ( M/ F n ( M )).Since ∗ ℓ A ( M/ F n +1 ( M )) − ∗ ℓ A ( M/ F n ( M )) = ∗ ℓ A ( F n ( M ) / F n +1 ( M )), the ∆ operator applied to the*Samuel function is polynomial-like, so Lemma 7.4. If M ∈ grmod ( A ) and I is a GIOD for M , the *Samuel function for the graded I -bonnefiltration F ( M ) is polynomial-like. To set notation, the polynomial that calculates ∗ ℓ A ( M/ F n ( M )) for n >> ∗ p ( M, F , n ),and if F is the I -adic filtration on M , we will instead write ∗ p ( M, I , n ) . The following lemma incorporates graded versions of results in II.B.4 of [15].
Lemma 7.5.
Suppose that M ∈ grmod ( A ) and F ( M ) is a graded I -bonne filtration of M for someGIOD I for M . Then a) For every d ∈ Z , I is a GIOD for M ( d ) , F ( d )( M ( d )) is an I -bonne filtration of the graded A -module M ( d ) and ∗ p ( M ( d ) , F ( d ) , n ) = ∗ p ( M, F , n ) . b) ∗ p ( M, I , n ) = ∗ p ( M, F , n ) + R ( n ) , where R is a polynomial with nonnegative leading coefficientand degree strictly less than that of the degree of ∗ p ( M, I , n ) . c) If ( Ann A ( M )+ I ) /Ann A ( M ) is generated by r homogeneous elements, then the degree of ∗ p ( M, I , n ) is less than or equal to r , and ∆ r ( ∗ p ) is a constant less than or equal to ∗ ℓ A ( M/ I M ) . d) If → N → M → P → is a short exact sequence in grmod ( A ) , and I is a GIOD for M , then I is a GIOD for both N and P and ∗ p ( M, I , n ) + R ( n ) = ∗ p ( N, I , n ) + ∗ p ( P, I , n ) , where R is a polynomial with nonnegative leading coefficient and degree strictly less than that of ∗ p ( N, I , n ) . e) If I and ˆ I are two GIODs for M such that ∗ V ( I + Ann A ( M )) = ∗ V (ˆ I + Ann A ( M )) , then thedegree of ∗ p ( M, I , n ) equals the degree of ∗ p ( M, ˆ I , n ) . Proof.
We’ve already noted that I ( F ( d ) n ( M ( d ))) = ( IF n ( M ))( d ); so that F ( d )( M ( d )) is an I -bonne fil-tration of M ( d ). The *Samuel polynomials are identical since M ( d ) / F ( d ) n ( M ( d )) = M ( d ) / ( F n ( M )( d )) =( M/ ( F n ( M ))( d ) , for every n . The proofs of b)-e) follow exactly the proofs in Section II.B.4 of Lemma3 and Propositions 10 and 11 of [15], adapted with clear notational changes to the graded case, and arenot given here. (cid:3) Since we will be interested in the leading coefficient of *Samuel polynomials, b) above tells us that wemay as well just consider I -adic filtrations and suppress all talk about I -bonne filtrations; the need toconsider general I -bonne filtrations F is indicated in the proof of d), even though we haven’t given it,since the proof of d) uses the Artin-Rees lemma, which also holds in the graded context. Definition 7.6.
Suppose that M ∈ grmod ( A ), I is a GIOD for M and d ∈ Z , d ≥ deg( ∗ p ( M, I , n )).The *Samuel multiplicity of M with respect to I is defined as ∗ e ( M, I , d ) . = ∆ d ( ∗ p ( M, I , n )) . By properties of the finite difference operator ∆, we see that ∗ e ( M, I , d ) = 0 whenever d > deg( ∗ p ( M, I , n )).When d = deg( ∗ p ( M, I , n )), ∗ e ( M, I , d ) is a positive integer, and one may compute that ∗ p ( M, I , n ) = ∗ e ( M, I , d ) d ! n d + lower order terms.Using Lemma 7.5d), we see that if 0 → N → M → P → grmod ( A ), I is a GIOD for M and d ≥ deg( ∗ p ( M, I , n )), then both ∗ e ( N, I , d )and ∗ e ( P, I , d ) exist and ∗ e ( M, I , d ) = ∗ e ( N, I , d ) + ∗ e ( P, I , d ) . Therefore, using Lemma 7.5a) as well, we have
Corollary 7.7.
Suppose that M ∈ grmod ( A ) , I is a GIOD for M and M • is a graded filtration of M such that M ⊂ M ⊂ · · · M N − ⊂ M N = M, and, for each N ≥ i ≥ , there are graded primeideals p i in A , integers d i and graded isomorphisms of A -modules ( A/ p i )( d i ) ∼ = M i /M i − . Then, i) I is a GIOD for A/ p i and ∗ p ( A/ p i , I , n ) exists, for ≤ i ≤ N . i) If D . = max { deg( ∗ p ( A/ p i , I , n )) . = d i | ≤ i ≤ N } and D ( M • ) . = { p j | d j = D } , ∗ e ( M, I , D ) = X p ∈D ( M • ) n p ( M • )( ∗ e ( A/ p , I , D )) , where n p ( M • ) is equal to the number of times A/ p , possibly suspended, occurs as an A -moduleisomorphic to a subquotient of the filtration M • . Furthermore, all of the integers on both sidesof the equation are strictly positive. Finally, we point out some scenarios in which *Samuel multiplicities equal those computed in theungraded category.
Theorem 7.8.
The positively graded case.
