Asymptotic behavior of Ext for pairs of modules of large complexity over graded complete intersections
aa r X i v : . [ m a t h . A C ] D ec ASYMPTOTIC BEHAVIOR OF EXT FOR PAIRS OF MODULESOF LARGE COMPLEXITY OVER GRADED COMPLETEINTERSECTIONS
DAVID A. JORGENSEN, LIANA M. S¸EGA, AND PEDER THOMPSON
Abstract.
Let M and N be finitely generated graded modules over a gradedcomplete intersection R such that Ext iR ( M, N ) has finite length for all i ≫ iR ( M, N ) for all large even i and all large odd i , have the same degreeand leading coefficient whenever the highest degree of these polynomials is atleast the dimension of M or N . Refinements of this result are given when R is regular in small codimensions. introduction Let Q be a regular local ring with maximal ideal n and f = f , . . . , f c be a Q -regularsequence contained in n . Set R = Q/ ( f ) and let M and N be finitely generated R -modules. In this case R is called a complete intersection and Gulliksen [14] hasshown that Ext R ( M, N ) has the structure of a finitely generated graded module overthe polynomial ring R [ χ , . . . , χ c ], where deg χ i = 2 for 1 ≤ i ≤ c . Consequently,Ext R ( M, N ) decomposes as a direct sum of two graded submodules, one consistingof the Ext iR ( M, N ) for i even, and the other of the Ext iR ( M, N ) for i odd.Further assume that Ext iR ( M, N ) has finite length for all i ≫
0. The tails ofthe even and odd Ext modules are thus finitely generated over the polynomial ring R/ n n [ χ , . . . , χ c ] for some integer n ≥
1. Standard commutative algebra then tellsus that there exist polynomials P
M,N even ( t ) and P M,N odd ( t ) with rational coefficients,which we call the even and odd Hilbert polynomials of Ext R ( M, N ), such thatP
M,N even ( i ) = λ (Ext iR ( M, N )) for all even i ≫ M,N odd ( i ) = λ (Ext iR ( M, N )) forall odd i ≫
0, where λ ( − ) denotes length.A natural question to ask is whether P M,N even and P
M,N odd have the same degreeand, if so, the same leading coefficient. There are easy and well-known examplesshowing that this is not always the case. For example, consider the ring R = k J x , x K / ( x x ) and the R -modules M = R/ ( x ) and N = R/ ( x ). With thesechoices, P M,N odd = 1 and P
M,N even = 0. However, for N = k , where k = Q/ n (so that theintegers λ (Ext iR ( M, N )) are the Betti numbers of M ), the polynomials P M,N even andP
M,N odd do have the same degree and leading coefficient, for example by the proof ofAvramov’s [1, Theorem 9.2.1(1)]; see also Celikbas and Dao [9, Proposition 3.2].The above statements translate—as is standard—to the graded setting, and inthis paper we establish equality of degree and leading coefficient of the Hilbert
Date : December 19, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Complete intersection, complexity, graded ring, Hilbert series.This work was supported in part by a Simons Foundation grant ( polynomials P
M,N even and P
M,N odd for a broad contingent of pairs of finitely generatedgraded modules over a graded complete intersection. The maximum degree of P
M,N even and P
M,N odd is one fewer than the complexity of the pair (
M, N ), denoted cx R ( M, N ),and we formulate our results in terms of this established invariant; see Section 1.The complexity cx R ( M, N ) is also the dimension of the support variety of M and N ; see Avramov and Buchweitz [2]. The following is part of Theorem 4.3 below. Theorem 1.
Let R be a graded complete intersection and let M and N be finitelygenerated graded R -modules with λ (Ext iR ( M, N )) < ∞ for all i ≫ . If cx R ( M, N ) > max { dim R/ (ann M + ann N ) , dim M + dim N − dim R } then P M,N even and P M,N odd have the same degree and leading coefficient.
In particular, the conclusion holds when cx R ( M, N ) > min { dim M, dim N } . Italso holds for possibly smaller values of cx R ( M, N ), provided R p is regular for allhomogeneous prime ideals p of small codimension, see Theorem 4.3(3) and Theorem4.2 below.Before describing the paper further, we comment that its results complementthose of several recent articles on certain homological invariants for pairs of modulesover hypersurfaces and, more generally, over complete intersections. The Herbranddifference introduced by Buchweitz [8] for hypersurfaces, and generalized to com-plete intersections of higher codimension in [9], is one such invariant; see (1.2.1)below for the definition of the j th Herbrand difference h Rj ( M, N ).For the remainder of the introduction, unless otherwise specified, R is a gradedcomplete intersection of codimension c , and M and N are finitely generated graded R -modules with λ (Ext iR ( M, N )) < ∞ for all i ≫
0. Set r = cx R ( M, N ). As ispointed out by Moore, Piepmeyer, Spiroff, and Walker [19], when r >
M,N even and P
M,N odd have the same degree and leading coefficient if and only if h Rr ( M, N ) = 0. Several recent papers establish the vanishing of h Rc ( M, N ) undercertain conditions on the ring and (or) on the modules, see [7, 10, 11, 12, 18, 19, 22,23]. Most of these results are formulated in terms of vanishing of higher codimensionversions of Hochster’s theta pairing, such as Dao’s eta-invariant η Rr ( M, N ) in [10];these invariants are defined similarly, only using Tor instead of Ext. When R is anisolated singularity, the vanishing of η Rc ( − , − ) for all pairs of finitely generated R -modules is equivalent to the vanishing of h Rc ( − , − ) for all pairs of finitely generated R -modules, due to a result of Dao [22, Lemma 5.9].The origin of these invariants goes back to Serre and his work on intersectionmultiplicities [21]. Hochster defined his theta invariant in order to extend Serre’swork to hypersurfaces. He showed in [15] that his theta invariant for a pair offinitely generated modules M and N satisfying λ ( M ⊗ R N ) < ∞ over a so-calledadmissible hypersurface vanishes if and only if dim M + dim N ≤ dim R . Theorem1 above provides a cohomological extension of Hochster’s result to graded completeintersections of arbitrary codimension. Corollary.
Let R be a graded complete intersection and let M and N be finitelygenerated graded R -modules with cx R ( M, N ) > dim R/ (ann M + ann N ) and suchthat λ (Ext iR ( M, N )) < ∞ for all i ≫ . Then P M,N even and P M,N odd have the samedegree and leading coefficient if and only if dim M + dim N − dim R < cx R ( M, N ) . This is Corollary 4.4 below. If λ ( M ⊗ R N ) < ∞ , then dim R/ (ann M +ann N ) = 0and λ (Ext iR ( M, N )) < ∞ for all i ≫
0. Thus the hypothesis of Hochster’s result
SYMPTOTIC BEHAVIOR OF EXT 3 implies the hypothesis of the corollary when cx R ( M, N ) = 1, and in this case theconclusion of the corollary gives the conclusion of Hochster’s result; see also Remark4.5 in case cx R ( M, N ) = 0.A simple fact from commutative algebra is that r ≤ c , where c is the codimensionof R and r = cx R ( M, N ) as above. Furthermore, pairs of finitely generated graded R -modules ( M , N ), neither of which is k , can be chosen so that r takes any givenvalue between 1 and c (or −∞ ), see Proposition 5.1. Since the results in the recentliterature only establish the vanishing of h Rr ( M, N ) when r = c , such results have nobearing on the question of whether the degree and leading coefficients of P M,N even andP
M,N odd are equal when r < c . One of the major ingredients used in the proofs of theexisting results is that, when R is an isolated singularity, the pairings h Rc ( − , − ) and η Rc ( − , − ) are biadditive on short exact sequence of finitely generated R -modules,and this allows for work in Grothendieck groups. On the other hand, if 1 ≤ r < c ,the pairing h Rr ( − , − ) is only biadditive on certain short exact sequences, see [9,Theorem 3.4(2)].The main ingredient in our work consists of uncovering a connection betweenP M,N even and P
M,N odd having the same degree and leading coefficient and certain invari-ants in work of Avramov, Buchweitz, and Sally [3]. Let H ( M, t ) be the Hilbert seriesof M and for each j ≥ ρ jR ( M, N )( t ) denote the following rational function: ρ jR ( M, N )( t ) = j X i =0 ( − i H (Ext iR ( M, N ) , t ) − H ( M, t − ) H ( N, t ) H ( R, t − ) . Further, let o ( ρ jR ( M, N )) denote the order of the Laurent series expansion around t = 1 of ρ jR ( M, N )( t ), see Section 1 for details. The following is Theorem 4.1 below. Theorem 2.
