Lengths of Tors Determined by Killing Powers of Ideals in a Local Ring

Journal of Algebra 247, 104᎐133 Ž2002. doi:10.1006rjabr.2001.9020, available online at http:rrwww.idealibrary.com on

Lengths of Tors Determined by Killing Powers of Ideals in a Local Ring J. Bruce Fields Uni¨ ersity of Michigan, Ann Arbor, Michigan E-mail: [email protected] Communicated by Craig Huneke Received February 12, 2001

Given a local ring R and n ideals whose sum is primary to the maximal ideal of R, one may define a function which takes an n-tuple of exponents Ž a1 , . . . , a n . to the length of Tork Ž RrI1 , . . . , RrIn .. These functions are shown to have rational generating functions in certain cases. 䊚 2002 Elsevier Science

In this paper we investigate functions f k s length Ž TorkR Ž M1rI1a1 M1 , . . . , MnrIna n Mn . . , where R is a Žnoetherian . local ring, Mi is a finitely generated R-module and Ii an ideal of R, for 1 F i F n, and I1 q ⭈⭈⭈ qIn q Ann M1 q ⭈⭈⭈ qAnn Mn is primary to the maximal ideal of R. Special cases of these functions are interesting in their own right; see, for example, the discussion below of Hilbert᎐Kunz functions. It is also hoped that a better understanding of these functions may lead to insights into the theory of intersection multiplicities; for example, it seems reasonable to attempt to calculate the intersection multiplicity of varieties defined by ideals I1 and I2 using only properties of the function f 0 Ž a1 , a2 . s length Ž Rr Ž I1a1 q I2a 2 . . , although this has not yet proved possible. We are, however, able to establish certain properties of the functions f k in some generality. In 104 0021-8693r02 $35.00 䊚 2002 Elsevier Science All rights reserved.

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LENGTHS OF TORS

particular, they are shown to have rational generating functions in the following cases: Ž1. Ž2.

n s 2, M1 s M2 s R, k G 2; or n s 2, M1 s M2 s R, k is arbitrary, R is regular, and

[

a1 , a 2G0

Ž I1a l I2a . t1a t 2a ; R w t1 , t 2 x 1

2

1

2

is a finitely generated R-algebra; or Ž3. n is arbitrary, M1 s ⭈⭈⭈ s Mn s R, k s 0, R s K ww x 1 , . . . , x d xx Ž K any field., and I1 , . . . , In are generated by monomials. 1. BASIC DEFINITIONS All rings are commutative, and local rings are in addition assumed to be Noetherian. DEFINITION 1.1. Given n modules M1 , . . . , Mn over a ring R, define the R-module TorkR Ž M1 , . . . , Mn . by Ž1. choosing a projective resolution ⭈⭈⭈ ª Pi , 3 ª Pi , 2 ª Pi , 1 ª Pi , 0 ª Mi ª 0 for each Mi , Ž2. tensoring together all of these projective resolutions Žwith the modules Mi removed. and taking the total complex, ⭈⭈⭈ ª

[

Ž i1 , . . . , i n . Ý j i js2

P1, i1 m ⭈⭈⭈ m Pn , i n ª

[

Ž i1 , . . . , i n . Ý j i js1

P1, i1 m ⭈⭈⭈ m Pn , i n

ª P1, 0 m ⭈⭈⭈ m Pn , 0 ª 0, and Ž3. taking the kth homology of this complex. This defines a functor, covariant in each of the n variables, which specializes for n s 2 to the usual Tor. It shares most of the basic properties with the usual 2-variable Tor; for example. Ž1. A short exact sequence in any of the variables gives the usual long exact sequence. Ž2. Any element of R that kills one of the Mi also kills Tork Ž M1 , . . . , Mn ..

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Ž3. Tor0R Ž M1 , . . . , Mn . s M1 mR ⭈⭈⭈ mR Mn . Ž4. If all but one of the Mi is flat, then Tork Ž M1 , . . . , Mn . s 0 for all k ) 0. Ž5. If the Mi are all finitely generated over R, so is Tori Ž M1 , . . . , Mn .. DEFINITION 1.2. Let R be a Noetherian ring, let n be a positive integer, let M1 , . . . , Mn be finitely generated R-modules, and let I1 , . . . , In be ideals of R. Let J s I1 q ⭈⭈⭈ qIn q Ann R Ž M1 . q ⭈⭈⭈ qAnn R Ž Mn ., and assume that RrJ has finite length. For k a non-negative integer, define f k Ž a1 , . . . , a n . s length Ž Tork Ž M1rI1a1 M1 , . . . , MnrIn b a n Mn . . . Note that the modules Tork Ž M1rI1a1 M1 , . . . , MnrIna n Mn . do have finite length, since they are finitely generated modules that are killed by the ideal J⬘ s I1a1 q ⭈⭈⭈ qIna n q Ann M1 q ⭈⭈⭈ qAnn Mn , and hence are finitely generated modules over RrJ⬘, which has finite length. ŽThis is because J⬘ has the same radical as the ideal J above, and RrJ is assumed to be finite length.. In this paper, the ring R will always be local, and the modules M1 , . . . , Mn will almost always be equal to R. We conclude this section with some examples. Ž1. If n s 1, then f 0 is the usual Hilbert function and is eventually a polynomial of degree d s dim R. Ž2. In w2, 3x, Brown studies f 0 in the case n s 2 and M s R; he calls this function the Hilbert function of I1 and I2 . In w2x, he gives sufficient conditions for f 0 to be eventually polynomial Žin other words, for there to exist Ž b1 , b 2 . such that f 0 Ž a1 , a2 . agrees with a polynomial for all Ž a1 , a2 . G Ž b1 , b 2 ... In w3x, he considers the case where f 0 is eventually a polynomial in a1 , a2 , and minŽ a1 , a2 .. Both papers also provide several examples of such functions, including examples which show that f 0 is not necessarily eventually a polynomial in a1 , a2 , and minŽ a1 , a2 .. He also provides counterexamples to a more general conjecture that f 0 is eventually polynomial on regions bordered by parallel lines. Ž3. If M1 s ⭈⭈⭈ s MN s M, I1 , . . . , In are all principal ideals, and R has characteristic p, for p ) 0 a prime, then the function HK M , I : a ¬ f 0 Ž p a, . . . , p a . is called the Hilbert᎐Kunz function on M of the ideal I s I1 q ⭈⭈⭈ qIn . The more standard definition is in terms of bracket powers, Iw p x s Ž i p
e

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LENGTHS OF TORS

With this notation, HK M , I Ž a. s lengthŽ MrI w p x M .. Because R has chare e e acteristic p, I w p x s I1p q ⭈⭈⭈ qInp , which shows that HK M , I is dependent only on I and not on the choice of I1 , . . . , In . Monsky has shown w15x that this function has the form e

e HK Ž M, I . p d a q O Ž p aŽ dy1. . , where d s dim M and e HK Ž M, I . is a positive real constant. The constant e HK Ž R, m. may be written e HK Ž R ., and HK R, m may be written HK R and called just the Hilbert᎐Kunz function of R. It is not even known in general whether e HK Ž R . is rational. However, many results have been proved in particular cases; see w4, 6, 10, 15᎐18, 22x. The Hilbert᎐Kunz function also has an important connection with the theory of tight closure. If I ; J are m-primary ideals of R, then I* s J * if and only if e HK Ž R, I . s e HK Ž R, J .. This statement is true when R is a complete local domain, and in somewhat more generality. ŽSee w13, Theorem 8-17x, or see the more thorough discussion in w12x.. Ž4. In w7x, Contessa considers the generalization of the previous situation to the case of arbitrary k. She is able to use results about the higher Tors to determine the form of Hilbert᎐Kunz functions of modules over regular rings of dimension at most two.

2. QUASIPOLYNOMIAL FUNCTIONS Given an ⺞ n-graded module M having finite length in each graded piece, there is a natural notion of a Hilbert function ⺞ n ª ⺞ given by

Ž a1 , . . . , an . ¬ length Ž MŽ a1 , . . . , a n . . . For n s 1 this is the usual Hilbert function. We will see that certain ⺞ n-graded algebraic objects Žincluding, for example, finitely generated modules over finitely generated algebras over fields. have Hilbert functions which have rational generating functions. In fact, these Hilbert functions are quasipolynomial, a stronger condition. This fact will be used to prove that in some cases the functions f k are also quasipolynomial. Most of the results of this section are known, though not perhaps in the form we need them; see, for example, w14x. Most of the functions considered here are defined on ⺞ n. However, it will be convenient to identify such functions with functions defined on all of ⺪ n that are zero for elements of ⺪ n not in ⺞ n. DEFINITION 2.1. Given linearly independent vectors ␣ 1 , . . . , ␣ e g ⺞ n, say that a function f : ⺞ n ª ⺪ is periodic with respect to ␣ 1 , . . . , ␣ e if the