Suppose that S is a positively graded Noetherian ring with S Artinian, M ∈ grmod ( S ) and I a GIOD for M . Then, the “ungraded” Samuel polynomial p ( M, I , n ) exists, ∗ p ( M, I , n ) = p ( M, I , n ) and, for every d , ∗ e ( M, I , d ) = e ( M, I , d ) . Proof.
Lemma 3.11 tells us that, when we forget the grading, I has the property that ℓ S ( M/ I n M ) = ∗ ℓ S ( M/ I n M ) < ∞ . Therefore, the “ungraded” Samuel polynomial p ( M, I , n ) exists ( p ( M, I , n ) iscomputed after forgetting the grading) and ∗ p ( M, I , n ) = p ( M, I , n ) . So, if d ≥ deg( ∗ p ( M, I , n )) =deg( p ( M, I , n )), ∗ e ( M, I , d ) is the exact same multiplicity e ( M, I , d ) defined in [15], after forgetting thegrading. (cid:3) Theorem 7.9.
The *local case in which A − N has a homogeneous element of degree 1. Suppose that ( A, N ) is a *local Noetherian ring, M ∈ grmod ( A ) and A − N has a homogeneous elementof degree 1. Then, I is such that ℓ A ( M / ( I ) n M ) < ∞ , ∗ p ( M, I , n ) = p ( M , I , n ) and for every d , e ( M , I , d ) = ∗ e ( M, I , d ) . Proof.
First, note that for any graded ideal J in A , and every X ∈ grmod ( A ), it turns out in this casethat ( J X ) = J X : the containment “ ⊇ ” is clear. For the remaining containment, let T ∈ A − N be any element of degree one. Now, every element of ( J X ) has the form P j a j x j where a j ∈ J and x j ∈ X and deg( a j ) + deg( x j ) = 0 whenever a j x j = 0. However, T is invertible in A , and P j a j x j = P j ( a j T − deg( a j ) )( T deg( a j ) x j ) ∈ J X . Using this result for powers of J , and induction, wesee that ( J n ) = ( J ) n , for every n ≥ X ∈ grmod ( A ) is such that ∗ V ( X ) = {N } (or equivalently, ∗ ℓ A ( X ) < ∞ ), since A − N has a homogeneous element of degree 1, Proposition 4.13 tells us that for every j , ℓ A ( X j ) = ∗ ℓ A ( X ) forevery j .Putting all this together, if I is a GIOD for M , and X = M/ I n M , then we have X = ( M/ I n M ) = M / ( I n ) M = M / ( I ) n M and thus ℓ A ( M / I n M ) = ∗ ℓ A ( M/ I n M ) < ∞ for every n . Therefore, I is an ideal such that the ordinary Samuel polynomial p ( M , I , n ), constructed in the ungradedcase for the A -module M , is defined and ∗ p ( M, I , n ) = p ( M , I , n ). Therefore, in this case, for d ≥ deg( ∗ p ( M, I , n )) = deg( p ( M, I , n )), ∗ e ( M, I , d ) is equal to the multiplicity e ( M , I , d ) defined inthe ungraded case. (cid:3) We do not make a comparison if ( A, N ) is a *local Noetherian ring with no homogeneous elements ofdegree 1 in A − N .7.3. *Dimension, *Samuel polynomials and GSOPs for *local rings. In this section, A is a*local Noetherian graded ring with unique *maximal graded ideal N . Here we present an analoguein the graded category to the fundamental theorem of dimension theory for local rings. This theoremshows the relationship between *Krull dimension, graded systems of parameters, and the degree of the*Samuel polynomial. Applying the results to the category grmod ( R ), for R positively graded and R a field, we combine the fundamental dimension theorem for *local rings to Smoke’s dimension theorem(5.16). In this case, the order of the pole of the Poincare series at t = 1, equals the measures fromthe fundamental *local dimension theorem, which in turn equal the ungraded Krull dimension. This issummarized in corollary 7.11. Returning to the theory of multiplicities, we conclude the section with asum decomposition of the *Samuel multiplicity by minimal primes (corollary 7.13).We start by supposing that I is a GIOD for M ; since A is *local, we’ve seen that this is true if andonly if ∗ V ( M/ I M ) = {N } . The previous section shows that the degree of the *Samuel polynomial of M ith respect to the I -adic filtration does not depend on the choice of I . We call this degree ∗ d ( M ). Ofcourse, N is always a GIOD for M .If M ∈ grmod ( A ), M = 0, ∗ s ( M ) is defined to be the least s such that there exist homogeneouselements w , . . . , w s ∈ N such that the graded A -module M/ ( w , . . . , w s ) M has finite *length over A .Note that ∗ s ( M ) = 0 if and only if ∗ ℓ A ( M ) < ∞ .The fundamental theorem for *local dimension theory is: Theorem 7.10. If ( A, N ) is a *local Noetherian ring and M ∈ grmod ( A ) , then ∗ dim A ( M ) = ∗ d ( M ) = ∗ s ( M ) . Proof.