Let R be a graded complete intersection and let M and N be finitelygenerated graded R -modules. Set ℓ = inf { i | λ (Ext jR ( M, N )) < ∞ for all j ≥ i } and r = cx R ( M, N ) . If ℓ < ∞ and r ≥ , then the following hold: (1) o ( ρ ℓR ( M, N )) ≥ − r , and (2) The polynomials P M,N even and P M,N odd have the same degree and leading coeffi-cient if and only if o ( ρ ℓR ( M, N )) > − r . The inequality in (1) of Theorem 2 can be interpreted in terms of relations be-tween the Laurent coefficients of the Laurent series expansions around t = 1 of H (Ext iR ( M, N ) , t ) for i = 1 , . . . , ℓ , and those of H ( M, t ), H ( N, t ), and H ( R, t ), asin [3, Theorem 7]. Part (2) of Theorem 2 provides the main ingredient in provingour results about the odd and even Hilbert polynomials of Ext R ( M, N ) and it leadsdirectly to the proof of Theorem 1. In addition, the proofs of Theorem 4.3(3) andTheorem 4.2 use several results from [3].The structure of the paper is as follows. Section 1 introduces notation and recordsproperties of Laurent coefficients, and Section 2 further investigates such propertiesunder the additional condition that R p is regular for homogeneous prime ideals p ofcertain codimension. Section 3 presents a construction, which is possible when thebase field is infinite, of a module K such that the maximum degree of the even andodd Hilbert polynomials of the pair ( K, N ) is one less than that of the pair (
M, N ),and the even and odd Hilbert polynomials of the pair (
M, N ) have the same degreeand leading coefficient if and only if the same is true for the pair (
K, N ); this is astandard construction of superficial elements that exploits the finite generation of
DAVID A. JORGENSEN, LIANA M. S¸EGA, AND PEDER THOMPSON
Ext R ( M, N ) over the ring R [ χ , . . . , χ c ] of cohomology operators. Section 4 givesa proof of Theorem 2, using computations with Hilbert series, Laurent expansions,and a repeated application of the construction from Section 3, which then alsoyields a proof of Theorem 1. Section 5 contains examples of pairs of moduleshaving arbitrary complexity, and shows that the regularity hypotheses in Theorem4.2 are necessary for pairs of modules of smaller complexity.1. Notation and properties of Laurent series
Setting.
Let Q = k [ x , . . . , x ν ] be a polynomial ring over a field k , withvariables x i of positive degree, and R = Q/I be a graded ring, with I a homogeneousideal such that I ⊆ ( x , . . . , x ν ) . Let M and N be finitely generated graded R -modules.We denote length by λ ( − ) and we setf R ( M, N ) = inf { n ∈ Z | λ R (Ext iR ( M, N )) < ∞ for all i > n } and β Ri ( M, N ) = λ R (Ext iR ( M, N )) , for each i ≥ f R ( M, N ) . We say R is a graded complete intersection (of codimension c ) if R is as in 1.1,with I = ( f , . . . , f c ) for a homogeneous Q -regular sequence f , . . . , f c . Defini-tions and results for local complete intersections translate to the graded settingin a standard manner. In particular, if R is a graded complete intersection withf R ( M, N ) < ∞ then there exist polynomials P M,N even and P
M,N odd with rational coeffi-cients such thatP
M,N even (2 i ) = β R i ( M, N ) and P
M,N odd (2 i + 1) = β R i +1 ( M, N ) for all i ≫ R ( M, N ) < ∞ . The complexity of a pair ( M, N ) of finitely generatedgraded R -modules is defined ascx R ( M, N ) = inf ( b ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β Rn ( M, N ) ≤ an b − for somereal number a and for all n ≫ ) . This definition of complexity agrees, by [9, Proposition 2.2], with the more gen-eral one given by Avramov and Buchweitz [2]. As noted in the introduction,cx R ( M, N ) = 1 + max n deg P M,N even , deg P M,N odd o .We next define an invariant that measures when P M,N even and P
M,N odd have the samedegree and leading coefficient.
Definition 1.2.
Assume R is a graded complete intersection. Let M and N be finitely generated graded R -modules such that f R ( M, N ) < ∞ and set r =cx R ( M, N ). Since r < ∞ , we may writeP M,N even ( t ) = a r − t r − + · · · + a t + a , P M,N odd ( t ) = b r − t r − + · · · + b t + b , where at least one of a r − and b r − is nonzero. We define h R ( M, N ) := a r − − b r − . SYMPTOTIC BEHAVIOR OF EXT 5
The j th (generalized) Herbrand difference of M and N is defined in [9] as(1.2.1) h Rj ( M, N ) = lim n →∞ P ni =f R ( M,N )+1 ( − i β Ri ( M, N ) n j . Remark 1.3.
The invariant h R ( M, N ) equals the classical Herbrand differencefrom [8] when c = r = 1. Looking at the proof of [9, Theorem 3.4(1)] and theformula in [9, Proposition 3.2(2)], we see h R ( M, N ) = 2 rh Rr ( M, N ). Thus when r >
0, the invariants h R ( M, N ) and h Rr ( M, N ) vanish simultaneously.
Remark 1.4.
The condition f R ( M, N ) < ∞ holds for any pair of finitely generatedgraded R -modules if R is an isolated singularity. Unlike most results in the recentliterature, we do not require that R is an isolated singularity.For the remainder of this section we do not need to assume that R is a completeintersection. Next we introduce invariants defined in terms of the coefficients of theLaurent series expansions around t = 1 of certain rational functions involving theHilbert series of the modules M and N and their Ext modules, and discuss theirproperties. We begin by discussing Laurent series expansions of rational functionsin general.Let ϕ ( t ) ∈ R ( t ) be a rational function. Laurent series.
Let ∞ X n = −∞ a n ( t − n be the Laurent series expansion of ϕ ( t ) around t = 1. The order of this Laurentseries is denoted o ( ϕ ) and is defined as o ( ϕ ) = inf { n ∈ Z | a n = 0 } . When ϕ = 0, the order o ( ϕ ) is defined to be ∞ . For each integer n we set g n ( ϕ ) = ( − n a − n . With this notation, we have o ( ϕ ) = − sup { n ∈ Z | g n ( ϕ ) = 0 } and the Laurent series of ϕ ( t ) can be rewritten as X n ≤− o ( ϕ ) g n ( ϕ )(1 − t ) n . Poles and orders.
Since ϕ is a rational function, we have o ( ϕ ) > −∞ . We saythat ϕ has a pole at t = 1 if 1 is a zero of the denominator of ϕ when ϕ is writtenin reduced form. If ϕ has a pole at t = 1, we say that the pole has order m if ϕ ( t ) = Q ( t )(1 − t ) m for some rational function Q ( t ) with no pole at t = 1 and Q (1) = 0.If ϕ does not have a pole at t = 1 then o ( ϕ ) ≥
0. Otherwise, we see that ϕ hasa pole of order m > t = 1 if and only if o ( ϕ ) = − m . In this case, we have g m ( ϕ ) = Q (1) and g a ( ϕ ) = 0 for all a > m , where Q ( t ) is as above.Note that o ( ϕ ) = 0 if and only if ϕ does not have a pole at t = 1 and ϕ (1) = 0.Also, o ( ϕ ) ≥ ϕ does not have a pole at t = 1 and ϕ (1) = 0. DAVID A. JORGENSEN, LIANA M. S¸EGA, AND PEDER THOMPSON
Sums and products.