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function ␣ ¬ f Ž ␣ q ␥ . y f Ž ␣ . is identically zero on ⺞ n, for any ␥ g ⺞ ␣ 1 q ⭈⭈⭈ q⺞ ␣ e . DEFINITION 2.2. Given ␤ g ⺞ n and linearly independent vectors ␣ 1 , . . . , ␣ e g ⺞ n, the cone with ¨ ertex ␤ generated by ␣ 1 , . . . , ␣ e is C␤ , ␣ 1 , . . . , ␣ e s ␤ q ⺢ G 0 ␣ 1 q ⭈⭈⭈ q⺢ G 0 ␣ e ; ⺢ nG 0 . Given x s Ž x 1 , . . . , x n . and a s Ž a1 , . . . , a n ., x a will be used in the following as a shorthand for x 1a1 x 2a 2 ⭈⭈⭈ x na n . DEFINITION 2.3. Let ␣ 1 , . . . , ␣ e g ⺞ n be linearly independent vectors in ⺢ n, and let ␤ be another element of ⺞ n. Let C s C␤ , ␣ 1 , . . . , ␣ e be the cone with vertex ␤ generated by ␣ 1 , . . . , ␣ e . Call f : ⺞ n ª ⺪ simple quasipolynomial of polynomial degree d on ␤ , ␣ 1 , . . . , ␣ e if f Ž x . s 0 for x f C and f Ž x. s

Ý

␣g⺞

n

c␣ Ž x . x ␣

for x g C, where each c␣ , considered as a function of x, is periodic with respect to ␣ 1 , . . . , ␣ e and is identically zero for all ␣ s Ž a1 , . . . , a n . such that Ýa i ) d, and where there exists an ␣ satisfying Ýa i s d such that c␣ Ž x ., as a function of x, is not identically zero. If ␤ s 0, e s n, and ␣ 1 , . . . , ␣ e g ⺞ n are generators of the lattice ⺪ n, then the functions c␣ Ž x . do not depend on x, and the simple quasipolynomials of polynomial degree d on ␤ , ␣ 1 , . . . , ␣ e are exactly the integervalued polynomials of degree d on ⺞ n. We use the term ‘‘polynomial degree’’ and not just ‘‘degree’’ because there is also a second notion of degree: DEFINITION 2.4. The cumulati¨ e degree of a simple quasipolynomial of polynomial degree d on ␤ , ␣ 1 , . . . , ␣ e is d q e. The cumulative degree will turn out to be the more useful number for our purposes. It will also be convenient to allow simple quasipolynomial functions of cumulative degree 0, defined as follows. DEFINITION 2.5. A function f : ⺞ n ª ⺪ is simple quasipolynomial of cumulati¨ e degree 0 if it is nonzero on at most a single element of ⺞ n. LEMMA 2.6. If f 1 and f 2 are both simple quasipolynomial functions on ␤ , ␣ , . . . , ␣ e , of cumulati¨ e degree d1 and d 2 , respecti¨ ely, then f 1 q f 2 is also simple quasipolynomial on ␤ , ␣ 1 , . . . , ␣ e , of cumulati¨ e degree at most maxŽ d1 , d 2 .. Proof. The proof is immediate from the definition of a simple quasipolynomial function.

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LENGTHS OF TORS

DEFINITION 2.7. Given a function f : ⺞ n ª ⺪, recall that the generating function of f is the power series F g ⺪w x 1 , . . . , x n x defined by FŽ x. s

Ý f Ž ␣ . x ␣.

␣g⺞ n

LEMMA 2.8. Let F Ž x 1 , . . . , x n . be the generating function of f : ⺞ n ª ⺪, fix Ž ␤ g ⺞ n, and fix a linearly independent set of ¨ ectors ␣ 1 , . . . , ␣ e g ⺞ n. Suppose 1 F Ž x1 , . . . , x n . s e d Ł js1 Ž 1 y x ␣ j . j with Ý ejs1 d j s d. Then f is the simple quasipolynomial function of cumulati¨ e degree d gi¨ en by

¡ ¢

Ł f Ž a , . . . , a . s~ 1

n

j

ž

dj y 1 q mj mj

if Ž a1 , . . . , a n . s Ý j m j ␣ j ,

/

otherwise.

0

Proof. Given ␣ g ⺞ , there is at most one way of writing ␣ as an ⺞-linear combination of ␣ 1 , . . . , ␣ e . This means that the coefficient of x ␣ in F is nonzero if and only if ␣ can be written as the ⺞-linear combination of ␣ 1 , . . . , ␣ e . If ␣ is such an exponent, write ␣ s Ým j ␣ j . Then the coefficient of x ␣ in F is the product of the coefficients of x m j ␣ j in 1rŽ1 y x ␣ j . d j , and it is easy to verify that these coefficients are just the binomial coefficients given in the description of f above. n

DEFINITION 2.9. Given a polynomial P Ž x 1 , . . . , x n ., define Supp P, the support of P, to be the set of all ␣ g ⺞ n such that the coefficient of x ␣ in P is nonzero. DEFINITION 2.10. Given linearly independent vectors ␣ 1 , . . . , ␣ e g ⺞ n and another vector ␤ g ⺞ n, let

½

⌸␤ , ␣ 1 , . . . , ␣ e s ␣ g ⺞ n ␣ s ␤ q

Ý mi ␣i , 0 F mi - 1 i

5

.

Given any ␣ g C␤ , ␣ 1 , . . . , ␣ e l ⺞ n, there is a unique representative of ␣ modulo ␣ 1 , . . . , ␣ e in ⌸␤ , ␣ 1 , . . . , ␣ e; in the following, we use ␣ to denote that unique representative. LEMMA 2.11. Let F Ž x 1 , . . . , x n . be the generating function of f : ⺞ n ª ⺪, fix ␤ g ⺞ n, and fix a linearly independent set of ¨ ectors ␣ 1 , . . . , ␣ e g ⺞ n. Suppose P F Ž x1 , . . . , x n . s e d Ł js1 Ž 1 y x ␣ j . j

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with Ý ejs1 d j s d and Supp P ; ⌸ s ⌸␤ , ␣ 1 , . . . , ␣ e. Write P s Ý ␣ g ⌸ c␣ x ␣ , with each c␣ g ⺪ a constant. Then f is the simple quasipolynomial function of cumulati¨ e degree d gi¨ en by fŽ␣. s

Ł c␣ j

ž

dj y 1 q mj mj

/

ž

where ␣ s ␣ q

Ý mj ␣j j

/

.

Proof. The generating function F can be written as the sum of the fractions c␣ x ␣ Ł ejs1 Ž 1 y x ␣ j .

dj

.

The result then follows from the previous lemma. LEMMA 2.12. Fix ␤ g ⺞ n, and fix linearly independent ¨ ectors ␣ 1 , . . . , ␣ e g ⺞ n. Then functions of the form x␣

F Ž x1 , . . . , x n . s

Ł ejs1 Ž 1 y x ␣ j .

dj

with ␣ g ⌸ s ⌸␤ , ␣ 1 , . . . , ␣ e, and Ý ejs1 d j F d, form a ⺪-basis for the generating functions of the simple quasipolynomial functions of cumulati¨ e degree at most d on ␤ , ␣ 1 , . . . , ␣ e . Proof. This is a consequence of the previous lemma and of the fact that the functions

Ž a1 , . . . , a e . ¬ Ł j

ž

d j y 1 q aj , aj

/

with ÝŽ d j y 1. at most d, form a basis over ⺪ for the polynomials of degree at most d. ŽSee, e.g., Theorem 11.1.12 of w5x.. It will prove useful Žin particular, in Theorem 2.25. to know something about the restriction of a simple quasipolynomial function to a smaller cone with different generators. LEMMA 2.13. Let f 1 be simple quasipolynomial on ␤ , ␣ 1 , . . . , ␣ e of X cumulati¨ e degree d. Pick ␤ ⬘ g ⺞ n and linearly independent ␣ 1X , . . . , ␣ e⬘ such that C⬘ s C␤ ⬘, ␣ 1X , . . . , ␣ Xe⬘ : C s C␤ , ␣ 1 , . . . , ␣ e .

LENGTHS OF TORS

111

Define a new function f ⬘ such that f ⬘Ž ␣ . s ␣ for ␣ g C⬘ and f ⬘Ž ␣ . s 0 for ␣ ⬘ g C⬘. Then there exist positi¨ e integers m i such that f ⬘ is simple quasipolyX nomial of cumulati¨ e degree at most d on ␤ ⬘, m1 ␣ 1X , . . . , m e⬘ ␣ e⬘ . Proof. For each ␣ iX , choose m i such that m i ␣ iX g ⺞ ␣ 1 q ⭈⭈⭈ q⺞ ␣ e . Since f is simple quasipolynomial, we know that f Ž x. s

Ý

␣g⺞

n

c␣ Ž x . x ␣

for x g C, where c␣ is periodic with respect to ␣ 1 , . . . , ␣ e and is identically zero for all ␣ s Ž a1 , . . . , a n . such that Ýa i ) d y e. Since the c␣ ’s are periodic with respect to ␣ 1 , . . . , ␣ e , they are also periodic with respect to m1 ␣ 1X , . . . , m e⬘ ␣ e⬘. Therefore f ⬘ is also simple quasipolynomial. The bound on the degrees of the terms x ␣ bounds the polynomial degree of f ⬘ above by d y e. The cumulative degree of f ⬘ is therefore at most d y e q e⬘ F d. The next definition, and the following theorem, make it possible to reduce the proofs of theorems about quasipolynomial functions defined on ⺞ n to the n s 1 case. DEFINITION 2.14. Given a function f ⺞ n ª ⺞, define a new function f * by f * Ž a. s

Ý

a1q ⭈⭈⭈ qa n sa

f Ž a1 , . . . , a n . .