The proof of this mimics the proof of the analogous theorem in the ungraded, local case given in[15] in Section III.B.2, Theorem 1, but we give a sketch anyway. First, if x is a homogeneous element of N , let x M be the graded A -module consisting of all elements m of M such that xm = 0. If deg( x ) = d ,then there are short exact sequences in grmod ( A )0 → x M ( − d ) → M ( − d ) · x → xM → , → xM → M → M/xM → . If I is a GIOD for M , it is also a GIOD for every module in the exact sequences above. Furthermore,the short exact sequences and Lemma 7.5 say that ∗ p ( x M, I , n ) − ∗ p ( M/xM, I , n ) is a polynomial ofdegree strictly less than ∗ d ( M ). It’s straightforward to see that ∗ s ( M ) ≤ ∗ s ( M/xM ) + 1.We may as well assume that the GIOD we are using to calculate ∗ d ( M ) is N .Next, set D ( M ) to be the (finite) set of all p in ∗ V ( M ) with the property that ∗ dim A ( M ) = ∗ dim A ( A/ p ) = ∗ dim( A/ p ); it’s important to note that D ( M ) could also be defined as the set of allprimes in V ( M ) with dim A ( M ) = dim A ( A/ p ) since the minimal elements in the sets ∗ V ( M ) and V ( M )are exactly the same. If a homogeneous element x is not in any prime of D ( M ), then ∗ dim A ( M/xM ) < ∗ dim A ( M ); this is true for exactly the same reason as in the ungraded case: ∗ V ( M/xM ) = ∗ V (( x ) + Ann A ( M )).Finally, one proceeds to the proof by first arguing that ∗ dim A ( M ) ≤ ∗ d ( M ), then ∗ d ( M ) ≤ ∗ s ( M ),and lastly, ∗ s ( M ) ≤ ∗ dim A ( M ).For the first inequality one uses induction on ∗ d ( M ). Note that ∗ d ( M ) = 0 means that there is a q such that ∗ ℓ A ( M/ N i M ) = ∗ ℓ A ( M/ N i +1 M ) for all i ≥ q . But this forces N q M = N q +1 M and gradedNakayama’s lemma says that N q M = 0, so that ∗ V ( M ) has exactly one ideal, N in it. By definition, ∗ dim A ( M ) = 0. Supposing that ∗ d ( M ) ≥
1, as in [15], we reduce to the case M = A/ p for some gradedprime ideal p properly contained in N . Taking a chain of graded prime ideals p . = p ⊂ p ⊂ · · · ⊂ p n in A , we may suppose that n ≥
1, and thus may choose a homogeneous element x in p that is not in p .Since x / ∈ p , but x ∈ p , the chain p ⊂ · · · ⊂ p n corresponds to a chain of primes in ∗ V ( M/xM ). Since M = A/ p , and x / ∈ p , x M = 0, so that ∗ p ( M/xM, N , n ) has degree strictly less than ∗ d ( M ), and byinduction, ∗ dim A ( M/xM ) ≤ ∗ d ( M/xM ). Thus, n − ≤ ∗ dim A ( M/xM ) ≤ ∗ d ( M ) − n ≤ ∗ d ( M ).This forces ∗ dim A ( M ) ≤ ∗ d ( M ).For the second inequality, if x , . . . , x k is a list of homogeneous elements of N that generate a GIOD I for M , we must have that ∗ V ( I + Ann A ( M )) contains only N , so that ∗ p ( M, I , n ) and ∗ p ( M, N , n )have the same degree ∗ d ( M ). But, Lemma 7.5 says that ∗ p ( M, I , n ) has degree less than or equal to k .Thus, ∗ d ( M ) ≤ ∗ s ( M ).For the third inequality, use induction on ∗ dim A ( M ), which we may assume to be at least 1, since ∗ dim A ( M ) = 0 if and only if N is the only prime in ∗ V ( M ), so that M has finite *length and ∗ s ( M ) = 0by definition. If ∗ dim A ( M ) ≥
1, none of the primes in D ( M ) are *maximal, so there is a homogeneouselement x ∈ N such that x is not in any of the primes in D ( M ). We’ve noted above that ∗ s ( M ) ≤∗ s ( M/xM ) + 1 and ∗ dim A ( M ) ≥ ∗ dim A ( M/xM ) + 1 . These inequalities plus the induction hypothesesgive us the result. (cid:3) If R is a positively graded ring with R = k a field, then ( R, R + ) is a *local ring, so we may applythe fundamental theorem for *local dimension. On the other hand, recall Smoke’s dimension theorem(theorem 5.16). For any M ∈ grmod ( R ) the hypotheses for Smoke’s dimension theorem are satisfied, andwe may therefore combine the two dimension theorems. Corollary 7.11. If R is a positively graded Noetherian ring with R a field and M ∈ grmod ( R ) , then ∗ dim R ( M ) = ∗ d ( M ) = ∗ s ( M ) = s ( M ) = d ( M ) = dim R ( M ) . Going back to the definition of a GSOP for the A -module M , as a corollary to Theorem 7.10. we have orollary 7.12. If ( A, N ) is a *local graded Noetherian ring and M ∈ grmod ( A ) , then a GSOP existsfor M , and the length of every GSOP is equal to ∗ dim A ( M ) = ∗ d ( M ) = ∗ s ( M ) . Moreover, if A − N has a homogeneous element T of degree 1, necessary invertible, and d ( M ) is the degree of the ordinarySamuel polynomial p ( M , N , n ) , then d ( M ) = ∗ d ( M ) , and if x , . . . , x D is a GSOP for M , where D = ∗ dim A ( M ) = ∗ d ( M ) = d ( M ) , then x T − e , . . . , x D T − e D is an ordinary system of parameters for M as an A -module, if e i = deg( x i ) .Proof. The first statement is clear using 7.10; for the second use Theorem 7.9 to see that ∗ d ( M ) = d ( M );if x , . . . , x D generate a GIOD I for M , then I is generated by x T − e , . . . , x D T − e D . Therefore, theungraded dimension theorem ensures that D = d ( M ) = dim A ( M ), so x T − e , . . . , x D T − e D is an ordinary system of parameters for M . (cid:3) We also have a corollary to Corollary 7.7; here D ( M ) is defined as the set of minimal primes ofmaximal dimension (as in the proof of Theorem 7.10) Corollary 7.13.