Let ψ be another rational function. Set m = − o ( ϕ ) and n = − o ( ψ ). When adding Laurent series, one simply adds the coefficients, and wehave(1.7.1) o ( ϕ + ψ ) ≥ min { o ( ϕ ) , o ( ψ ) } , with strict inequality if and only if m = n and g m ( ϕ ) + g m ( ψ ) = 0.In general, Laurent series cannot be multiplied because computing the coeffi-cients of a product leads to infinite sums. However, the problem is avoided whenthe series have order different than −∞ , as is the case for rational functions:(1.7.2) o ( ϕψ ) = o ( ϕ ) + o ( ψ ) and o (cid:18) ϕψ (cid:19) = o ( ϕ ) − o ( ψ ) when ψ = 0.The coefficients of the Laurent series of ϕψ can be computed using the formula g a ( ϕψ ) = m X i = a − n g i ( ϕ ) g a − i ( ψ ) . In particular, we have g m + n ( ϕψ ) = g m ( ϕ ) g n ( ψ ) , and g m − n ( ϕψ ) = g m ( ϕ ) g n ( ψ ) when ψ = 0 . Hilbert series.
The Hilbert series of a graded R -module M = ⊕ i ≥ b M i is H ( M, t ) = X i ≥ b dim k ( M i ) t i ∈ Z (( t )) . Let a be an integer and let M ( a ) denote the graded R -module with M ( a ) i = M a + i .Note that H ( M ( a ) , t ) = t − a H ( M, t ) . The following is a standard result in dimension theory, for example see Marley’sproof in [17, Proposition 3.2.1]: If dim M = m , then there exist positive integers s , . . . , s m and h ( t ) ∈ Z [ t, t − ] where h (1) = 0 such that H ( M, t ) = h ( t ) Q mi =1 (1 − t s i ) . In particular, the Hilbert series H ( M, t ) is a rational function that can be furtherwritten as H ( M, t ) = Q M ( t )(1 − t ) m , where Q M ( t ) is a rational function with no pole at t = 1 and Q M (1) = 0. To simplify notation, we set for each integer ig i ( M ) = g i ( H ( M, t )) and o ( M ) = o ( H ( M, t )) . We have:(1.8.1) o ( M ) = − dim M and g dim M ( M ) = Q M (1)and the Laurent series expansion of H ( M, t ) around t = 1 is X n ≤ dim M g n ( M )(1 − t ) n . In particular, note that g n ( M ) = 0 for n > dim M .When R is standard graded, meaning that R is generated as a k -algebra by R , we have Q M ( t ) ∈ Z [ t, t − ], and Q M (1) is equal to the multiplicity of M . The SYMPTOTIC BEHAVIOR OF EXT 7 next formula is well-known in this case, and the same proof works when R is notnecessarily standard graded, see [3, Lemma 9]:(1.8.2) g m ( M ) = X p ∈ Proj( R ) , dim R/ p = m λ R p ( M p ) g m ( R/ p ) . Here, m = dim M and Proj( R ) is the set of homogeneous prime ideals of R . New invariants.
Let j ≥ φ R ( M, N )( t ) = H ( M, t − ) H ( N, t ) H ( R, t − ) ; ω jR ( M, N )( t ) = j X i =0 ( − i H (Ext iR ( M, N ) , t ) ; ρ jR ( M, N )( t ) = ω jR ( M, N )( t ) − φ R ( M, N )( t ) . Note that φ R ( − , − ) is biadditive on short exact sequences.Let n be an integer. We are interested in the Laurent coefficient g n ( ρ jR ( M, N )) = j X i =0 ( − i g n (Ext iR ( M, N )) − g n ( φ R ( M, N )) . Assume g n (Ext iR ( M, N )) = 0 for all i ≫ ℓ n = inf { j ∈ Z | g n (Ext iR ( M, N )) = 0 for all i > j } . Further, define(1.9.2) γ nR ( M, N ) = g n ( ρ ℓ n R ( M, N )) , and note that(1.9.3) γ nR ( M, N ) = g n ( ρ jR ( M, N )) for all j ≥ ℓ n . In particular, if ℓ = f R ( M, N ) < ∞ , then dim R Ext iR ( M, N ) = 0 for all i > ℓ . Thus,if n >
0, then g n (Ext iR ( M, N )) = 0 for all i > ℓ , and hence ℓ n ≤ ℓ . Hence(1.9.4) γ nR ( M, N ) = g n ( ρ ℓR ( M, N )) = g n ( ρ jR ( M, N )) for all n > j ≥ ℓ . We see from the above that, if r ≥ ℓ = f R ( M, N ) < ∞ and j ≥ ℓ ,then the following are equivalent:(1) γ nR ( M, N ) = 0 for all n > r ;(2) o ( ρ ℓR ( M, N )) ≥ − r ;(3) o ( ρ jR ( M, N )) ≥ − r . Shifts.
For all integers a, b and all j ≥ φ R ( M ( a ) , N ( b )) ( t ) = t a − b φ R ( M, N )( t ) ; ω jR ( M ( a ) , N ( b ))( t ) = t a − b ω jR ( M, N )( t ) ; ρ jR ( M ( a ) , N ( b ))( t ) = t a − b ρ jR ( M, N )( t ) . In view of (1.7.2), the rational functions ρ jR ( M ( a ) , N ( b ))( t ) and ρ jR ( M, N )( t ) havethe same order for any integers a and b , since o ( t a − b ) = 0 per 1.6. DAVID A. JORGENSEN, LIANA M. S¸EGA, AND PEDER THOMPSON
Vanishing of Ext.
If Ext iR ( M, N ) = 0 for all i > j , then φ R ( M, N ) = ω jR ( M, N ) by [3, Theorem 1], and hence ρ jR ( M, N ) = 0. In particular, if M is freethen ρ jR ( M, N ) = 0 for all j ≥ H (Hom R ( M, N ) , t ) = φ R ( M, N )( t ). Also, if R is Gorenstein and N is free, then ρ jR ( M, N ) = 0 for all j ≥ dim R . Bounds on orders.
We have dim R Ext iR ( M, N ) ≤ dim R/ (ann M + ann N )for all i , since ann N and ann M both annihilate Ext iR ( M, N ). Thus by (1.7.1) and(1.8.1), it follows that(1.12.1) o ( ω jR ( M, N )) ≥ − dim R/ (ann M + ann N ) for all j ≥ . Using (1.7.2) and (1.8.1), we see o ( φ R ( M, N )) = − (dim M + dim N − dim R ) . (1.12.2)Further, for u = dim M + dim N − dim R , one has g u ( φ R ( M, N )) = g dim M ( M ) g dim N ( N ) g dim R ( R ) . (1.12.3)It follows from (1.7.1), (1.12.1), and (1.12.2) that(1.12.4) o ( ρ jR ( M, N )) ≥ − max { dim R/ (ann M +ann N ) , dim M +dim N − dim R } for all j . In particular, as dim R/ (ann M + ann N ) and dim M + dim N − dim R areboth at most min { dim M, dim N } , we obtain(1.12.5) o ( ρ jR ( M, N )) ≥ − min { dim M, dim N } ≥ − dim R for all j. Since we understand when equality holds in (1.7.1), we also obtain(1.12.6) o ( ρ jR ( M, N )) = − (dim M + dim N − dim R )whenever dim M + dim N − dim R > dim R/ (ann M + ann N ).2. Consequences of regularity conditions
We keep the notation of Section 1. For a prime ideal p of R , the codimension of p is codim p = dim R − dim R/ p . If u is an integer, saying that R is regular incodimension u means that R p is regular for all p ∈ Proj( R ) with codim p ≤ u . Inthis section we continue exploring the Laurent series introduced earlier, with ourprimary focus on bounds for the order of the series ρ jR ( M, N ) for certain values of j , under additional regularity hypotheses. Propositions 2.5 and 2.6 are the mainresults in this section; they will be used to prove parts of Theorems 4.2 and 4.3. A formula for γ nR ( M, N ) . Let u be an integer with u < dim R and assume R is regular in codimension u . Recall that the notation γ nR ( M, N ) was introducedin 1.9.2. We show here a formula for computing this invariant using the regularitycondition. We first claim:(2.1.1) g n (Ext iR ( M, N )) = 0 for all n, i with n ≥ dim R − u and i > dim R − n .The proof of this statement can be read off the statement of [3, Lemma 10]. Sincethis proof is rather short, we give it here for the convenience of the reader: We set L = Ext iR ( M, N ) and let n , i be as in (2.1.1). To show g n ( L ) = 0, we need to showdim L < n . Since n ≥ dim R − u , it suffices to assume dim L ≥ dim R − u . We havedim L = dim R/ p for some homogeneous prime ideal p ∈ Supp L and codim p =dim R − dim L ≤ u , so we know that R p is regular. Since 0 = L p = Ext iR p ( M p , N p ), SYMPTOTIC BEHAVIOR OF EXT 9 we see that i ≤ depth R p ≤ codim p = dim R − dim L . Since i > dim R − n , weconclude dim L < n . This finishes the proof of (2.1.1).In view of (2.1.1), if n ≥ dim R − u , then the integer ℓ n defined by (1.9.1) satisfies ℓ n ≤ dim R − n . Using (1.9.2) and (1.9.3) we obtain:(2.1.2) γ nR ( M, N ) = g n ( ρ uR ( M, N )) = g n ( ρ jR ( M, N ))for all n, j with n ≥ dim R − u and j ≥ dim R − n and in particular for all j ≥ u . Terminology interpretation.