THEOREM 2.15. If f : ⺞ n ª ⺞ is simple quasipolynomial of cumulati¨ e degree d, then f *: ⺞ ª ⺞ is also simple quasipolynomial of cumulati¨ e degree d. Proof. Write F for the generating function associated with f, and write F* for the generating function associated with f *. Then F*Ž y . s F Ž y, . . . , y .. By Lemma 2.12, F has the form PŽ x. Ł ejs1

Ž1 y x ␣j .

dj

,

so by substituting Ž y, . . . , y . for Ž x 1 , . . . , x n . s x, we see that F* has the form P⬘ Ž y . Ł ejs1 Ž 1 y y a j .

dj

,

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J. BRUCE FIELDS

where by a j we mean the sum of the coordinates of the vector ␣ j . By Lemma 2.12 again, this is the generating function of a quasipolynomial function of cumulative degree at most d, so f * must be quasipolynomial of cumulative degree at most d. It remains only to show that the cumulative degree of f * is not less than d. Let C, ␤ , ␣ 1 , . . . , ␣ e , and the functions c␣ be as in the definition of a simple quasipolynomial function ŽDefinition 2.3.. Without loss of generality we may assume that ␤ s 0. Let ⌳* s ⺪ ␣ 1 q ⭈⭈⭈ q⺪ ␣ e , and let ⌳qs ⺞ ␣ 1 q ⭈⭈⭈ q⺞ ␣ e . We can also assume without loss of generality that c␣ Ž x . s 0 for any x f ⌳q. Since c␣ Ž x . is periodic with respect to ␣ 1 , . . . , ␣ e , f may now be written in the form f Ž x. s

½

Ý ␣ c␣ x ␣ 0

for x g ⌳q for x f ⌳q,

where the c␣ ’s are constants not depending on x. Write pŽ x . for the polynomial function pŽ x . s Ý ␣ c␣ x ␣. Then f * Ž a. s

Ý

a1q ⭈⭈⭈ qa n sa

s

Ý

f Ž a1 , . . . , a n .

a1q ⭈⭈⭈ qa n sa Ž a1 , . . . , a n .g⌳ q

p Ž a1 , . . . , a n . .

Since f has cumulative degree d, p is a polynomial of degree d y e. To prove that f * has cumulative degree at least d, it will suffice to show that f * is bounded below by a polynomial of degree d y 1. Let pdy e be the d y e degree part of p, and let f dye s pdye < ⌳ q be the corresponding polynomial degree d y e part of f. It will suffice to show that Ž f dy e .* is bounded below by a polynomial of degree d y 1. If pdy e Ž a1 , . . . , a n . were negative for some Ž a1 , . . . , a n . g ⌳q, then the polynomial function a ¬ p Ž aa1 , . . . , aan . s pdye Ž aa1 , . . . , aan . q lower order terms s pdy e Ž a1 , . . . , a n . a d q lower order terms would have a negative leading term, and p would eventually take on negative values. In particular, since multiples of Ž a1 , . . . , a n . are also contained in ⌳q, p would take on negative values on ⌳q. But this does not happen, so pdy e must be nonnegative on ⌳q. It also cannot be the case that pdy e is identically zero on ⌳q. So pdye must be positive for some ␤ s Ž b1 , . . . , bn . g ⌳q.

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LENGTHS OF TORS

Let ␲ : ⺢ n ª ⺢ be the mapping defined by ␲ Ž a1 , . . . , a n . s Ý i a i , and write b s ␲ Ž ␤ .. The subset of ␲y1 Ž b . on which pdy e is zero is a proper algebraic subset of ␲y1 Ž b ., so one can find a positive real number c and an Ž n y 1.-dimensional disk ⌬ ; ␲y1 Ž b . contained in the ⺢q-span of ␣ 1 , . . . , ␣ e such that pdye Ž ␣ . G c for any ␣ g ⌬. Note that any positive a and any ␣ g ⌬, pdy e Ž a ␣ . s a dy e pdye Ž ␣ . G ca dy e, so pdye Ž ␣ . G ca dy e for any ␣ g a⌬. Now for any a g ⺞,

Ž f dy e . * Ž a. s

pdye Ž ␣ .

Ý

␣ g ␲ y1 Ž a .l⌳ q

G

Ý

pdy e Ž ␣ .

Ý

c

a ␣ g Ž ⌬ .l⌳ q b

G

a ␣ g Ž ⌬ .l⌳ q b

s噛

a

dy e

ž / b

⌬ l ⌳q c

žž / b

a

a

/ž / b

dy e

,

a

where the notation 噛ŽŽ ⌬ . l ⌳q . is used to denote the number of b

a

members of the set Ž ⌬ . l ⌳q. b Therefore it suffices to show that the cardinality of the set Ž a⌬ . l ⌳q is bounded below by a polynomial in a of degree e y 1. Note that the intersection of a⌬ with the ⺡ G 0-span of the ␣ i ’s is dense in a⌬. Therefore we can choose linearly independent ␣ 1X , . . . , ␣ eX in a⌬ such that each ␣ iX is a linear combination of the ␣ i ’s with coefficients in ⺡ G 0 . Clear denominators, letting m g ⺞ be such that m ␣ iX g ⌳q for all i, and write ␣ iY for m ␣ iX . Then ␣ 1Y , . . . , ␣ eY g m⌬ l ⌳q, and the ␣ 1Y , . . . , ␣ eY are linearly independent. Write ⌳Yq for the set of ⺞-linear combinations of the ␣ iY . Then clearly Ž ma⌬ . l ⌳⬙ q; Ž ma⌬ . l ⌳q for any a, so it suffices to prove that Ž ma⌬ . l ⌳⬙ q is a polynomial of degree e y 1 in a. Finally, Ž ma⌬ . l ⌳Yq is exactly the set  Ý i a i ␣ i N Ý i a i s a4 , and the cardinality of this set is Ž e y 1a q a ., a polynomial of degree e y 1. DEFINITION 2.16. A function f : ⺞ n ª ⺪ is quasipolynomial of degree d if it can be written as a finite sum of simple quasipolynomial functions of cumulative degree at most d, and it cannot be written as a finite sum of simple quasipolynomial functions of cumulative degree less than d.

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Sometimes we will say that a function f : ⺞ n ª ⺪ is quasipolynomial on ␣ 1 , . . . , ␣ e g ⺞ n ; by this we mean that f can be written as a finite sum of simple quasipolynomial functions on subsets of ␣ 1 , . . . , ␣ e . Note that this definition does not agree with the definition of the term quasipolynomial given, for example, in 4.4 of w25x. It is obvious from this definition that simple quasipolynomial functions of cumulative degree d are also quasipolynomial functions of degree d. LEMMA 2.17. If f 1: ⺞ n ª ⺪ and f 2 : ⺞ n ª ⺪ are quasipolynomial, of degrees d1 and d 2 respecti¨ ely, then f 1 q f 2 is quasipolynomial of degree at most maxŽ d1 , d 2 .. Also, if d1 / d 2 , then the degree of f 1 q f 2 is exactly maxŽ d1 , d 2 .. Proof. Immediate from the definition. The following lemmas provide a few examples of quasipolynomial functions. LEMMA 2.18. If f Ž ␣ . is zero for all but finitely many ¨ alues of ␣ , then f is quasipolynomial of degree 0. Proof. The proof follows immediately from the definition of a simple quasipolynomial of degree 0. LEMMA 2.19. A function f : ⺞ ª ⺪ is quasipolynomial on 1 Ž in other words, is the sum of simple quasipolynomial functions on ␤ , 1 for ¨ arious ␤ ’s . of degree d iff it is e¨ entually polynomial of degree d y 1. Proof. A simple quasipolynomial function f Ž ␣ . on ␤ , 1 is a function that is zero for ␣ - ␤ and that agrees for ␣ G ␤ with a polynomial with coefficients that are periodic with respect to 1 and therefore are constant. Such a function is clearly eventually polynomial, as is the sum of such functions. For the converse, let f ⬘ be the polynomial function that f eventually agrees with. A polynomial of degree d y 1 is simple quasipolynomial of degree d. The difference between f and f ⬘ is then nonzero for only finitely many values of ⺞ and is accounted for by the previous lemma. Let ⑀ 1 , . . . , ⑀ n g ⺞ n be the standard basis for ⺞ n, so that ⑀ i is the vector having a one in the ith coordinate and zeros elsewhere. In the following, wherever we write ␣ G ␤ , for ␣ s Ž a1 , . . . , a n . and ␤ s Ž b1 , . . . , bn ., we mean that a i G bi for all i g Ž1, 2, . . . , n. LEMMA 2.20. If a function f : ⺞ n ª ⺪ is quasipolynomial on ⑀ 1 , . . . , ⑀ n of degree d then there exists ␣ ⬘ g ⺞ n such that f Ž ␣ . agrees with a polynomial of degree d y n for all ␣ G ␣ ⬘. Ž Here we take a polynomial of negati¨ e degree to be identically zero..