Suppose that ( A, N ) is a *local graded Noetherian ring, M ∈ grmod ( A ) , I is a GIODfor M and M • is a graded filtration of M such that M ⊂ M ⊂ · · · M N − ⊂ M N = M, and, foreach N ≥ i ≥ , there are graded prime ideals p i in A , integers d i and graded isomorphisms of A -modules ( A/ p i )( d i ) ∼ = M i /M i − . Then, if D . = ∗ dim A ( M ) , ∗ e ( M, I , D ) = X p ∈D ( M ) ∗ ℓ A [ p ] ( M [ p ] )( ∗ e ( A/ p , I , D )) . Proof.
Lemma 2.5 tells us that, for every minimal prime p for M , there is at least one subquotient ofthe filtration isomorphic to the graded A -module A/ p , possibly shifted. Therefore, adding the *localhypothesis, • D ( M • ) = D ( M ) since we now know that, for every shift d , the degree of ∗ p (( A/ p )( d ) , I , n ) = ∗ p ( A/ p , I , n ) is independent of the choice of I and is equal to ∗ dim A ( A/ p ). Theorem 7.10 alsotells us that the D in this corollary is exactly the same D as in Corollary 7.7. • Moreover, for every prime p in D ( M ), A/ p , possibly shifted, occurs exactly ∗ ℓ A [ p ] ( M [ p ] ) = ℓ A p ( M p ) times (using Theorem 4.10 and Corollary 7.7) as a subquotient of the filtration M • , sothat n p ( M • ) = ∗ ℓ A [ p ] ( M [ p ] ) = ℓ A p ( M p ). (cid:3) Koszul complexes in grmod(A) and *Koszul multiplicities
In this section, A is any graded Noetherian ring and M ∈ grmod ( A ). We again follow the discussionin [15].The definition of a complex of modules in grmod ( A ) is as usual: this is a sequence ( M , ∂ ) · · · → M j ∂ j → M j − ∂ j − → · · · → ∂ → M of objects and morphisms in grmod ( A ), such that ∂∂ = 0 everywhere. The sequence of morphisms ∂ above is called the differential for the complex.The subscripts j seem assigned ambiguously, but here’s what we mean: If ( M , ∂ ) is a complex in grmod ( A ) as above, then the set of elements of M j of degree i is equal to( M j ) i . = M j,i . In other words, when speaking of a complex in grmod ( A ), a single integer subscript denotes thesequential index of the complex, and a doubly-indexed subscript is read as “first index is the complexindex, second is the graded-module index”. We will often suppress the internal gradings, so if there isjust one subscript, it refers to the “complex index”. Hopefully this won’t be too confusing.To further set notation, we will regard any M in grmod ( A ) as a “complex concentrated in degree0”–this is the complex with all differentials equal to zero, with M i = 0, if the “complex-index” i = 0,and M = M , for “complex-index ” 0.The homology groups of a complex ( M , ∂ ) are defined as “ ker ∂/ im ∂ ” of course, and are also in grmod ( A ): H j ( M ) i = ker( ∂ : M j,i → M j − ,i ) / im( ∂ : M j +1 ,i → M j,i ) . Morphisms of graded complexes and short exact sequences of graded complexes are defined in theusual way. short exact sequence of graded complexes in grmod ( A ) gives rise to a long exact sequence onhomology: if 0 → A α −→ B β −→ C → grmod ( A ), thereexists a graded morphism ω of complex degree − · · · ω j +1 −−−→ H j ( A ) α ∗ −−→ H j ( B ) β ∗ −→ H j ( C ) ω j −→ H j − ( A ) α ∗ −−→ · · · is an exact sequence in grmod ( A ). Definition 8.1.
The *Euler characteristic of a complex ( M , ∂ ) in grmod ( A ) is defined when ( M , ∂ )is such that each A -module M i has ∗ ℓ A ( M i ) < ∞ and for all but finitely many i , ∗ ℓ A ( M i ) = 0 . Giventhese conditions, the following sum is well defined: ∗ χ ( M ) . = Σ i ( − i ∗ ℓ A ( M i ) . Since ∗ ℓ sums over short exact sequences, we get the following lemma. Lemma 8.2.
Let → A → B → C → be a short exact sequence of graded complexes in grmod ( A ) .Then, the *Euler characteristic of B is defined if and only if the *Euler characteristics of A and C are,and ∗ χ ( B ) = ∗ χ ( A ) + ∗ χ ( C ) . If the two conditions for a well-defined *Euler characteristic of a complex are not met, there may bea way to salvage the situation by passing to homology.
Definition 8.3.
Let ( M , ∂ ) be a complex in grmod ( A ) such that for every i , H i ( M ) has finite *lengthover A , and for i >> ∗ ℓ ( H i ( M )) = 0. We define the *Euler characteristic of the homology to be ∗ χ ( H ( M )) . = Σ i ( − i ∗ ℓ A ( H i ( M )).With the proof exactly analogous to that in the ungraded case, we have Theorem 8.4.
When the *Euler characteristic ∗ χ ( M ) is defined, then ∗ χ ( H ( M )) is also defined, andwe have that ∗ χ ( M ) = ∗ χ ( H ( M )) . Note however that the converse is not necessarily true; i.e. ∗ χ of the homology may be defined but ∗ χ of the complex not.Using the additivity of ∗ ℓ , and the long exact sequence on homology, if A B ։ C is a short exactsequence of graded complexes in grmod ( A ) such that the *Euler characteristic of the homology of eachcomplex is defined, then, ∗ χ ( H ( B )) = ∗ χ ( H ( A )) + ∗ χ ( H ( C )) . If A is a graded ring, we may do homological algebra in grmod ( A ) quite analogously to how it’s donein the ungraded case. In particular, graded A -modules T or Ai ( M, N ) ∈ grmod ( A ) for every i ≥
0, and
M, N ∈ grmod ( A ) may be defined mimicking the constructions and definitions in the ungraded category:beginning with the graded tensor product M ⊗ A N . (For the definition of the graded tensor product ofgraded modules over a graded ring, see [8].) The tensor product M ⊗ A N has a natural grading on it: if m ∈ M i and n ∈ N j are homogeneous elements, then deg( m ⊗ n ) = i + j . Then, one proceeds to talkabout projective resolutions, and arrives at the definition of T or Ai ( M, N ) ∈ grmod ( A ). We do not givefurther details here.8.1. The Koszul complex.