In [3], an invariant ε jR ( M, N ) was defined forevery j . Using our notation, this invariant can be defined by the formula ε jR ( M, N ) = g dim R − j ( ω jR ( M, N )) = j X i =0 ( − i g dim R − j (Ext iR ( M, N )) . If R is regular in codimension u < dim R , then, in view of (2.1.2) we have(2.2.1) γ dim R − uR ( M, N ) = g dim R − u ( ρ uR ( M, N ))= ε uR ( M, N ) − g dim R − u ( φ R ( M, N )) . We now interpret the terminology introduced after [3, Proposition 6] using ournotation. Set d = dim R and let u be an integer. In loc. cit. the sequence ofcoefficients of the Laurent series expansion of φ R ( M, N ) around t = 1 and thesequence { ε jR ( M, N ) } j are said to agree up to level u if(2.2.2) φ R ( M, N )( t ) = u X j =0 ε jR ( M, N )(1 − t ) d − j + O (cid:18) − t ) d − u − (cid:19) . Set(2.2.3) ϕ ( t ) = φ R ( M, N ) − u X j =0 ε jR ( M, N )(1 − t ) d − j = φ R ( M, N ) − d X n = d − u ε d − nR ( M, N )(1 − t ) n . The equation (2.2.2) holds if and only if o ( ϕ ) > − ( d − u ). Since o ( φ R ( M, N )) ≥ − d ,this is equivalent to g n ( ϕ ) = 0 for all n with d − u ≤ n ≤ d . Note that g n ( ϕ ) = g n ( φ R ( M, N )) − ε d − nR ( M, N )= g n ( φ R ( M, N )) − g n ( ω d − nR ( M, N ))= g n ( ρ d − nR ( M, N )) . Consequently, the sequence of coefficients of the Laurent expansion of φ R ( M, N )around t = 1 and the sequence { ε jR ( M, N ) } j agree up to level u if and only if g n ( ρ d − nR ( M, N )) = 0 for all n with d − u ≤ n ≤ d . Equivalent conditions for bounds on orders.
Assume now that R is regular incodimension u < dim R . In view of (2.1.2), the following statements are equivalent:(1) The sequence of coefficients of the Laurent series expansion of φ R ( M, N )around t = 1 and the sequence { ε jR ( M, N ) } j agree up to level u ;(2) γ nR ( M, N ) = 0 for all n ≥ dim R − u ;(3) o ( ρ uR ( M, N )) > − (dim R − u );(4) o ( ρ jR ( M, N )) > − (dim R − u ) for all j ≥ u .If ℓ = f R ( M, N ) < ∞ , use (1.9.4) to see that these conditions are also equivalent to(5) o ( ρ ℓR ( M, N )) > − (dim R − u ). Biadditivity. If R is regular in codimension u < dim R , then ε uR ( − , − ) isbiadditive on short exact sequences, see [3, Lemma 11]. Since g dim R − u ( φ R ( M, N ))is also biadditive, we conclude from (2.2.1) that γ dim R − uR ( − , − ) is biadditive onshort exact sequences.In view of the interpretation in 2.3, further translation of [3, Theorem 7] yields: Proposition 2.5.
Let R be a graded ring as in 1.1 and let M and N be finitelygenerated graded R -modules. The following hold: (1) If R has a unique prime p of codimension and R p is a field, then o ( ρ R ( M, N )) > − dim R. (2) If R is an integral domain that is regular in codimension , then o ( ρ R ( M, N )) > − (dim R − . (3) If R is a unique factorization domain that is regular in codimension , then o ( ρ R ( M, N )) > − (dim R − . (cid:3) The next result is inspired by results of [3] recorded above, and improves on thefirst inequality in (1.12.5), under additional hypotheses. Let K be the total ring offractions of R . We say an R -module M has rank r if M ⊗ R K is a free K -moduleof rank r . We say M has a rank if there is an integer r such that M has rank r . Proposition 2.6.
Let R be a graded ring as in 1.1, and assume it is Gorenstein.Let M and N be finitely generated graded R -modules. Set v = min { dim M, dim N } and assume v > , R is regular in codimension dim R − v and, if v = dim R , then M or N has a rank.Then o ( ρ dim R − vR ( M, N )) > − v .Proof. Set d = dim R and u = d − v . Since v >
0, we have u < d .By (1.12.5), we know o ( ρ uR ( M, N )) ≥ − v , and hence g n ( ρ uR ( M, N )) = 0 for all n > v . The inequality o ( ρ uR ( M, N )) > − v is thus equivalent to g v ( ρ uR ( M, N )) = 0.Recall that γ vR ( M, N ) = g v ( ρ uR ( M, N )) by (2.1.2). Since R is regular in codimension u , we know that γ vR ( − , − ) is biadditive on short exact sequences by 2.4. Case 1.
Assume dim N = v and if v = d (that is, u = 0) then M has a rank.If u >
0, then, since R is regular in codimension u , it is regular in codimension1, hence it is a (normal) domain, and thus M has a rank in this case as well.If M is free, then ρ uR ( M, N ) = 0 by 1.11, hence γ vR ( M, N ) = 0. In general, weobserve that it suffices to assume that M is maximal Cohen-Macaulay, by replacing M with a high enough syzygy. Indeed, this follows from the fact that the biaddi-tivity of γ vR ( − , − ) on short exact sequences gives that if M ′ is a syzygy of M , then γ vR ( M, N ) = 0 if and only if γ vR ( M ′ , N ) = 0. The hypothesis that M has a rank ispreserved when replacing M with a syzygy.Thus we may assume that M is maximal Cohen-Macaulay and has rank a . Since R is regular in codimension u , we see that M p is free for all homogeneous primeideals with codim p ≤ u , hence [3, Lemma 10] implies g jR (Ext iR ( M, N )) = 0 forall j ≥ v and all i >
0. We conclude γ vR ( M, N ) = g v ( ρ R ( M, N )), see (1.9.2) and(1.9.3). It suffices thus to show that g v (Hom R ( M, N )) = g v ( φ R ( M, N )).Since M has rank a and M p is free for all p with codim p = u , we conclude M p ∼ = ( R p ) a for all p with codim p = u . SYMPTOTIC BEHAVIOR OF EXT 11
Since dim M = d , we use formula (1.8.2) to obtain:(2.6.1) g d ( M ) = X p ∈ Proj( R ) , codim p = u λ R p ( M p ) g d ( R/ p )= X p ∈ Proj( R ) , codim p = u aλ R p ( R p ) g d ( R/ p )= ag d ( R ) . Using then (1.12.3), we have g v ( φ R ( M, N )) = g d ( M ) g v ( N ) g d ( R ) = ag v ( N ) . As dim N = v , we can choose p ∈ Supp R N with codim p = u . We have M p ∼ = ( R p ) a ,hence Hom R p ( M p , N p ) ∼ = ( N p ) a = 0, so p is in Supp R (Hom R ( M, N )). It followsthat dim R (Hom R ( M, N )) ≥ v . We conclude dim R (Hom R ( M, N )) = v , since thereverse inequality also holds. We use then (1.8.2) again as follows: g v (Hom R ( M, N )) = X p ∈ Proj( R ) , codim p = u λ R p (Hom R p ( M p , N p )) g v ( R/ p )= X p ∈ Proj( R ) , codim p = u aλ R p ( N p )) g v ( R/ p )= ag v ( N ) . We conclude g v (Hom R ( M, N )) = g v ( φ R ( M, N )), and hence γ vR ( M, N ) = 0.