115

LENGTHS OF TORS

Proof. A simple quasipolynomial function on a proper subset of e1 , . . . , e n is easily seen to be eventually zero, whereas a simple quasipolynomial function on ⑀ 1 , . . . , ⑀ n agrees eventually with a polynomial with constant coefficients, as in the previous proof. The sum of such functions is, therefore, eventually a polynomial, and the statement about the degree follows from the definition of cumulative degree. The converse to this theorem is false; for example, if f has non-polynomial behavior on along a line extending parallel to one of the coordinate axes, then f is not quasipolynomial, even though there may exist an ␣ ⬘ such that ifŽ ␣ . agrees with a polynomial for all ␣ G ␣ ⬘. LEMMA 2.21. The product of two quasipolynomial functions in distinct sets of ¨ ariables is quasipolynomial of degree equal to the sum of the degrees of the two quasipolynomial functions. Proof. Let f Ž x 1 , . . . , x m . and g Ž y 1 , . . . , yn . be the two functions. Let f s f 1 q ⭈⭈⭈ qf r , with each f i simple quasipolynomial, and with f 1 having degree equal to the degree of f. Choose g 1 , . . . , g s similarly. Then fg s

Ý fi g j , i, j

so it is clear that fg is quasipolynomial if the product of two simple quasipolynomial functions is simple quasipolynomial. If, in addition, such products have degrees equal to the sums of the degrees of their factors, then the statement about degrees will also be proved, since f 1 g 1 would then have the correct degree, with all the other products having equal or lesser degrees. Thus we may assume without loss of generality that f and g are simple quasipolynomial. Write F Ž x 1 , . . . , x m . and G Ž y 1 , . . . , yn . for the generating functions of f and g, respectively. The generating function of fg is just FG and, by Lemma 2.12, FG is the generating function of a simple quasipolynomial function of the correct degree. Theorem 2.24 will demonstrate that quasipolynomial functions have a very simple characterization in terms of generating functions. Some preliminary lemmas will be required before the proof of this fact. LEMMA 2.22. Assume

Let F Ž x 1 , . . . , x n . be the generating function of f : ⺞ n ª ⺪.

F Ž x1 , . . . , x n . s

P Ž x1 , . . . , x n . Ł ejs1 Ž 1 y x ␣ j .

dj

with ␣ 1 , . . . , ␣ d g ⺞ n ¨ ectors that are linearly independent in ⺢ n and P a polynomial. Then f is quasipolynomial of cumulati¨ e degree at most Ý ejs1 d j .

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J. BRUCE FIELDS

Proof. The function F is the sum of fractions of the form x␥ Ł ejs1 Ž 1 y x ␣ j .

.

dj

Each of these is the generating function of a simple quasipolynomial function of degree d, by Lemma 2.12. LEMMA 2.23. form

Let F Ž x . be a rational function in ⺞Ž x 1 , . . . , x n . of the FŽ x. s

PŽ x. Ł ejs1

Ž1 y x ␣j .

dj

,

with P Ž x . g ⺞w k 1 , . . . , k n x, ␣ j g ⺞ n, and d j g ⺞. If the ␣ j ’s are linearly dependent, then F Ž x . can be rewritten in the form FŽ x. s

Ý i

Pi Ž x . Ł ejs1

Ž 1 y x ␣ i, j .

di, j

,

for some ␣ i, j g ⺞ n, Pi g ⺞w x 1 , . . . , x n x, and d i, j , e i g ⺞, such that each e i is strictly less than e. Proof. Choose a linear dependence relation among the ␣ j ’s. By clearing denominators and renumbering the ␣ j ’s if necessary, this relation can be written m1 ␣ 1 q ⭈⭈⭈ qm k ␣ k s m kq1 ␣ kq1 q ⭈⭈⭈ qm e ␣ e , where each m k is a nonnegative integer. Write ␥ for the element of ⺞ n that both sides of this equation are equal to. Let d s 1 q Ý kjs1Ž d j y 1., and let d⬘ s 1 q Ý ejskq1Ž d j y 1.. Since

Ž 1 y x m j ␣ j . s Ž 1 y x ␣ j . Ž 1 q x ␣ j q x 2 ␣ j q ⭈⭈⭈ qx Ž m jy1 . ␣ j . , F can be rewritten in the form P⬘ Ž x . Ł ejs1

Ž1 y x mj␣j .

dj

,

with P⬘ a new polynomial. Next write

Ž 2.1.

FŽ x. s

P⬘ Ž 1 y x ␥ .

Ž 1 y x␥ .

dq d⬘

dq d⬘

Ł ejs1 Ž 1 y x m j ␣ j .

dj

,

117

LENGTHS OF TORS

and note that

Ž 1 y x ␥ . s Ž 1 y x Ý is 1Ž m i ␣ i . . s Ž 1 y x m 1 ␣ 1 . x Ý is 2 Ž m i ␣ i . k

k

q Ž 1 y x m 2 ␣ 2 . x Ý is 3Ž m i ␣ i . k

q ⭈⭈⭈ q Ž 1 y x m ky 1 ␣ ky 1 . x m k ␣ k qŽ 1 y x m k ␣k . , which shows that Ž1 y x ␥ . is contained in the ideal generated by Ž1 y x m 1 ., . . . , Ž1 y x m k .. Similarly, Ž1 y x ␥ . is also contained in the ideal generated by Ž1 y x m kq 1 ., . . . , Ž1 y x m e .. This fact, together with our choice of d and d⬘, allows us to rewrite Ž1 y x ␥ . dq d⬘ as the sum of terms each of which is contained one of both Ž1 y x m i . d i for 1 F i F k and Ž1 y x m i . d i for k - i F e as factors. Finally we are able to write F in the desired form by expanding the numerator of Eq. Ž2.1., breaking up the fraction into a sum of fractions, and canceling these factors. THEOREM 2.24. Let F Ž x 1 , . . . , x n . be the generating function of f : ⺞ n ª ⺪. Then f is quasipolynomial iff F can be written in the form FŽ x. s

PŽ x. Ł ejs1

Ž1 y x ␣j .

mj

.

Proof. If F is in the given form, then we can use repeated applications of the previous lemma to rewrite F as the sum of terms each of which is in the form specified in Lemma 2.22. Thus f is the sum of quasipolynomial functions and must itself be quasipolynomial. Conversely, if f is quasipolynomial, then f can be written as the sum of simple quasipolynomial functions. The form of the generating function of each of these simple quasipolynomial functions is given by Lemma 2.12, and the sum of such functions can clearly be written in the form required. THEOREM 2.25.

Let f : ⺞ n ª ⺞ be quasipolynomial of degree d. Then f * Ž a. s

Ý

a1q ⭈⭈⭈ qa n sa

f Ž a1 , . . . , a n .

is also quasipolynomial of degree d. Proof. If f is quasipolynomial, and F Ž x 1 , . . . , x n . is its generating function, then the generating function of f * is F*Ž x, . . . , x .. The fact that f * is quasipolynomial then follows immediately from Theorem 2.24. It is also clear from this theorem that the degree of f * is at most d. The only remaining difficulty is to show that the degree cannot be less than d.

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There must exist ␤ , ␣ 1 , . . . , ␣ e such that f restricted to C␤ , ␣ 1 , . . . , ␣ e is simple quasipolynomial of cumulative degree d. If not, then we could cover ⺞ n by cones in such a way that on each cone f was simple quasipolynomial Žusing Theorem 2.13. and had degree less than d. This would express f as the sum of simple quasipolynomial functions of cumulative degree less than d, contradicting the fact that f has degree d. Let f ⬘ be the restriction of f to C␤ , ␣ 1 , . . . , ␣ e. Then f ⬘Ž ␣ . F f Ž ␣ . for all ␣ g ⺞ n, so f ⬘*Ž a. F f *Ž a. for all a g ⺞. By Theorem 2.15, f ⬘* has degree d; it follows immediately from the definition of simple quasipolynomial that f * must have degree at least d. LEMMA 2.26. If f, g: ⺞ n ª ⺞ are quasipolynomial and f Ž ␣ . F g Ž ␣ . for all ␣ g ⺞ n, then deg f F deg g. Proof. The previous theorem allows us to reduce to the case n s 1. A quasipolynomial function of degree 1 is eventually equal to a polynomial with periodic coefficients, and the result is trivial for such functions. Our eventual goal is to show that the Hilbert functions of certain ⺞ n-graded algebraic objects are quasipolynomial functions. The next theorem will allow us to make arguments that use induction on the degree of those algebraic objects. THEOREM 2.27. Let f : ⺞ n ª ⺪, g: ⺞ n ª ⺪, and ␤ g ⺞ n be such that Ž f ␣ q ␤ . y f Ž ␣ . s g Ž ␣ .. If g is quasipolynomial, then so is f. In addition, if f : ⺞ n ª ⺞, so all the ¨ alues of f are nonnegati¨ e, then the degree of f is one greater than that of g. Proof. Let F and G be the generating functions of f and g, respectively. The relation f Ž ␣ q ␤ . y f Ž ␣ . s g Ž ␣ . is equivalent to the relation F s G Ž1 y x ␤ .. It is clear, then, that F is the generating function of a quasipolynomial function if G is. It remains to show that the degree increases by one when the values of f are known to be nonnegative. Let b be the sum of the coordinates of the vector ␤ . Then

F* s

G* 1 y xb

,

where F* and G* are the generating functions associated with, respectively, f * and g*. Thus by the previous theorem and by Theorem 2.25 we can assume without loss of generality that n s 1. It is not hard to see that in the case n s 1 a quasipolynomial function g agrees eventually with a polynomial with periodic coefficients. Such a

119

LENGTHS OF TORS

function is eventually bounded below by a function with generating function of the form c

Ž1 y x N .

d

,

where c a positive real number and d is the degree of g. Therefore f is bounded below by a function with generating function c d

Ž1 y x N . Ž1 y x b.