Standard properties of the Koszul complex in the ungraded case may befound in [15], Chapter IV. We use Serre’s notation: if ¯ x . = x , . . . , x u is a sequence of elements in A , thenthe Koszul complex is denoted by K (¯ x, A ).If we pass to the graded category, with A a graded ring, and choose a sequence ¯ x . = x , . . . , x u ofhomogeneous elements of A , the definition of the graded Koszul complex is briefly summarized as follows.Recall that the tensor product of graded complexes C ⊗ A D is defined exactly analogously to the ungradedcase, and is again a graded complex; keep in mind in particular the definition of the differential of atensor product of complexes: if c ∈ C i and d ∈ D j , then ∂ C ⊗ A D ( c ⊗ d ) = ∂ C ( c ) ⊗ d + ( − i c ⊗ ∂ D ( c ) . Starting with the case u = 1, K ( x , A ) is the two-term complex in grmod ( A ) K ( x , A ) = A ( − d ) · x → K ( x , A ) = A, where d is the degree of x . Then, if ¯ x = x , . . . , x u is a sequence of homogeneous elements in A , K (¯ x, A ) . = K ( x , A ) ⊗ A · · · ⊗ A K ( x u , A ) . If M ∈ grmod ( A ), the Koszul complex associated to the graded A -module M and ¯ x is: K A (¯ x, M ) . = K (¯ x, A ) ⊗ A M. f we’re always regarding a graded abelian group M as an A -module, for a fixed graded ring A , we willoften delete the superscript A .Setting notation, K ( x i , A ) is identified with A as a free, graded A -module (in other words, the freegenerator lies in degree zero, and is identified with 1 ∈ A ). For K , choose e x i of deg( x i ) and identify K ( x i , A ) with the free graded A -module generated by the homogeneous element e x i . Then, K p (¯ x, A )is the free graded A -module isomorphic to the free graded A -module generated by the homogeneouselements e x i ⊗ · · · ⊗ e x ip of degree deg( x i ) + · · · + deg( x i p ) where i < · · · < i p , so is isomorphic to thegraded exterior product Λ p ( A ( − deg( x )) ⊕ · · · ⊕ A ( − deg( x u ))) . In addition, both the p th part of the Koszul complex K A (¯ x, M ), and its differential have exactly thesame form as described in [15], IV.A.2 in the ungraded case. A particular consequence is that, as agraded A -module, K Ap (¯ x, M ) is a direct sum of (cid:0) up (cid:1) copies of M , each shifted: the copy associated to themulti-index i < · · · i p looks like M ( − (deg( x i ) + · · · + deg( x i p ))); if I is the graded ideal of A generatedby x , . . . , x u , and k ≥
0, then K Ap (¯ x, M ) / I k K Ap (¯ x, M ) is, as a graded A -module, isomorphic to (cid:0) up (cid:1) copies of M/ I k M , each shifted as described above.The p th homology group of the graded Koszul complex K A (¯ x, M ) is denoted by H p (¯ x, M ) , or H Ap (¯ x, M )if we need to emphasize the role of A . These homology groups are also graded A -modules. Definition 8.5.
Suppose that x , . . . , x u is a sequence of nonzero nonunit homogeneous elements in A and M ∈ grmod ( A ). This sequence is a M -sequence if and only if x is not a zero-divisor on M , and foreach i > x i is not a zero-divisor on M/ ( x , . . . , x i − ) M .The following may all be proved as in the ungraded case (see [15], Chapter IV): Proposition 8.6.
Let A be a graded ring and M ∈ grmod ( A ) . If ¯ x is a M -sequence, then the Koszulcomplex K A (¯ x, M ) is acyclic. As in the ungraded case, H A (¯ x, M ) = M/ ( x , . . . , x u ) M . Conversely, in the *local Noetherian case one has
Proposition 8.7. If ( A, N ) is a *local Noetherian ring, and M ∈ grmod ( A ) , then the following areequivalent, for a sequence of homogeneous elements ¯ x . = x , . . . , x u of N : • H Ap (¯ x, M ) = 0 , for p ≥ . • H A (¯ x, M ) = 0 . • ¯ x is an M -sequence in A . The proofs of the above Propositions are exactly analogous as that of IV.A.2, Propositions 2, 3 in [15],replacing any use of Nakayama’s lemma with the graded Nakayama’s lemma (Lemma 4.12); similarly,IV.A.2, Corollary 2 yields, in the graded case,
Corollary 8.8. If ( A, N ) is a *local Noetherian ring, M ∈ grmod ( A ) , and ¯ x = x , . . . , x u are homo-geneous elements of N that form an A -sequence for A , then there is a natural isomorphism of graded A -modules ψ : H Ai (¯ x, M ) → T or Ai ( A/ (¯ x ) , M ) . Finally, IV.A.2, Proposition 4, has the analogous
Proposition 8.9.
Suppose that ( A, N ) is a *local graded Noetherian ring and M ∈ grmod ( A ) . If x , . . . , x u are homogeneous elements of N , then (¯ x ) + Ann A ( M ) ⊆ Ann A ( H Ai (¯ x, M )) . As a corollary,
Proposition 8.10.