Case 2.
Assume dim M = v and if v = d (so u = 0) then N has a rank.As in Case 1, we see that N has a rank when v < d as well, since R is a domainin this case.If N is free, then we have ρ jR ( M, N ) = 0 for all j > dim R by 1.11, since R isGorenstein. Since γ vR ( M, N ) = g v ( ρ jR ( M, N )) for all j > u by (1.9.3), we see that γ vR ( M, N ) = 0 in this case. In general, we observe that it suffices to assume that N is maximal Cohen-Macaulay, by replacing N with a high enough syzygy. Indeed,this follows from the fact that the biadditivity of γ vR ( − , − ) on short exact sequencesgives that if N ′ is a syzygy of N , then γ vR ( M, N ) = 0 if and only if γ vR ( M, N ′ ) = 0.The hypothesis that N has a rank is preserved when replacing N with a syzygy.Thus we may assume that N is maximal Cohen-Macaulay and has rank b . Sincedim M = v , we can find a sequence x of length u in ann M that is regular on N . Inparticular, it follows that Ext iR ( M, N ) = 0 for all i < u . To show γ vR ( M, N ) = 0, itsuffices thus to show that g v (Ext uR ( M, N )) = g v ( φ R ( M, N )).The module M has a filtration by modules of the form R/ q ( i ) with q ∈ Proj( R )and ht q ≥ u . Since γ vR ( − , − ) is biadditive on short exact sequences, it suffices toprove γ vR ( R/ q ( i ) , N ) = 0 for all i and all q ∈ Proj( R ) with codim q ≥ u . If codim q >u , then dim R/ q ( i ) > v and we have o ( ρ u ( R/ q ( i ) , N )) > − v by 1.12.5, and hence γ vR ( R/ q ( i ) , N ) = 0. We may assume thus codim q = u , so that dim R/ q ( i ) = v . Wemay therefore assume that M = R/ q ( i ).By 1.10, we have o ( ρ uR ( R/ q ( i ) , N )) = o ( ρ uR ( R/ q , N )). It suffices thus to assume M = R/ q with codim q = u . Note that (Ext uR ( R/ q , N )) q = Ext uR q ( k ( q ) , R q ) ∼ = k ( q ) = 0, hence q belongsto Supp(Ext uR ( R/ q , N )). Since we have dim R (Ext uR ( R/ q , N )) ≤ v , we concludedim R (Ext uR ( R/ q , N )) = v .Since N is maximal Cohen-Macaulay and R is regular in codimension u , we knowthat N p is free for all p with ht p = u . Since N has rank b , we have then N p = ( R p ) b for all p with codim p = u . We have: g v (Ext uR ( R/ q , N )) = X p ∈ Proj( R ) , codim p = u λ R p (Ext uR p (( R/ q ) p , N p )) g v ( R/ p )= bλ R q (Ext uR q ( k ( q ) , R q )) g v ( R/ q )= bg v ( R/ q ) . On the other hand, g d ( N ) = bg d ( R ) by an argument similar to that in (2.6.1).Using further (1.12.3), we have g v ( φ R ( R/ q , N )) = g v ( R/ q ) g d ( N ) g d ( R ) = bg vR ( R/ q ) = g v (Ext uR ( R/ q , N ))and this implies γ vR ( R/ q , N ) = 0, completing the proof of Case 2.Finally, note there are no remaining cases: if v = d then dim M = dim N = d ,and so if either M or N has a rank, then one of the above cases apply. (cid:3) Reducing complexity
As indicated by the title, the purpose of this section is to establish results whichwill later allow us to craft induction arguments by reducing the complexity of apair of modules.We maintain the setting in Section 1, and assume that R is a graded completeintersection and that M and N are finitely generated graded R -modules. In par-ticular, R = Q/ ( f , ..., f c ) where Q = k [ x , . . . , x ν ] is a polynomial ring over a field k with variables x i of positive degree and f , . . . , f c is a homogeneous Q -regularsequence. Let e i = deg( f i ) for each i . Graded Eisenbud operators.
Let (
F, ∂ ) be a minimal graded free resolution of M over the ring R :( F, ∂ ) = · · · → F i ∂ i −→ F i − → · · · → F ∂ −→ F and for each i ≥ Ri M = Coker( ∂ i +1 ), the i th syzygy of M . There are exactsequences(3.1.1) 0 → Ω Ri M → F i − → Ω Ri − M → . We consider the Eisenbud operators τ , . . . , τ c , as constructed in Eisenbud’s [13],see also [1, Construction 9.1.5]. In our graded setting, each operator τ n is a chainmap on F of bidegree ( − , − e n ): τ nj : F j +2 → F j ( − e n ) . These are constructed as follows: Let ( e F , e ∂ ) be a lifting of F to Q , so that ( F, ∂ ) =( e F ⊗ Q R, e ∂ ⊗ Q R ). The relation ∂ = 0 yields e ∂ ( e F ) ⊆ ( f , . . . , f c ) e F , hence we canchoose homological degree − e τ n : e F → e F ( − e n ) such that e ∂ = P cn =1 f n e τ n . Set τ n = e τ n ⊗ Q R . SYMPTOTIC BEHAVIOR OF EXT 13
The Eisenbud operators determine maps of graded R -modules χ jn : Ext jR ( M, N )( e n ) → Ext j +2 R ( M, N ) . These maps turn L j,i Ext jR ( M, N ) i into a bigraded module over a bigraded poly-nomial ring R [ χ , . . . , χ c ] with variables χ n of bidegree ( − , − e n ). Furthermore, L j,i Ext jR ( M, N ) i is a finitely generated R [ χ , . . . , χ c ]-module, as proved by Gul-liksen [14], see also [1, Theorem 9.1.4].The following lemma is a standard argument on the existence of superficial el-ements. We spell out a proof in order to properly address the fact that, in ourcontext, such elements can be chosen to be bihomogeneous. Lemma 3.2.
Assume that k is infinite. There exists a bihomogeneous element χ ∈ R [ χ , . . . , χ c ] of bidegree ( − , − e ) and an integer s ≥ such that χ is a non-zerodivisor on Ext > sR ( M, N ) .Proof. Set X = L j,i Ext jR ( M, N ) i and R = R [ χ , . . . , χ c ]. By the previous com-ments, X is a bigraded finitely generated R -module. We use here a primary decom-position argument. Since R is bigraded by the monoid ( − N ) × ( − N ), all ideals in thisproof are bihomogeneous. We refer to Northcott [20, Section 2.13] for a treatmentof primary decomposition in a bigraded setting. Choose a primary decomposition0 = q ′ ∩ · · · ∩ q ′ m ∩ q ∩ · · · ∩ q n of the zero submodule of X , where the associated primes p ′ i of q ′ i do not contain R − , ∗ and the associated primes p i of q i contain R − , ∗ . Set S i = p ′ i ∩ R − , ∗ .Since each S i is properly contained in R − , ∗ , Nakayama’s Lemma shows that ( S i + m R − , ∗ ) / m R − , ∗ is a proper subspace of R − , ∗ / m R − , ∗ . Since k is infinite, wehave ∪ ni =1 ( S i + m R − , ∗ ) / m R , ∗ = R − , ∗ / m R − , ∗ . There exists thus an integer e ≥ χ of bidegree ( − , − e ) such that χ / ∈ S i for all i , hence χ / ∈ p ′ i for all i . Since q ′ i is primary to p ′ i for each i , it follows that (0 : X χ ) ⊆ q ′ i for each i .Since each p i is finitely generated and p i = p ann( X/ q i ), there exists a i suchthat p a i i ⊆ ann( X/ q i ). As R − , ∗ ⊆ p i , we have ( R − , ∗ ) a i ⊆ ann( X/ q i ) for each i .Choose a large enough such that( R − , ∗ ) a ⊆ ann( X/ q ) ∩ · · · ∩ ann( X/ q n )and hence ( R − , ∗ ) a X ⊆ q ∩ · · · ∩ q n . Since X is finitely generated and the generators χ i of R + have bidegree ( − , − e i ),for j ≫ X j, ∗ ⊆ ( R − , ∗ ) a X for j sufficiently large. Thus(0 : X χ ) ∩ X j, ∗ ⊆ (0 : X χ ) ∩ ( R , ∗ ) a X ⊆ q ′ ∩ · · · ∩ q ′ m ∩ q ∩ · · · ∩ q n = 0for j ≫ (cid:3) Convention for the case when cx R ( M, N ) = 1 . Assume f R ( M, N ) < ∞ .If cx R ( M, N ) = 1, then there exists an integer t such that λ (Ext iR ( M, N )) = λ (Ext i +2 R ( M, N )) for all i ≥ t . Assume also that k is infinite. In this case, wemake the convention that the integer s in Lemma 3.2 is chosen so that s ≥ t .This ensures that the element χ coming from the lemma induces an isomorphismExt iR ( M, N )( e ) → Ext i +2 R ( M, N ) for all i ≥ s . Construction.