,

and it follows that f has degree d q 1. 3. ⺞ n-GRADED ALGEBRAS AND MODULES The quasipolynomial functions introduced in the previous section are useful because they occur as the Hilbert functions of certain ⺞ n-graded algebraic objects. THEOREM 3.1. Let S be an ⺞ n-graded algebra such that S0 is Artin and S is finitely generated as an algebra o¨ er S0 . Let M be a finitely generated, ⺞ n-graded S-module of dimension d. Then the Hilbert function f Ž ␣ . s lengthŽ M␣ . is quasipolynomial of degree d. Proof. The proof will be by induction on d. If d s 0 then M has finite length, and length Ž M␣ . is zero for all but finitely many ␣ , so f is quasipolynomial of degree 0. Now assume that d ) 0 and that the theorem is proved for all smaller d. Because the sum of quasipolynomial functions is quasipolynomial, we can take a primary filtration of M and assume without loss of generality that M is P-primary for some P g SpecŽ S .. We can map an ⺞ n-graded polynomial ring onto S and work over the polynomial ring instead of over S. So we may as well assume S is polynomial; say S s S0 w s1 , . . . , s e x, with degŽ si . s ␣ i g ⺞ n. Since d s dim M ) 0, there must be some si not contained in P; otherwise M would have finite length. That si must be a non-zero divisor on M. This means that the sequence si

0 ª MªM ª

M si M

ª0

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J. BRUCE FIELDS

is exact, so length Ž M␣ . ylength Ž M␣y ␣ i . s length

M

žž / / si M

␣

f Ž ␣ . y f Ž ␣ y ␣ i . s fM r si M Ž ␣ . . Since si is a non-zero divisor on M, Mrsi M has dimension d y 1. Therefore, by the induction assumption, Mrsi M is quasipolynomial of degree d y 1, and, by Theorem 2.27, f M is quasipolynomial of degree d. It should be clear from the proof that if s1 , . . . , s e are generators of S of degrees ␣ 1 , . . . , ␣ e , then the generating function of the Hilbert function can be written as a rational function with denominator Ł i Ž1 y x ␣ i .. A few examples might be helpful. Ž1. If M is a finitely generated module over an ⺞-graded algebra S that is generated by elements of degree 1, then the Hilbert function f M is quasipolynomial on  14 and hence is eventually polynomial, as expected. Ž2. If M is a finitely generated module over an ⺞ 2-graded algebra S, and if every algebra-generator of S over S0 has degree Ž0, 1. or Ž1, 0., then S is quasipolynomial of degree d s dim S on Ž0, 1. and Ž1, 0.. By Theorem 2.20, there exists a ␤ such that, for all ␣ G ␤ , f S Ž ␣ . is equal to a polynomial of degree at most d y 2. Thus we obtain Theorem 2 of w1x, that Hilbert functions of bigraded modules are eventually polynomial of degree at most Ždim M . y 2. More results about these functions can be found in w26x. Ž3. More generally, let M be a finitely generated module over an ⺞ n-graded algebra S, and assume every algebra-generator of S over S0 has degree e i , where e i is the vector in ⺞ n that is one in the ith position but zero elsewhere. Again, there is a ␤ such that f M Ž ␣ . agrees with a polynomial of degree at most d y n for all ␣ G ␤ . ŽSee w11x, where it is also shown that all the highest degree monomials of this polynomial have nonnegative coefficients.. Ž4. If M s S s k w s1 , . . . , s e x, and if the degree of si is ␣ i , then the generating function of the Hilbert function f S is 1 Ł i Ž 1 y x ␣i . and f S is quasipolynomial of degree d. Also for the twisted module M Žy␣ . Žrecall that M Ž ␣ .␤ s My ␣q ␤ ., the corresponding generating function is x␣ . Ł i Ž 1 y x ␣i .

LENGTHS OF TORS

121

It is not hard to see that any quasipolynomial function with a generating function of the form PŽ x. Ł i Ž 1 y x ␣i . such that P Ž x . has nonnegative coefficients can be written as the sum of such functions. Direct sums of modules have Hilbert functions that are the sums of the Hilbert functions of the individual modules. In this way, any quasipolynomial function of this form can be realized as the Hilbert function of some finitely generated module over such an S. In fact, this identifies exactly the set of all functions that arise as the Hilbert functions of ⺞ n-graded modules. THEOREM 3.2. Let S be an ⺞ n-graded, finitely generated, algebra o¨ er a local ring R with maximal ideal m, and let T be an ⺞ n-graded, finitely generated, S-algebra. Assume that S0 s T0 s R. Let N and M be finitelygenerated ⺞ n-graded modules o¨ er T and S, respecti¨ ely, with N > M, and suppose that for e¨ ery element of N there is a power of m multiplying that element into M. Then f N r M : ␣ ª lengthŽŽ NrM .␣ . is quasipolynomial of degree at most dimŽTrŽAnn T N ... Proof. We can filter NrM by NrTM and TMrM. The quotient NrTM is a T-module, not just an S-module. Also, NrTM is a finitely generated T-module, and since every element of N is multiplied into M Žand hence into TM . by a power of m, it follows that there exists an n such that m n kills every element of NrTM. Therefore NrTM can be thought of as a module over Trm n T. This module now satisfies the conditions of the previous theorem, so the Hilbert function of NrTM is quasipolynomial. Since Ann T NrTM > Ann T N, this function is in addition quasipolynomial of degree at most dim N. It remains only to show that TMrM has a quasipolynomial Hilbert function of degree at most dim N. So assume N s TM. Suppose that T is generated over S by r generators, so T s Sw t 1 , . . . , t r x. Suppose that r is at least 2, and that the theorem has already been proved for smaller values of r. Apply the theorem with Sw t 1 x replacing S and Sw t 1 x M replacing M, and with T and N as before. It is clear that the hypotheses of the theorem are still satisfied, and T is generated over S w t 1 x M by r y 1 elements, so by assumption the Hilbert function of NrSw t 1 x M is quasipolynomial. Similarly, apply the theorem with S and M as before, but T replaced by S w t 1 x and N by Sw t 1 x M, and the conclusion is that Sw t 1 x MrM also has a quasipolynomial Hilbert function. The Hilbert function of NrM is the sum of the Hilbert functions of NrSw t 1 x M and Sw t 1 x MrM, and sum of two quasipolynomial functions is quasipolynomial, so the result follows by induction if only we can deal with the case r s 1.

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So assume N s Sw t x M Žwhere t s t 1 .. Construct the S-module ⬁

N⬘ s

t iq1 M q t i M q ⭈⭈⭈ qtM q M

[ t Mqt i

is0

iy1

M q ⭈⭈⭈ qtM q M

.

This module has the same Hilbert series as NrM; to see this, note that the quotient modules that we are summing are just successive quotients of modules in the filtration M M

tM q M

;

M

<

t 2 M q tM q M M

; ⭈⭈⭈ ,

and note that ⬁

D

t i M q t iy1 M q ⭈⭈⭈ qM

is0

M

s

N M

,

thanks to our assumption that N s TM s Sw t x M. Note also that t Ž t i M q ⭈⭈⭈ qM . ; Ž t iq1 M q ⭈⭈⭈ qM ., so multiplication by t induces a well-defined surjection t i M q ⭈⭈⭈ qM t iy1 M q ⭈⭈⭈ qM

ª

t iq1 M q ⭈⭈⭈ M t i M q ⭈⭈⭈ qM

,

and this in turn induces a well-defined map N i ª N⬘. Thus multiplication by t makes sense on N⬘, making N⬘ a T-module as well as an S-module. Let m1 , . . . , m k be a set of generators for M as a module over S. Write tm1, . . . , tm k for the images of these generators in tMrM. An arbitrary element of t iq1 M q t i M q ⭈⭈⭈ qtM q M t i M q t iy1 M q ⭈⭈⭈ qtM q M can be written t i Ý j s jtm j, with s j g S. An arbitrary element of N⬘ can be written as a finite sum of such elements, so it follows that N⬘ is generated over T by m1 , . . . , m k , and hence that N⬘ is finitely generated over T. Any element of N⬘ is killed by a power of m because N⬘ is a direct sum of subquotients of NrM, which has the property that any element is killed by a power of m. Since N⬘ is a T-module, it must also be the case that any element of N⬘ is killed by a power of mT. Because N⬘ is finitely generated, it follows that N⬘ is killed by some fixed power of mT and hence is a finitely generated module over TrŽ mT . j for some j. The fact that N⬘, and hence NrM, has a quasipolynomial Hilbert function follows now from the previous theorem. Note also that Ann T N⬘ > Ann T N, so

123

LENGTHS OF TORS

dim N⬘ F dim N, and NrM has a quasipolynomial Hilbert function of degree at most dim N. 4. RESULTS ABOUT Tork Ž RrI1a1 , RrI2a 2 . FOR k G 2 Now we apply the results of the previous section to the problem of describing the functions f k Ž a1 , . . . , a n . s length Ž Tork Ž M1rI1a1 M1 , . . . , MnrIna n Mn . . in the case n s 2, M1 s M2 s R, k G 2. Let I be any ideal of a ring R, and let M be an R-module. The short exact sequence 0 ª I ª R ª RrI ª 0 gives rise to a long exact sequence

Ž 4.1. ⭈⭈⭈ ⭈⭈⭈

ª

Tork Ž I, M .

ª

0

ª

Tork Ž RrI, M .

ª

ª

Torky 1 Ž I, M .

ª

0

ª

Torky1 Ž RrI, M .

ª

⭈⭈⭈

ª

Tor0 Ž I, M .