Suppose that ( A, N ) is a *local graded Noetherian ring , and M ∈ grmod ( A ) . Let I be a GIOD for M , generated by the homogeneous sequence ¯ x = x , . . . , x u ∈ N . Then, H Aj (¯ x, M ) hasfinite *length over A for every j ≥ .Proof. Since I + Ann A ( M ) ⊆ Ann A ( H Aj (¯ x, M )), and {N } = ∗ V ( I + Ann A ( M )), if H Aj (¯ x, M ) = 0 , {N } = ∗ V ( Ann A ( H Aj (¯ x, M ))) . (cid:3) Thus, in the *local case the *Euler characteristic of the homology of the graded Koszul complex iswell defined: efinition 8.11. Suppose that ( A, N ) is a *local, Noetherian graded ring and M ∈ grmod ( A ). Let I be a GIOD for M ∈ grmod ( A ) generated by a homogeneous sequence ¯ x = x , . . . , x u . We define the* Koszul multiplicity ∗ χ A (¯ x, M ) to be the *Euler characteristic of the homology of the graded Koszulcomplex: ∗ χ A (¯ x, M ) = u X i =1 ( − i ∗ ℓ A ( H Ai (¯ x, M )) . Equality of *Samuel and *Koszul multiplicities.
As in the ungraded case [15], IV.A.3, the*Koszul multiplicity is equal to a certain *Samuel multiplicity. This section concludes our account ofthe theory of multiplicities adapted to the Z -graded category.Let ( A, N ) be a *local, Noetherian graded ring, M ∈ grmod A and ¯ x = x , . . . , x u a sequence ofhomogeneous elements contained in N . If I is the graded ideal of A generated by ¯ x , suppose also that I is a GIOD for M .One then filters the graded Koszul complex, yielding graded complexes F i K for every i , with F i K p . = I i − p K p for every p (we’ve dropped the arguments ¯ x, M for expediency). Notice we have three indicesnow: the filtration index, the complex index and the internal gradings of the various A -modules involved.We are suppressing the internal grading. This filtration defines the associated graded complex gr ( K ) . = ⊕ i F i K / F i +1 K .If gr ( A ) is the associated bigraded ring to the I -adic filtration, then denote the images of x , . . . , x u in gr ( A ) , ∗ by ξ , . . . , ξ u . Let gr ( M ) be the bigraded gr ( A )-module associated with the I -adic filtration of M . Then, there is an isomorphism of graded objects gr ( K ) ∼ = K ( ¯ ξ, gr ( M )). Moreover, one argues thatthe homology modules H p ( ¯ ξ, gr ( M )) have finite *length over gr ( A ), for all p , since A/ ( I + Ann A ( M )) is*Artinian. This in turn, enables one to argue that there exists an m ≥ u such that the graded homologygroups of the complex F i K / F i +1 K all vanish for i > m , and so one sees that the graded homologygroups the complex F i K all vanish if i > m .Continuing as in [15], IV.A.3, (which is really a spectral sequence argument), this means there is an m such that if i > m , then H p ( K ) ∼ = H p ( K / F i K ) for i > m and for all p .Using the fact that *Euler characteristics don’t change when passing to homology, ∗ χ (¯ x, M ) = P p ( − p ∗ ℓ ( H p ( K / F i ( K )) = ∗ χ ( K / F i K ), for i > m . As noted in the previous section, ( K / F i K ) p is isomorphic as a graded A -module to a direct sum of (cid:0) up (cid:1) copies of M/ I i − p M , shifted appropriately,and since the *length of a shifted A -module M ( d ) is the same as the *length of M , the remainder ofthe proof is argued exactly as in [15], IV.A.3, with length replaced by *length, p (a Samuel polynomial)replaced by ∗ p and e replaced by ∗ e .Thus, we have Theorem 8.12.
Let ( A, N ) be a *local Noetherian ring. Let x , . . . , x u ∈ N be homogeneous elementsgenerating a graded ideal of definition I for M ∈ grmod ( A ) . Then, ∗ χ A (¯ x, M ) = ∗ e ( M, I , u ) , so ∗ χ A (¯ x, M ) is a strictly positive integer if ∗ dim A ( M ) = u , and ∗ χ A (¯ x, M ) = 0 if u > ∗ dim A ( M ) . Multiplicities and Degree for Positively Graded Rings
In this section, we specialize to the case of a positively graded Noetherian ring R with R a field k ; allgraded modules are in grmod ( R ) . Set m = R + and note that ( R, m ) is then a *local graded Noetherianring. We do not want to make the assumption that R is generated by elements in degree 1.In this chapter we may use the * notation even though we could just as well omit the * (e.g. If M ∈ grmod ( R ), then ∗ dim R ( M ) = dim R ( M ).) This is done to emphasize the fact that all computationsmay be done in the graded category using the theory developed in the previous two chapters (which isoften simpler than the ungraded theory.)We introduce the degree of a graded module, show how it relates to *multiplicity (Theorem 10.7), andgive a sum decomposition of degree by a certain set of minimal primes (Theorem 10.5.)Since R = k , ℓ k ( M i ) = vdim k ( M i ) for every i , so the Poincar´e series for M is equal to P M ( t ) = X i ∈ Z vdim k ( M i ) t i . Furthermore, this Laurent series has a pole at t = 1 using the Hilbert-Serre theorem, and the order ofthe pole d ( M ) at t = 1 is, by Smoke’s dimension theorem, is exactly ∗ dim R ( M ).This leads to the definition of deg R ( M ): efinition 9.1. If R is a positively graded Noetherian ring with R = k a field, M ∈ grmod ( R ) ,M = 0 and D ( M ) = ∗ dim R ( M ), thendeg R ( M ) . = lim t → (1 − t ) D ( M ) P M ( t )is a well-defined, strictly positive, rational number. For convenience, define deg R (0) = 0 . Often we delete the subscript R and just write deg( M ). We use the (somewhat ambiguous) name of“degree” for this rational number in deference to the nomenclature already used in [3]. For equivariantcohomology, this “degree” was first studied by Maiorana [10].10. Multiplicities and Euler-Poincar´e series If X ∈ grmod ( R ) has finite *length as an R -module, since each R i is finite-dimensional as a vectorspace over k , we may use Lemmas 3.11 and 3.12 to conclude that ℓ R ( X ) = ∗ ℓ R ( X ) = vdim k ( X ), wherevdim k ( X ) is the total dimension P j vdim k ( X j ) of the graded k -vector space X . We may then prove: Lemma 10.1.