Assume f R ( M, N ) < ∞ and k is infinite. Let χ and s beas in Lemma 3.2. Let n > s . Since χ has homological degree −
2, it is a linearcombination of the elements χ i , and it comes thus from a chain map τ on F with τ j : F j +2 → F j ( − e )for all j ≥
0. As in the proof of [5, Proposition 2.2(i)], with the added observationthat all maps involved are maps of graded R -modules, this chain map induces amap Ω Rn +2 M → Ω Rn M ( − e ) and a commutative pushout diagram with exact rows(3.4.1) 0 / / Ω Rn +2 M / / (cid:15) (cid:15) F n +1 / / (cid:15) (cid:15) Ω Rn +1 M / / / / Ω Rn M ( − e ) / / K / / Ω Rn +1 M / / χ :Ext i − R (Ω Rn M ( − e ) , N ) χ / / Ext iR (Ω Rn +1 M , N )Ext i + n − R ( M, N )( e ) Ext i + n +1 R ( M, N )for all i >
1. Since n ≥ s + 1, these maps are injective for all i > n > s there exists a graded R -module K (depending on n )such that we have exact sequences(3.4.2) 0 / / Hom R (Ω Rn +1 M , N ) / / Hom R ( K, N ) / / Hom R (Ω Rn M , N )( e ) EDBCGF@A . . ❫❫❫❫ Ext n +2 R ( M, N ) / / Ext R ( K, N ) / / i > → Ext i + n − R ( M, N )( e ) → Ext i + n +1 R ( M, N ) → Ext iR ( K, N ) → . When cx R ( M, N ) = 1, with the convention in 3.3 in place, we get(3.4.4) Ext iR ( K, N ) = 0 and Ext i + n − R ( M, N ) ∼ = Ext i + n +1 R ( M, N )for all i > Reducing complexity.
Assume ℓ = f R ( M, N ) < ∞ and that k is infinite.Choose s as in Lemma 3.2, let n > max { s, ℓ } , and let K be as constructed in 3.4.A length count in the exact sequence (3.4.3) gives f R ( K, N ) ≤ β Ri ( K, N ) = β Ri + n +1 ( M, N ) − β Ri + n − ( M, N )for all i >
1, yielding equalitiescx R ( K, N ) = cx R ( M, N ) − h R ( K, N ) = 2( − n +1 (cx R ( M, N ) − h R ( M, N ) . (3.5.3) SYMPTOTIC BEHAVIOR OF EXT 15
We note that (3.5.2) follows directly from the definition of complexity. We givebelow an explanation for (3.5.3). Indeed, set r = cx R ( M, N ). As described inSection 1, there exists an integer i such that β R i ( M, N ) = a r − (2 i ) r − + · · · + a (2 i ) + a β R i +1 ( M, N ) = b r − (2 i + 1) r − + · · · + b (2 i + 1) + b for all i ≥ i , and h R ( M, N ) = a r − − b r − . If n is even, then using (3.5.1), wehave for all j > i : β R j ( K, N ) = b r − (cid:0) (2 j + n + 1) r − − (2 j + n − r − (cid:1) + · · · = 2( r − b r − (2 j ) r − + · · · , and β R j +1 ( K, N ) = a r − (cid:0) (2 j + n + 2) r − − (2 j + n ) r − (cid:1) + · · · = 2( r − a r − (2 j + 1) r − + · · · . These equalities show h R ( K, N ) = 2( r − b r − − a r − ) = − r − h R ( M, N ) . If n is odd, we similarly obtain h R ( K, N ) = 2( r − a r − − b r − ) = 2( r − h R ( M, N ) . We now set the stage for the proof of Theorem 2 in the introduction.
Proposition 3.6.
Adopt the notation and conventions in . Let n > s and let K be the R -module (depending on n ) constructed in (3.4.1) . The following holds: ρ R ( K, N )( t ) =( − n ( t − e − ρ nR ( M, N )( t ) + H (Ext n +1 R ( M, N ) , t ) − H (Ext n +2 R ( M, N ) , t ) . Proof.
Identifying Ext R (Ω Ri − M , N ) with Ext iR ( M, N ), the short exact sequence(3.1.1) gives an exact sequence0 / / Hom R (Ω Ri − M , N ) / / Hom R ( F i − , N ) / / Hom R (Ω Ri M , N ) / / Ext iR ( M, N ) / / . Recall we have an equality H (Hom R ( F i − , N ) , t ) = φ R ( F i − , N )( t ), see 1.11. Inview of this equality, a Hilbert series count on this exact sequence gives(3.6.1) H (Hom R (Ω Ri M , N ) , t ) = − H (Hom R (Ω Ri − M , N ) , t ) + H (Ext iR ( M, N ) , t ) + φ R ( F i − , N )( t ) . As φ R ( − , − ) is biadditive on short exact sequences, (3.1.1) gives(3.6.2) φ R (Ω Ri M , N )( t ) + φ R (Ω Ri − M , N )( t ) = φ R ( F i − , N )( t ) . Combining (3.6.2) and (3.6.1), we have:(3.6.3) H (Hom R (Ω Ri M , N ) , t ) − φ R (Ω Ri M , N )( t ) = − (cid:0) H (Hom R (Ω Ri − M , N ) , t ) − φ R (Ω Ri − M , N )( t ) (cid:1) + H (Ext iR ( M, N ) , t ) . Applying this repeatedly, starting with i = n and ending with i = 1, we obtain:(3.6.4) H (Hom R (Ω Rn M , N ) , t ) − φ R (Ω Rn M , N )( t ) = ( − n ρ nR ( M, N )( t ) . The exact sequence (3.4.2) gives(3.6.5) H (Hom R ( K, N ) , t ) − H (Ext R ( K, N ) , t ) = H (Hom R (Ω Rn +1 M , N ) , t ) + t − e H (Hom R (Ω Rn M , N ) , t ) − H (Ext n +2 R ( M, N ) , t ) . In addition, the bottom short exact sequence in the diagram (3.4.1) gives(3.6.6) φ R ( K, N )( t ) = φ R (Ω Rn +1 M , N )( t ) + t − e φ R (Ω Rn M , N )( t ) . Combining (3.6.5) and (3.6.6), we have H (Hom R ( K, N ) , t ) − H (Ext R ( K, N ) , t ) − φ R ( K, N )( t ) = H (Hom R (Ω Rn +1 M , N ) , t ) − φ R (Ω Rn +1 M , N )( t )+ t − e (cid:0) H (Hom R (Ω Rn M , N ) , t ) − φ R (Ω Rn M , N )( t ) (cid:1) − H (Ext n +2 R ( M, N ) , t ) . The left-hand side of this equation is ρ R ( K, N ). Using also (3.6.3), this yields ρ R ( K, N ) = ( t − e − (cid:0) H (Hom R (Ω Rn M , N ) , t ) − φ R (Ω Rn M , N )( t ) (cid:1) + H (Ext n +1 R ( M, N ) , t ) − H (Ext n +2 R ( M, N ) , t ) . Finally, using (3.6.4) to rewrite the right-hand side of this expression produces thesought after equality. (cid:3)
We end the section by pointing out that the assumption that k is infinite is notessential to the arguments elsewhere in this paper. Remark 3.7. If k is not infinite, we may consider a faithfully flat ring extension R → R ′ , where R ′ = k ′ [ x , . . . , x ν ] / ( f , ..., f c ) for an infinite field extension k ′ of k .Let M ′ = R ′ ⊗ R M and N ′ = R ′ ⊗ R N . Length is invariant under faithfully flatring extensions, so that f R ( M, N ) = f R ′ ( M ′ , N ′ ) and β Ri ( M, N ) = β R ′ i ( M ′ , N ′ ) forall i ≥ f R ( M, N ). Thus cx R ( M, N ) = cx R ′ ( M ′ , N ′ ) and h R ( M, N ) = h R ′ ( M ′ , N ′ ).Furthermore, the Hilbert series of a given R ′ -module is the same as its Hilbertseries when considered as an R -module, so in particular, ρ jR ′ ( M ′ , N ′ ) = ρ jR ( M, N )for any integer j ≥
0. 4.