ª

M

ª

MrIM

ª

0,

which yields an isomorphism Tork Ž RrI, M . ( Torky1Ž I, M . for i G 2 and an injection Tor1Ž RrI, M . ª Tor0 Ž I, M .. If I1 and I2 are two ideals of R, then Tork Ž RrI1 , RrI2 . ( Torky2 Ž I1 , I2 . for k G 3, by two applications of the above isomorphisms. Since direct sums commute with Tor, there is also an isomorphism

[

a1 , a 2G0

R TorkR Ž RrI1a1 , RrI2a 2 . ( Torky2

ž[ a1G0

I1a1 ,

[I a 2G0

a2 2

/

for k G 3. Let t 1 and t 2 be indeterminates; then we can replace [a i G 0 Iia i by the algebra Rw Ii t x s [a i G 0 Iia i t a i . This allows us to impose the structure of an algebra on [a1 , a 2 G 0 TorkR Ž RrI1a1 , RrI2a 2 .. LEMMA 4.1.

Let I1 and I2 be ideals of a ring R. Let Si s Rw I, t i x s

[a G 0 Iia t ia . Then the R-modules i

i

i

[

a1 , a 2G0

TorkR Ž RrI1a1 , RrI2a 2 .

can be gi¨ en the structure of ⺞ 2-graded, finitely generated S1 mR S2-modules, for k G 2. Proof. For k ) 2, we know that this direct sum is isomorphic to R Ž . w x Torky 2 S1 , S2 . Let r i, 1 , . . . , r i, m i generate Ii . Let T1 s R x 1, 1 , . . . , x 1, m 1

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J. BRUCE FIELDS

and T2 s Rw x 2, 1 , . . . , x 2, m 2 x be polynomial rings, and map Ti onto S i by mapping x i, j onto ri, j t i . This makes Si into a Ti-module. Choose a resolution of S i by finitely generated projective Ti-modules, ⭈⭈⭈ ª Pi , 1 ª Pi , 0 ª Si . Since Ti is a free R-module, this resolution is also a resolution for Si over R. Therefore TorkR Ž S1 , S2 . is the homology of the total complex ⭈⭈⭈ ª P1, 1 mR P2, 0 [ P1 , 0 mR P2, 1 ª P1, 0 mR P1, 0 ª 0. The R-modules in this complex are easily seen to also be finitely generated T1 mR T2-modules, and the maps T1 m T2-maps. Therefore the homology modules TorkR Ž S1 , S2 . are also finitely generated modules over T1 m T2 . In addition, note that these modules must be killed by the kernels of the maps Ti ª S i ; thus, each TorkR Ž S1 , S2 . is also a finitely generated S1 m S2module. For k s 2, apply direct sums to the end of the long exact sequence of Eq. Ž4.1. above to get 0ª

[

a1 , a 2G0

Tor2R Ž RrI1a1 , RrI2a 2 . ª R w I1 t 1 x mR R w I2 t 2 x

ª R w I1 t 1 x mR R w t 2 x . The map Rw I1 t 2 x mR Rw I2 t 2 x ª Rw I1 t 1 x mR Rw t 2 x is clearly a map of S1 mR S2-modules, so the kernel is an S1 mR S2-module. THEOREM 4.2. Let I1 and I2 be ideals of a local ring Ž R, m. of dimension d such that I1 q I2 is m-primary. Then for k G 2 the function f k Ž a1 , a2 . s length Ž Tork Ž RrI1a1 , RrI2a 2 . . is quasipolynomial of degree at most 2 d. Proof. Fix k G 2. Then [a1 , a 2 Tork Ž RrI1a1 , RrI2a 2 . is isomorphic to Žor, in the case of k s 2, is at least a submodule of. Torky 2 Ž S1 , S2 .. Since I1 q I2 is m-primary, I1a1 q I2a 2 is also m-primary, so some power of m is contained in I1a1 q I2a 2 and some power of m kills Tork Ž RrI1a1 , RrI2a 2 .. Therefore every graded piece of Torky 2 Ž S1 , S2 . is killed by a power of m. By the previous lemma, Torky 2 Ž S1 , S2 . is a finitely generated S1 m S2module. It follows that every element of Torky 2 Ž S1 , S2 . is also killed by a power of mŽ S1 m S2 . and, in fact, that a fixed power m N Ž S1 m S2 . kills every element of the module. Therefore Torky 2 Ž S1 , S2 . is actually a module over Ž Rrm N . mR Ž S1 m S2 ..

LENGTHS OF TORS

125

Thus the module [a1 , a 2 Tork Ž RrI1a1 , RrI2a 2 . is isomorphic, by an isomorphism that preserves the grading, to a finitely generated, bigraded module over Ž Rrm N . mR Ž S1 m S2 .. This makes f k the Hilbert function of a finitely generated bigraded module over the bigraded ring Ž Rrm N . mR Ž S1 m S2 .. By Theorem 3.1 this Hilbert function is quasipolynomial of degree equal to the dimension of the ring. So it remains only to calculate the dimension of Ž Rrm N . mR Ž S1 m S2 .. The ideal mŽ S1 m S2 . is nilpotent, so the ring Ž Rrm. mR Ž S1 m S2 . has the same dimension, and

Ž Rrm . mR Ž S1 m S2 . ( Ž Ž Rrm . mR S1 . mR r m Ž Ž Rrm . mR S2 . . The dimension of Ž Rrm. mR Si s Ž Rrm. mR Rw Ii t x is the analytic spread of Ii , which is at most d. The claimed result follows. The module Tork Ž S1 , S2 . whose Hilbert function we calculate in the above proof is a module over an algebra all of whose multidegrees are either Ž0, 1. or Ž1, 0.. By the second example of the previous section, such a function is eventually polynomial of degree at most 2 less than the dimension of the module. Therefore COROLLARY 4.3. Let I1 and I2 be ideals of a local ring Ž R, m. such that I1 q I2 is m-primary. Then there exists a Ž b1 , b 2 . g ⺞ 2 such that, for k G 2 and Ž a1 , a2 . G Ž b1 , b 2 ., the function f k Ž a1 , a2 . s length Ž Tork Ž RrI1a1 , RrI2a 2 . . is polynomial of degree at most 2 d y 2. Note that it is possible that the Tor modules could have dimension less than the sums of the analytic spreads of the two ideals; for example, if I1 and I2 are generated by disjoint parts of a system of parameters, then the higher Tor’s are all zero. However, I know of no examples showing that nonzero polynomials of smaller degree may occur. 5. Tor1Ž RrI1a1 , RrI2a 2 . The next goal is to establish some results about the functions f 1 and f 0 . As far as we know, the following may be true. CONJECTURE 5.1. Let I1 and I2 be ideals of a local ring Ž R, m. such that I1 q I2 is m-primary. Then the function f 1 Ž a1 , a2 . s length Ž Tor1 Ž RrI1a1 , RrI2a 2 . . is quasipolynomial of degree at most dim R q 2.

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J. BRUCE FIELDS

However, all we know now is this: THEOREM 5.2. Let I1 and I2 be ideals of a local ring Ž R, m. such that I1 q I2 is m-primary, and such that the R-algebra Ts

[ŽI a1 , a 2

a1 1

l I2a 2 . x 1a1 x 2a 2 ; R w x 1 , x 2 x

is finitely generated as an algebra o¨ er R. Then the function f Ž a1 , a2 . s length Ž Tor1 Ž RrI1a1 , RrI2a 2 . . is quasipolynomial of degree at most dim R q 2. Proof. Let T be the doubly graded R-algebra described above. By assumption it is finitely generated. Let S be the doubly graded R algebra S s R w I1 x 1 , I2 x 2 x s

[I a1 , a 2

a1 a 2 1 I2

x 1a1 x 2a 2 .

Since S ; T, T is also an S-algebra. Now observe that TrS s

[ a1 , a 2

I1a1 l I2a 2 I1a1 I2a 2

x 1a1 x 2a 2

and recall that Tor1R Ž RrI1a1 , RrI2a 2 . s Ž I1a1 l I2a 2 .rI1a1 I2a 2 . Apply Theorem 3.2, with N s T, M s S, and the conclusion follows; all that remains is to calculate the upper bound on the dimension of R. Let M ; T be the ideal generated by m and by Ž I1a1 l I2a 2 . x 1a1 x 2a 2 , for all a1 , a2 satisfying a1 q a2 G 1. Clearly M is maximal, with TrM ( Rrm. Also, dim T s height M. Let P be a minimal prime such that dim TrP s dim T, and let p s P l R. Apply the dimension formula to get dim T q 0 s height M q tr. deg R r m R TrMT F height Ž m R Ž Rrp . . q tr. deg R r p T F dim R q 2 DEFINITION 5.3. Given a pair Ž I1 , I2 . of ideals of the local ring Ž R, m., call the algebra [a1 , a 2Ž I1a1 l I2a 2 . x 1a1 x 2a 2 the intersection algebra of I 1 and I 2 . If this algebra is finitely generated over R, we will say that I1 and I2 have finite intersection algebra. The previous theorem can now be restated as follows. COROLLARY 5.4. If I1 and I2 , ideals of the local ring Ž R, m., ha¨ e finite intersection algebra, then the function f Ž a1 , a2 . s length Ž Tor1 Ž RrI1a1 , RrI2a 2 . . is quasipolynomial of degree at most dim R q 2.