Suppose R is a positively graded Noetherian ring with R = k , a field, and X ∈ grmod ( R ) is such that ∗ ℓ R ( X ) < ∞ . If B is a graded subring of R , Noetherian or not, with B = k = R , then X ∈ grmod ( B ) , ∗ ℓ B ( X ) < ∞ and ∗ ℓ B ( X ) = ∗ ℓ R ( X ) = ℓ R ( X ) = vdim k ( X ) < ∞ .Proof. For, using Lemma 3.12 applied to R , each X i is finite-dimensional over k , and there are integers t and J such that t ≤ J with X = ⊕ Jj = t X j . Also, ∗ ℓ R ( X ) = vdim k ( X ). However, since k ⊆ B ⊆ R , X is a finitely generated B -module. Whether B is Noetherian or not, since B j ⊆ R j for every j , and R j is finite-dimensional over k , so is B j . Thus, using Lemma 3.12 applied to B , ∗ ℓ B ( X ) = vdim k ( X ) aswell. (cid:3) If M ∈ grmod ( R ), then m is a GIOD for M , and we may then calculate a *Samuel polynomial ∗ p R ( M, m , n ) for M ; Theorem 7.8 says that this is the ordinary Samuel polynomial p R ( M, m , n ); Corollary7.11 says that the degree of this polynomial is D ( M ) . = ∗ d ( M ) = ∗ s ( M ) = s ( M ) = d ( M ) = ∗ dim R ( M ) = dim R ( M ) . Now, suppose that ¯ x = x , . . . , x D ( M ) is a GSOP for the R -module M . If I is the graded ideal in R generated by ¯ x , I is a GIOD for M . We can change rings to B . = k h x , . . . , x D ( M ) i , note that this is agraded polynomial ring over k in the indicated variables (Proposition 6.4). The ideal ˆ I generated by ¯ x in B is also a GIOD in B since clearly ˆ I n M = I n M for every n . Therefore Theorem 7.8 and the previouslemma guarantee that, for every n , the polynomials below are all equal, as indicated: ∗ p R ( M, I , n ) = p R ( M, I , n ) = p B ( M, ˆ I , n ) = ∗ p B ( M, ˆ I , n );in particular, they all have the same degree D ( M ), and the following positive integers are also all equal: ∗ e R ( M, I , D ( M )) = e R ( M, I , D ( M )) = e B ( M, ˆ I , D ( M )) = ∗ e B ( M, ˆ I , D ( M )) . Euler-Poincar´e series.
The following lemma is found in Avramov and Buchweitz [1]; [14] con-tains a similar result.
Lemma 10.2. (Lemma 7 of [1] ) If
M, N ∈ grmod ( R ) , then a. For each i , the graded R -module T or Ri ( M, N ) has finite dimensional (over k = R ) homogenouscomponents T or Ri ( M, N ) j , for every j ; also, for every i , T or Ri ( M, N ) j = 0 for j << . Thusone may form the Laurent series P T or Ri ( M,N ) ( t ) . = X j ∈ Z vdim k ( T or Ri ( M, N )) j t j . b. Furthermore,the alternating sum χ R ( M, N )( t ) . = X i ≥ ( − i P T or Ri ( M,N ) ( t ) , which is by definition the Euler-Poincar´e series of M, N , is a well-defined Laurent series withinteger coefficients and P R ( t ) χ R ( M, N )( t ) = P M ( t ) P N ( t ) . f a GSOP ¯ x is given for M ∈ grmod ( R ), B . = k h ¯ x i ⊆ R is then a graded polynomial ring over k (Proposition 6.4), and M ∈ grmod ( B ), using Lemma 6.3. Whether we consider M ∈ grmod ( R ), or M ∈ grmod ( B ), the Poincar`e series of M does not change.Hilbert’s Syzygy Theorem tells us that the graded Koszul complex K k h ¯ x i (¯ x, k ) is acyclic, thus is afree, finite graded resolution of k as a graded k h ¯ x i -module. In particular, we may tensor this resolutionwith M and use it to compute T or k h ¯ x i i ( k, M ) = T or k h ¯ x i i ( M, k ) , showing that T or k h ¯ x i i ( M, k ) = H k h ¯ x i i (¯ x, M ) . Lemma 10.3.