Vanishing of h R ( M, N )In this section we prove the theorems announced in the introduction.
Theorem 4.1.
Let R be a graded complete intersection and let M and N be finitelygenerated graded R -modules with f R ( M, N ) < ∞ and cx R ( M, N ) ≥ . Set r =cx R ( M, N ) and ℓ = f R ( M, N ) .The following hold: (1) o ( ρ ℓR ( M, N )) ≥ − r , and (2) h R ( M, N ) = 0 if and only if o ( ρ ℓR ( M, N )) > − r .Proof. In view of Remark 3.7, we may assume that the underlying field is infinite,so that the results of Section 3 apply. We use the notation in 3.4 and we choose n so that n > max { s, ℓ } .The equivalence (2) ⇔ (3) in 1.9 shows that the inequality o ( ρ ℓR ( M, N )) ≥ − r is equivalent to o ( ρ nR ( M, N )) ≥ − r and also, since r ≥
1, that the inequality o ( ρ ℓR ( M, N )) > − r is equivalent to o ( ρ nR ( M, N )) > − r . SYMPTOTIC BEHAVIOR OF EXT 17
The proof relies on using the formula in Proposition 3.6. Set A ( t ) = H (Ext n +1 R ( M, N ) , t ) − H (Ext n +2 R ( M, N ) , t ) . With this notation, the formula becomes:(4.1.1) ρ R ( K, N )( t ) = ( − n ( t − e − ρ nR ( M, N )( t ) + A ( t ) . Note that o ( t − e −
1) = 1. Since n > ℓ , we have A ( t ) ∈ Z [ t, t − ], so o ( A ) ≥
0, and o ( A ) ≥ A (1) = 0, cf. 1.6.We prove both statements by induction on r .First assume r = 1. We have then ρ R ( K, N ) = 0 by 1.11 and (3.4.4), henceformula (4.1.1) becomes( − n ( t − e − ρ nR ( M, N )( t ) = − A ( t ) . Since o ( t − e −
1) = 1 and o ( A ) ≥
0, we see from this that o ( ρ nR ( M, N )) ≥ −
1. Thisproves the base case of (1). For (2), note that o ( ρ nR ( M, N )) ≥ o ( A ) ≥
1, and that o ( A ) ≥ A (1) = 0. Thus (2) holds for r = 1 ifand only if β Rn +1 ( M, N ) = β Rn +2 ( M, N ). In view of the isomorphisms in (3.4.4), thisequality is equivalent to β Rj ( M, N ) = β Rj +1 ( M, N ) for all j ≫
0, or equivalently, h R ( M, N ) = 0. Thus the base case of (2) also holds.Next assume r > r −
1. By 3.5, we know cx R ( K, N ) = r − ℓ ′ = f R ( K, N ) ≤
1. Theinduction hypothesis gives that o ( ρ ℓ ′ R ( K, N )) ≥ − r + 1, and o ( ρ ℓ ′ R ( K, N )) > − r + 1 ifand only if h R ( K, N ) = 0. As before, we can replace o ( ρ ℓ ′ R ( K, N )) by o ( ρ R ( K, N ))in these statements.Since o ( A ) is non-negative, o ( t − e −
1) = 1, and o ( ρ R ( K, N )) ≥ − r + 1, it nowfollows from (4.1.1) that o ( ρ nR ( M, N )) ≥ − r , completing (1). It also follows that o ( ρ R ( K, N )) > − r + 1 if and only if o ( ρ nR ( M, N )) > − r . Since h R ( M, N ) = 0 ifand only if h R ( K, N ) = 0 by (3.5.3), this completes the proof of (2). (cid:3)
We now prove our main results on the vanishing of h R ( M, N ) for pairs of moduleswith large complexity relative to dimension; the first gives bounds that depend ondim R , while the bounds in Theorem 4.3 depend on dim M and dim N as well. Theorem 4.2.
Let R be a graded complete intersection and let M and N be finitelygenerated graded R -modules such that f R ( M, N ) < ∞ and cx R ( M, N ) ≥ .The equality h R ( M, N ) = 0 holds under any of the following conditions: (1) cx R ( M, N ) > dim R . (2) cx R ( M, N ) = dim R and R has a unique prime ideal p of codimension and R p is a field. (3) cx R ( M, N ) = dim R − and the ring R is regular in codimension . (4) cx R ( M, N ) = dim R − and R is a unique factorization domain which isregular in codimension .Proof. Set r = cx R ( M, N ) and ℓ = f R ( M, N ). By Theorem 4.1, it is sufficient toshow o ( ρ ℓR ( M, N )) > − r .The inequality o ( ρ ℓR ( M, N )) > − r follows from (1.12.5) under the hypothesison complexity in (1). The remaining parts (2), (3), and (4) follow from [3], asinterpreted in Proposition 2.5 in view of 2.3. (cid:3) Finally, a proof of Theorem 1 from the introduction is contained in following:
Theorem 4.3.
Let R be a graded complete intersection and let M and N be finitelygenerated graded R -modules such that f R ( M, N ) < ∞ and cx R ( M, N ) ≥ .The equality h R ( M, N ) = 0 holds under any of the following conditions: (1) cx R ( M, N ) > max { dim R/ (ann M + ann N ) , dim M + dim N − dim R } . (2) cx R ( M, N ) > min { dim M, dim N } . (3) cx R ( M, N ) = min { dim M, dim N } , the ring R is regular in codimension dim R − cx R ( M, N ) , and, if dim R = cx R ( M, N ) , then M or N has a rank.Proof. Set r = cx R ( M, N ) and ℓ = f R ( M, N ). By Theorem 4.1, it is sufficient toshow o ( ρ ℓR ( M, N )) > − r . Under the hypotheses in (1) and (2), this follows from(1.12.4) and (1.12.5), and under those in (3), it follows from Proposition 2.6. (cid:3) The following corollary gives a cohomological extension, in the graded setting,of Hochster’s result for hypersurfaces mentioned in the introduction.
Corollary 4.4.
Let R be a graded complete intersection and let M and N befinitely generated graded R -modules with cx R ( M, N ) > dim R/ (ann M + ann N ) and f R ( M, N ) < ∞ . The following are then equivalent: (1) h R ( M, N ) = 0 ; (2) dim M + dim N − dim R < cx R ( M, N ) .Proof. (2) = ⇒ (1) follows directly from Theorem 4.3 (1).Assume now h R ( M, N ) = 0 and dim M + dim N − dim R ≥ cx R ( M, N ). Thehypothesis implies dim M + dim N − dim R > dim R/ (ann M + ann N ). In view of(1.12.6), we have o ( ρ jR ( M, N )) = − (dim M + dim N − dim R ) for all j ≥ − (dim M + dim N − dim R ) > − cx R ( M, N )which is a contradiction with our assumption. Hence (2) must hold. (cid:3)
Remark 4.5.
In the case cx R ( M, N ) = 0, then h R ( M, N ) = 0 automatically holds,and moreover, dim M +dim N − dim R ≤ dim R/ (ann M +ann N ) also holds. Indeed,if dim M + dim N − dim R > dim R/ (ann M + ann N ), then in view of (1.12.6), wehave o ( ρ jR ( M, N )) = − (dim M + dim N − dim R ) for all j ≥
0, contradicting 1.11.5.
Pairs of modules with arbitrary complexity
Given a local complete intersection R of codimension c and with residue field k , itis known that for any integer r between 1 and c there exists an R -module M ofcomplexity r , see Avramov, Gasharov, and Peeva [4, (5.7)]. Consequently, one hascx R ( M, k ) = cx R ( M ) = r . One can in fact construct pairs of modules of any givencomplexity without having to assume that one of the modules is the residue field.We present below a more general construction that makes this point, and also givean example that shows the additional hypotheses in Theorem 4.3(3) are necessary. Proposition 5.1.