LENGTHS OF TORS

127

Note that there are examples of ideals I1 , I2 in local rings Ž R, m. that do not have finite intersection algebras. Such examples can be constructed from examples existing in the literature of non-finitely generated symbolic power algebras. ŽSee w9, 20, 21x. In particular, given an ideal P ; R such that the algebra R [ P Ž1. [ P Ž2. [ ⭈⭈⭈ is not finitely generated, it is possible to find a f g R such that the ideals Ž f . and P do not have finite intersection algebra. To do this, choose f so that Ž P a : f a . s P Ž a., for all a. LEMMA 5.5. If R regular and P ; R prime, then there exists f g R such that Ž P a : f a . s P Ž a. for all a. Proof. Since R is a regular local ring, so is R P . The associated graded ring of a regular local ring is a polynomial ring, so, in particular, grP R PŽ R P . s Ž R y P .y1 grP Ž R . is a domain. Let I be the kernel of the natural map grP Ž R . ª Ž R y P .y1 grP Ž R .. Then Ž R y P .y1 I s 0, and, since I is finitely generated, there exists f g R y P such that fI s 0. ŽWe are implicitly identifying elements in R with their images in grP Ž R . via the natural map R ª RrP ; grP Ž R ... This means that fg g P nq 1 for any element g of P n which can be multiplied into P nq 1 by an element of R y P. More generally, any element g g P n which can be multiplied into P nq m by an element of R y P satisfies f m g g P nqm . The result follows. LEMMA 5.6. If f and P are chosen as suggested abo¨ e, then the intersection algebra [a1 , a 2 G 0 Ž P a1 l Ž f a 2 .. x 1a1 x 2a 2 is not finitely generated. Proof. We will prove the contrapositive. Assume that the intersection algebra is finitely generated. Note that Ž P a1 : f a 2 . f a 2 s P a1 l Ž f a 2 ., so the algebra can be rewritten as [Ž P a1 : f a 2 . x 1a1 Ž x 2 f . a 2 . Map this to the symbolic power algebra [a P Ž a. x a by mapping tf to 1. This is an ungraded ring homomorphism. So the symbolic power is finitely generated, since it can be written as a homomorphic image of the intersection algebra, which we assumed to be finitely generated. The examples in the papers cited above give rise in this way to a number of quite concrete examples of pairs of ideals with non-finite intersection algebras. Studying the functions f 1 that arise in such examples would be an interesting subject for further research. 6. Tor0 Ž RrI1a1 , RrI2a 2 . Throughout this section, Ž R, m. is a regular local ring of dimension d.

128

J. BRUCE FIELDS

Let I1 and I2 be ideals of R such that I1 q I2 is m-primary, and let f␹ Ž a1 , a2 . be the Euler characteristic f␹ Ž a1 , a2 . s ␹ Ž RrI1a1 , RrI2a 2 . s

Ý Ž y1. k length Ž TorkR Ž RrI1a , RrI2a . . . 1

2

k

This sum is well defined because the Tor’s are all zero for k ) dim R, and because the condition that I1 q I2 be m-primary forces all of the Tork ’s to have finite length. The function ␹ is biadditive Žso, for example, ␹ Ž M [ N, P . s ␹ Ž M, P . q ␹ Ž N, P .. because each of the Tork ’s are. Also, ␹ Ž M, N . is zero whenever M and N are such that dimŽ M . q dimŽ N . - d Žsee w8, 19x.. And if length Ž M m N . - ⬁, as is the case when M s RrI a1 and N s RrI a 2 , then dimŽ M . q dimŽ N . F d Žsee w23x.. THEOREM 6.1. For large a1 and a2 , f␹ Ž a1 , a2 . is either identically zero or is e¨ entually a polynomial of degree d. Proof. Fix a1 and a2 and let Pu, 1 , . . . , Pu, s u be the primes associated to Iua u . For u s 1 and u s 2, choose filtrations of RrIua u by prime cyclic modules, and let pu, i be the number of times that RrPu, i occurs in the corresponding filtration. Then by biadditivity,

␹ Ž RrI1a1 , RrI2a 2 . s

p1, i p 2, j ␹ Ž RrP1, i RrP2, j . .

Ý iFs 1 , jFs 2

Assume that the Pu, i ’s are ordered so that Pu, 1 , . . . , Pu, sXu have the least height of any of the Pu, i ’s, and let e u be this minimal height. Since each Pu,i contains Iua u , we know that P1, i q P2, j is m-primary for any i and j. Therefore e1 q e2 G d. This means that if i ) sX1 or j ) sX2 then heightŽ P1, i . q heightŽ P2, j . ) e1 q e 2 G d, so ␹ Ž RrP1, i RrP2, j . s 0. Therefore

␹

ž

R I1a1

,

R I2a 2

/

s

Ý

X

X

p1, i p 2, j ␹

iFs 1 , jFs 2

ž

R

,

R

P1, i P2, j

/

.

Note that pu, i is independent of the choice of filtration; in fact, pu, i is just the length of R P u, irIua u R P u, i . Also, the ideals Pu, 1 , . . . , Pu, sXu are also the associated primes of minimal height of any power of Iu . So, for any a1 and a2 , f␹ Ž a1 , a2 . s ␹ s

R

ž

I1a1

Ý

X

R

,

I2a 2

X

iFs 1 , jFs 2

/

length

ž

R P1 , i I1a1 R P1 , i

/ ž length

R P2 , j I a2 2 R P2 , j

/ž ␹

R

,

R

P1, i P2, j

/

.

129

LENGTHS OF TORS

Since Iu is primary to the maximal ideal of R P u, i , the functions a u ¬ length

ž

R Pu , i Iua u R P u , i

/

as just Hilbert functions and are eventually polynomial of degree e u . Therefore, the function f␹ Ž a1 , a2 . is polynomial of degree d for sufficiently large a1 and a1 , if e1 q e2 s d, and is identically zero if e1 q e2 ) d, since in that case all of the ␹ Ž RrP1, i , RrP2, j .’s are zero. We can also describe f␹ as follows. THEOREM 6.2. degree d q 2.

f␹ Ž a1 , a2 . is either identically zero or quasipolynomial of

Proof. By the proof of the previous theorem, f␹ Ž a1 , a2 . is the sum of terms of the form cg 1Ž a1 . g 2 Ž a2 ., where g 1 and g 2 are eventually polynomial and c is a nonnegative integer. By Lemma 2.17, the sum of quasipolynomial functions is quasipolynomial of degree equal to the maximum of the degrees of the summands. So it suffices to show that each such summand is either identically zero or is quasipolynomial of degree d q 2. By Lemma 2.19, g 1 and g 2 are each quasipolynomial of degree e1 q 1 and e2 q 1 Žwhere e1 and e2 are as in the proof of the previous theorem.. So Lemma 2.21 says that cg 1Ž a1 . g 2 Ž a2 . is a quasipolynomial function of a1 and a2 of degree e1 q e2 q 2 s d q 2, as long as c is nonzero. THEOREM 6.3. function

If I1 and I2 ha¨ e finite intersection algebra, then the f 0 Ž a1 , a2 . s length

ž

R I1a1

q I2a 2

/

is quasipolynomial of degree dim R q 2. Proof. By the previous theorems, we already know that f i is quasipolynomial for i ) 0, and we know that the alternating sum ␹ s f 0 y f 1 q f 2 q ⭈⭈⭈ is quasipolynomial. Since f 0 s ␹ q f 1 y f 2 q ⭈⭈⭈ , and since the f i ’s are eventually zero, this expresses f 0 as a finite sum of quasipolynomial functions. Thus f 0 is quasipolynomial. It remains to calculate the degree, but this is done in the following lemma. LEMMA 6.4.

If the function f 0 Ž a1 , a2 . s length

ž

R I1a1

q I2a 2

is quasipolynomial, then its degree is dim R q 2.

/

130

J. BRUCE FIELDS

Proof. Both I1 and I2 are contained in m, so I1a1 q I2a 2 ; mminŽ a1 , a 2 .. Therefore f 0 is bounded below by, for example, the function given by f Ž a1 , a2 . s

½

length Ž RrmŽ a1qa 2 .r3 . 0

if 1r2 F a1ra2 F 2 otherwise.

Since length Ž Rrm a1 . is eventually polynomial of degree d, this function is quasipolynomial of degree d q 2. So f 0 is bounded below by a quasipolynomial function of degree d q 2. Since I1 q I2 is m-primary, there exists N such that m N ; I1 q I2 . It follows that m NŽ a1qa 2 . ; I1a1 q I2a 2 , so f 0 Ž a1 , a2 . F length Ž Rrm NŽ a1qa 2 . . . Therefore f 0 Ž a1 , a2 . can also be bounded above by a polynomial of degree d, and hence by a quasipolynomial function of degree d q 2. The result now follows from Lemma 2.26.

7. MONOMIAL IDEALS Let K be a field, let R s K ww x 1 , . . . , x d xx, and let I1 , . . . , In be ideals of R generated by monomials. The monomials of R correspond to elements of Ž⺪ G 0 . d ; ⺢ d , under a map ‘‘log’’ which takes a monomial x ␣ to the integer vector ␣ determined by its exponents. Similarly, monomial ideals correspond to subsets of ⺢ d ; write log I for the discrete subset of ⺢ d corresponding to the exponents of the monomials contained in I. THEOREM 7.1. If I1 , . . . , In are monomial ideals in R s K ww x 1 , . . . , x d xx satisfying I1 q ⭈⭈⭈ qIn s Ž x 1 , . . . , x n ., then

'

f 0 Ž a1 , . . . , a n . s length

ž

R I1a1

q ⭈⭈⭈ qIna n

/

is quasipolynomial of degree d q n. Proof. The length f 0 Ž a1 , . . . , a n . is equal to the cardinality of logŽ R . _ logŽ I1a1 q ⭈⭈⭈ qIna n .. Pick N such that m N ; I1 q ⭈⭈⭈ qIn . Given Ž a1 , . . . , a n . g ⺪ n, note that

Ž 7.1.

m NŽ a1q ⭈⭈⭈ qa n . ; I1a1 q ⭈⭈⭈ qIna n .