Let ¯ x = x , . . . , x D ( M ) be a GSOP for M ∈ grmod ( R ) , and let I be the graded idealin R generated by ¯ x . For every i , P T or k h ¯ x i i ( M,k ) ( t ) ∈ Z [ t, t − ] , and therefore χ k h ¯ x i ( M, k )( t ) ∈ Z [ t, t − ] . Furthermore, χ k h ¯ x i ( M, k )( t ) = D ( M ) X j =0 ( − j P H k h ¯ x i j (¯ x,M ) ( t ) , and evaluating this Laurent polynomial at t = 1 , we compute χ k h ¯ x i ( M, k )(1) = ∗ χ k h ¯ x i (¯ x, M ) = ∗ e R ( I , M, D ( M )) = ∗ χ R (¯ x, M ) , where D ( M ) = ∗ dim R ( M ) . Proof.
Lemma 10.2 shows the first part of the statement, and since the resolution K k h ¯ x i (¯ x, k ) is in anycase zero for complex degree larger than D ( M ), T or k h ¯ x i i ( M, k ) is also zero for i > D ( M ), so χ k h ¯ x i ( M, k )( t ) . = X j ≥ ( − j P T or k h ¯ x i j ( M,k ) ( t ) = D ( M ) X j =0 ( − j P H k h ¯ x i j (¯ x,M ) ( t ) , being a finite sum of Laurent polynomials, is a Laurent polynomial.Setting B . = k h ¯ x i yields, ∗ ℓ B ( H Bi (¯ x, M )) = ℓ B ( H Bi (¯ x, M )) = vdim k ( H Bi (¯ x, M )) , so ∗ χ B (¯ x, M ) = D ( M ) X j =0 ( − j ℓ B ( H Bj (¯ x, M )) = D ( M ) X j =0 ( − j vdim k ( H Bj (¯ x, M )) . = χ B ( M, k )(1) . As noted at the beginning of this section, if ˆ I is the ideal generated by ¯ x in B , then ∗ e B ( M, ˆ I , D ( M )) = ∗ e R ( M, I , D ( M )) and Theorem 8.12 says that ∗ χ B (¯ x, M ) = ∗ e B ( M, ˆ I , D ( M )) = ∗ e R ( M, I , D ( M )) = ∗ χ R (¯ x, M ) . (cid:3) Degree of a Graded Module in grmod ( R ) . Given M ∈ grmod ( R ), deg( M ) >
0, if M = 0, wecan read off the degree of a module directly from the Poincare series if we expand it as a Laurent seriesabout t = 1: P R ( t ) = deg( M )(1 − t ) D ( M ) + ”higher order terms” . Lemma 10.4.
Suppose that → N → M → P → is an exact sequence in grmod ( R ) . Then, • D ( M ) = max { D ( N ) , D ( P ) } . • If D ( N ) < D ( M ) , then deg( M ) = deg( P ) . • If D ( P ) < D ( M ) , then deg( M ) = deg( N ) . • If D ( P ) = D ( N ) = D ( M ) , then deg( M ) = deg( N ) + deg( P ) . • deg( M ( d )) = deg( M ) , for every integer d . This immediately yields, as in [3]:
Theorem 10.5.
Let M ∈ grmod ( R ) , and D ( M ) be defined as in Theorem 7.10: this is the set of primeideals p in R , necessarily minimal primes for M and graded, such that ∗ dim R ( R/ p ) = ∗ dim R ( M ) . Then, deg( M ) = X p ∈D ( M ) ∗ ℓ R [ p ] ( M [ p ] ) · deg( R/ p ) . roof. Choose a graded filtration M • of M of the form in Lemma 2.5 We know that if p ∈ D ( M ), thenthe graded R -module R/ p , possibly shifted, occurs exactly ∗ ℓ R [ p ] ( M [ p ] ) = ℓ R p ( M p ) times (using Theorem4.10) as a subquotient in the filtration. The lemma above then gives the result. (cid:3) We want to compare degree to our previously studied multiplicities.Letting ¯ x be a GSOP for M ∈ grmod ( R ), we’ve seen that k h ¯ x i is a graded polynomial ring, and onedirectly calculates that P k h ¯ x i ( t ) = 1 Q D ( M ) i =1 (1 − t d i ) , where d i is the degree of the homogeneous element x i .Now, using M ∈ grmod ( k h ¯ x i ), and recalling that P M ( t ) is the same whether we consider M ∈ grmod ( R )or M ∈ grmod ( k h ¯ x i ), Lemma 10.2 gives that P k ( t ) P M ( t ) = P k h ¯ x i ( t ) χ k h ¯ x i ( M, k )( t ) . Also, χ k h ¯ x i ( M, k )( t ) ∈ Z [ t, t − ]. Since P k ( t ) = 1, we have Theorem 10.6. If M ∈ grmod ( R ) and ¯ x is a GSOP for M , then P M ( t ) = χ k h ¯ x i ( M, k )( t ) Q D ( M ) i =1 (1 − t d i ) , with χ k h ¯ x i ( M, k )( t ) ∈ Z [ t, t − ] . Since (1 − t ) D ( M ) P M ( t ) = χ k h ¯ x i ( M, k )( t ) Q D ( M ) i =1 (1 + t + · · · + t d i − ) , taking the limit as t approaches 1, and using Lemma 10.3, yields: Theorem 10.7. If M = 0 is in grmod ( R ) , and x , . . . , x D ( M ) of degrees d , . . . , d D ( M ) form a GSOPfor M , generating the graded ideal I of R , then deg( M ) = ∗ e R ( M, I , D ( M )) d · · · d D ( M ) = ∗ χ R (¯ x, M ) d · · · d D ( M ) . Thus, the ratio ∗ e R ( M, I , D ( M )) d · · · d D ( M ) is independent of the choice of system of parameters x , . . . , x D ( M ) for M . Note that we can delete the “stars” in the equalities of the above theorem and retain the equalities,using Theorem 7.8. The reader should compare this result to Proposition 5.2.2 of [13], which states asimilar result for rings with standard grading.
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