Let Q be a ring, let f , ..., f c be a Q -regular sequence, and set R = Q/ ( f , ..., f c ) . Assume that one of the following holds: (1) Q is a regular local ring with maximal ideal n and residue field k , and f , ..., f c ∈ n , or (2) Q is a polynomial ring over a field k with homogeneous maximal ideal n , and f , ..., f c is a homogeneous sequence in n . SYMPTOTIC BEHAVIOR OF EXT 19
Set R = Q/ ( f , . . . , f i ) and R = Q/ ( f j , . . . , f c ) with ≤ i, j ≤ c and consider thesyzygy modules f M = Ω R dim R k and e N = Ω R dim R k .The R -modules M = f M / ( f i +1 , . . . , f c ) f M and N = e N / ( f , . . . , f j − ) e N satisfy f R ( M, N ) < ∞ . If j ≤ i , then cx R ( M, N ) = i − j + 1 , otherwise cx R ( M, N ) = 0 .Proof.
Assume first that ( Q, n , k ) is a regular local ring and f , ..., f c ∈ n is a Q -regular sequence. Since f M is a syzygy of the residue field of R , it has maximalcomplexity over R , namely i . Moreover, as f M is a dim R -th syzygy over R , astandard depth lemma shows that f M is maximal Cohen-Macaulay as an R -module,and so f i +1 , . . . , f c is regular on f M . Similarly, e N is an R -module of complexity c − j + 1 such that f , . . . , f j − is regular on e N .Since f i +1 , . . . , f c is regular on f M and annihilates N , we have the standardisomorphisms Ext nR ( M, N ) ∼ = Ext nR ( f M , N ) for all n ≥
0, see for example [16,Lemma 2, p. 140]. As f M is a syzygy of the residue field of R , for any non-maximal prime p of R we have that f M p is a free ( R ) p -module. Thus for anynon-maximal prime p of R we obtain Ext nR ( M, N ) p = Ext n ( R ) p ( f M p , N p ) = 0 for all n >
0, and so f R ( M, N ) = 0.The remainder of the proof uses results and notation found in [6]. In particu-lar, V R ( M, N ) is the support variety of the pair (
M, N ) over R , and V R ( M ) =V R ( M, k ) is that of M . Recall from [2] that the dimension of the support vari-ety V R ( M, N ) is the complexity (as defined in [2]) of the pair (
M, N ). Set I =( f , . . . , f c ), J = ( f , . . . , f i ), and J = ( f j , . . . , f c ), and let ϕ J : J / n J → I/ n I and ϕ J : J / n J → I/ n I denote the natural maps.By assumption we have V R ( f M ) = J / n J and V R ( e N ) = J / n J . By [6,Corollary 5.2] we therefore have V R ( M ) = ϕ J ( J / n J ) and V R ( N ) = ϕ J ( J / n J ).Putting this together with [2, Theorem 5.6(8)] we find thatV R ( M, N ) = V R ( M ) ∩ V R ( N )= ϕ J ( J / n J ) ∩ ϕ J ( J / n J )is a linear subspace of dimension i − j + 1 if j ≤ i , and zero otherwise. This is thecomplexity of the pair ( M, N ) as defined in [2]. As these two notions of complexityagree [9, Proposition 2.2], we obtain the desired result in the local case.Finally, assume instead that Q is a polynomial ring over a field k , with homoge-neous maximal ideal n , and that f , . . . , f c is a homogeneous sequence in n . Thelocal ring R n contains its residue field k . The finite length modules Ext nR n ( M n , N n )are finite dimensional k -vector spaces; the same is true of the modules Ext nR ( M, N ),and dim k Ext nR ( M, N ) = dim k Ext nR n ( M n , N n ). Now the local result applies to showthe proposition is true in the graded case. (cid:3) The next example gives modules with cx R ( M, N ) = min { dim M, dim N } and h R ( M, N ) = 0. This shows that the additional conditions in part (3) of Theorem4.3 are necessary. Example 5.2.
Consider the ring R = k [ x , . . . , x d ] / ( x x d +1 , . . . , x d x d ), for afield k , and the R -modules M = R/ ( x , . . . , x r ), N = R/ ( x r +1 , . . . , x d ), where1 ≤ r ≤ d . We have then:(1) dim M = d = dim R and dim N = r ;(2) cx R ( M, N ) = r ; (3) h R ( M, N ) = 0;(4) R is regular in codimension 0, but it does not have a unique minimal prime,and hence it is not regular in codimension 1. Proof.
That dim R = d follows since dim k [ x , . . . , x d ] = 2 d , and x x d +1 , . . . , x d x d is a regular sequence of length d . Now one has that the module M ∼ = k [ x r +1 , . . . , x d ] / ( x r +1 x r +1+ d , . . . , x d x d )has dimension d since k [ x r +1 , . . . , x d ] has dimension 2 d − r , and the sequence x r +1 x r +1+ d , . . . , x d x d is a regular sequence of length d − r . Also, the dimension of N ∼ = k [ x , . . . , x r ] is evidently r .For 0 ≤ j ≤ r , define N j = N/ ( x j +1 , . . . , x r ) N . For 1 ≤ j ≤ r , there are exactsequences 0 / / N j x j / / N j / / N j − / / . Since x j annihilates Ext iR ( M, N j ) for all i , the resulting long exact sequence incohomology breaks into short exact sequences0 / / Ext iR ( M, N j ) / / Ext iR ( M, N j − ) / / Ext i +1 R ( M, N j ) / / . Set E j ( t ) = P ∞ i =0 dim k Ext iR ( M, N j ) t i . One has E j − ( t ) = E j ( t ) + t E j ( t ). Since N ∼ = k , we have E ( t ) = P RM ( t ), and so we need to compute this Poincar´e series.This can be done along the lines of [1, 9.3.7], but we include a different approachhere.The module M = k [ x , x d +1 ] / ( x ) has a very simple free resolution F over thering R = k [ x , x d +1 ] / ( x x d +1 ); it is a periodic resolution comprised of rank one freemodules where the maps alternate between multiplication by x and multiplicationby x d +1 . The same is true of the free resolutions F n of M n = k [ x n , x d + n ] / ( x n ) over R n = k [ x n , x d + n ] / ( x n x d + n ), for 1 < n ≤ r . Now we observe that F ⊗ k · · · ⊗ k F r ⊗ k k [ x r +1 ,...,x d ,x d + r +1 ,...,x d ]( x r +1 x d + r +1 ,...,x d x d ) is a free resolution of M = M ⊗ k · · · ⊗ k M r ⊗ k k [ x r +1 ,...,x d ,x d + r +1 ,...,x d ]( x r +1 x d + r +1 ,...,x d x d ) over R = R ⊗ k · · · ⊗ k R r ⊗ k k [ x r +1 ,...,x d ,x d + r +1 ,...,x d ]( x r +1 x d + r +1 ,...,x d x d ) . It follows that P RM ( t ) = Q rn =1 P R n M n ( t ) = 1 / (1 − t ) r . Therefore E r ( t ) = t r E ( t )(1 + t ) r = t r (1 − t ) r This shows that cx R ( M, N ) = r .From the expression for E r ( t ) one checks directly that Ext iR ( M, N ) is zero forall large even i and non-zero for all large odd i if r is odd, and vice versa if r iseven. This shows that h R ( M, N ) = 0.Property (4) is an easy exercise, having noted that the minimal primes of R areof the form ( x i , . . . , x i d ) where i j ∈ { j, d + j } . (cid:3) SYMPTOTIC BEHAVIOR OF EXT 21
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Department of Mathematics, University of Texas at Arlington, 411S. Nedderman Drive, Pickard Hall 429, Arlington, TX 76019, USA
Email address : [email protected] URL : (L. M. S¸ega) Department of Mathematics and Statistics, University of Missouri -Kansas City, 206 Haag Hall, 5100 Rockhill Road, Kansas City, MO 64110-2499, USA
Email address : [email protected] URL : http://s.web.umkc.edu/segal/ (P. Thompson) Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway
Email address : [email protected] URL ::