Write 噛Ž A. to mean the cardinality of a set A. The function Ž a1 , . . . , a n . ¬ 噛logŽ R . _ logŽ m NŽ a1q ⭈⭈⭈ qa n . .. is clearly quasipolynomial, so it suffices to show that the function

Ž 7.2.

Ž a1 , . . . , an . ¬ 噛 Ž log Ž I1a1 q ⭈⭈⭈ qIna n . _ log Ž m NŽ a1q ⭈⭈⭈ qa n . . .

131

LENGTHS OF TORS

is quasipolynomial. Let Ž x ␤ i, 1 , . . . , x ␤ i, k i . be a set of generators for Ii . The set S␣ s logŽ I1a1 q ⭈⭈⭈ qIna n . _ logŽ m NŽ a1q ⭈⭈⭈ qa n . . Žwhere ␣ s Ž a1 , . . . , a n .. can be defined as the set of all ␥ s Ž g 1 , . . . , g d . satisfying the ⺪-linear inequalities;

␥

Ý d i , j ␤i , j

G

i, j

Ý d1, j

s

l 1 a1

.. . s

l n an

s

1

-

N Ž a1 q ⭈⭈⭈ qa n .

j

Ý dn, j j

Ý li i

Ý gi i

for some l i , d i, j g ⺞. The next-to-last equation is equivalent to the requirement that l i⬘ s 1 for a single i⬘, and that l i s 0 for all i / i⬘. Together with the previous n equations, this implies that Ý j d i⬘, j s a i⬘ for some i⬘, and that d i, j s 0 for any i / i⬘. The first equation then degenerates to the statement that there exists an i⬘ such that ␥ G Ý i⬘, j d i⬘, j ␤i⬘, j for some d i⬘, j with Ý j d i⬘, j s a i⬘. This in turn is true iff x ␥ g Ii⬘a i⬘ for some i⬘. A monomial x ␥ is contained in a sum of monomial ideals iff it is contained in one of the ideals, so we see that ␥ satisfies all but the last of the equations above iff x ␥ g I1a1 q ⭈⭈⭈ qIna n . Finally, the last equation imposes the requirement that x ␥ not be contained in m NŽ a1 , . . . , a n .. We can introduce auxiliary variables ⑀ s Ž e1 , . . . , e d . g ⺞ d and f g ⺞ and rewrite these inequalities as equalities: y␥ q

Ý ␤i , j d i , j q ⑀

s

0

s

0

.. . s

0

s

0

s

0.

i, j

yl 1 a1 q

Ý d1, j j

yl n a n q

Ý dn, j j

y1 q

Ý li i

y Ý Nai q i

Ý gi q f q 1 i

132

J. BRUCE FIELDS

A solution to these equations is an element of ⺞ 2 nqÝ i k iq2 dq1 of the form

Ž a1 , . . . , an , I1 , . . . , l n , d1, 1 , . . . , d n , k

n

, e1 , . . . , e d , f , g 1 , . . . , g d . .

Let T ; ⺞ 2 nqÝ i k iq2 dq1 be the set of all such solutions. For brevity’s sake, we will write Ž ␣ , ␭, ␦ , ⑀ , f, ␥ . for a typical element of T. By a fundamental theorem of integer programming Žsee, e.g., Section I.3 of w24x., T is a finitely generated monoid, and the monoid algebra A s K w T x is Noetherian. Make A an ⺞ n-graded algebra by giving the element Ž ␣ , ␭, ␦ , ⑀ , f, ␥ . degree ␣ . Form the ideal I generated by all elements Ž ␣ , ␭, ␦ , ⑀ , f, ␥ . y Ž ␣ ⬘, ␭⬘, ␦ ⬘, ⑀ ⬘, f ⬘, ␥ ⬘. satisfying ␣ s ␣ ⬘ and ␥ s ␥ ⬘. This ideal is homogeneous, so the algebra ArI is also ⺞ n-graded. Given a degree ␣ ⬘ g ⺞ d , the degree-␣ ⬘ part of ArI has a K-basis consisting of elements of the form Ž ␣ , ␭, ␦ , ⑀ , f, ␥ . q I satisfying ␣ ' ␣ ⬘. Such a K-basis can be put in one-to-one correspondence with the elements of S␣ , with Ž ␣ , ␭, ␦ , ⑀ , f, ␥ . q I mapping to ␥ g S␣ . Therefore the function f 0 is the Hilbert function of ArI and is quasipolynomial by Theorem 3.1. The degree bound follows by the same argument used in Lemma 6.4. In the case n s 2, where there are only two ideals I1 and I2 , we know already that the functions f␹ and f k for k G 2 are quasipolynomial. COROLLARY 7.2. then

If I1 , I2 are monomial ideals in R s K ww x 1 , . . . , x d xx,

f k Ž a1 , a2 . s length Tork

ž ž

R I1a1

,

R I2a 2

//

is a quasipolynomial function for all k G 0.

REFERENCES 1. P. B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge Philos. Soc. 53 Ž1957., 568᎐575. 2. W. C. Brown, Hilbert functions for two ideals, J. Algebra, to appear. 3. W. C. Braun, ‘‘Hilbert Functions for Two Ideals ii,’’ preprint. 4. R.-O. Buchweitz and Q. Chen, Hilbert᎐Kunz functions of cubic curves and surfaces, J. Algebra 197 Ž1997., 246᎐267. 5. P.-J. Cahen and J.-L. Chabert, ‘‘Integer-Valued Polynomials,’’ Amer. Math. Soc., Providence, 1997. 6. A. Conca, Hilbert᎐Kunz function of monomial ideals and binomial hypersurfaces, Manuscripta Math. 90, No. 3 Ž1996., 287᎐300. 7. M. Contessa, On the Hilbert᎐Kunz function and Koszul homology, J. Algebra 175 Ž1995., 757᎐766. 8. H. Gillet and C. Soule, et nullite ´ K-theorie ´ ´ des multiplicites ´ d’intersection, C. R. Acad. Sci. Paris Ser. I Math. 300, No. 3 Ž1985., 71᎐74.

LENGTHS OF TORS

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9. S. Goto, K. Nishida, and K. Watanabe, Non-Cohen᎐Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsik’s question, Proc. Amer. Math. Soc. 120, No. 2 Ž1994., 383᎐392. 10. C. Han and P. Monsky, Some surprising Hilbert᎐Kunz functions, Math. Z. 214, No. 1 Ž1993., 119᎐135. 11. M. Herrmann, E. Hyry, J. Ribbe, and Z. Tang, Reduction numbers and multiplicities of multigraded structures, J. Algebra 197 Ž1997., 311᎐341. 12. M. Hochster, Tight closure in equal characteristic, big Cohen᎐Macaulay algebras, and solid closure, in ‘‘Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra ŽSouth Hadley, MA, 1992.,’’ pp. 173᎐196, Amer. Math. Soc., Providence, 1994. 13. M. Hochster and C. Huneke, Tight closure, invariant theory, and the Brianc¸on᎐Skoda theorem, J. Amer. Math. Soc. 3, No. 1 Ž1990., 31᎐116. 14. A. G. Khovanskiı, ˇ Sums of finite sets, orbits of commutative semigroups and Hilbert functions, Funktsional. Anal. i Prilozhen. 29 Ž1995., No. 2, 36᎐50, 95. 15. P. Monsky, The Hilbert᎐Kunz function, Math. Ann. 263, No. 1 Ž1983., 43᎐49. 16. P. Monsky, The Hilbert᎐Kunz function of a characteristic 2 cubic, J. Algebra 197 Ž1997., 268᎐277. 17. P. Monsky, Hilbert᎐Kunz functions in a family: Line-S 4 quartics, J. Algebra 208 Ž1998., 359᎐371. 18. P. Monsky, Hilbert᎐Kunz functions in a family: Point-S4 quartics, J. Algebra 208 Ž1998., 343᎐358. 19. P. Roberts, The vanishing of intersection multiplicities of perfect complexes, Bull. Amer. Math. Soc. Ž N.S.. 13, No. 2 Ž1985., 127᎐130. 20. P. Roberts, An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert’s fourteenth problem, J. Algebra 132 Ž1990., 461᎐473. 21. P. C. Roberts, A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian, Proc. Amer. Math. Soc. 94, No. 4 Ž1985., 589᎐592. 22. G. Seibert, The Hilbert᎐Kunz function of rings of finite Cohen᎐Macaulay type, Arch. Math. Ž Basel . 69, No. 4 Ž1997., 286᎐296. 23. J.-P. Serre, ‘‘Algebre locale. Multiplicites,’’ ` ´ Springer-Verlag, Berlin, 1965; Cours au College ` de France, 1957᎐1958, redige ´ ´ par Pierre Gabriel, 2nd ed., 1965; Lecture Notes in Mathematics, 11. 24. R. P. Stanley, ‘‘Combinatorics and Commutative Algebra,’’ 2nd ed., Birkhauser, Boston, ¨ 1996. 25. R. P. Stanley, ‘‘Enumerative Combinatorics,’’ Vol. 1, Cambridge Univ. Press, Cambridge, UK, 1997, with a foreword by Gian-Carlo Rota, corrected reprint of the 1986 original. 26. J. K. Verma, D. Katz, and S. Mandal, Hilbert functions of bigraded algebras, in ‘‘Commutative Algebra ŽTrieste, 1992.,’’ pp. 291᎐302, World Scientific, River Edge, NJ, 1994.