Length functions determined by killing powers of several ideals in a local ring

by

J. Bruce Fields

A dissertation submitted in partial fulfillment

of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2000

Doctoral Committee:

Professor Melvin Hochster, Chair Associate Professor Alexandre Barvinok Assistant Professor Claudia Miller Professor Lawrence Sklar Associate Professor Karen Smith

ACKNOWLEDGEMENTS

Thanks to Mel Hochster.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii CHAPTER

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

II. Quasipolynomial Functions and ℕ^{n}-graded modules . . . . . . . . . . . . . . 9 2.1 Quasipolynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 ℕ^{n}-graded algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

III. Results about Tor_{k}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}) . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1 Tor_{k}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}) for k ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Tor_{1}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Tor_{0}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

IV. Monomial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1 The case n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 The case n ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 A. Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

CHAPTER I

Introduction

1.1 Motivation

Intuition suggests that the parabola y = x^{2} intersects the line y = 0 more closely than does the line y = x. This intuition can be made more concrete by associating with each isolated point of intersection of two plane curves a number called the intersection multiplicity of the curves at that point. The intersection multiplicity at a point P is, roughly speaking, equal to the number of points of intersection you would expect to get near P if you perturbed the intersecting varieties slightly. For example, if ϵ is small (but nonzero), then y = x^{2} and y = 0 + ϵ intersect in 2 points nearby the origin, so the intersection multiplicity of these two curves is 2. On the other hand, y = x and y = ϵ only intersect once, so the intersection multiplicity is 1 in this case.

There is also a more algebraic definition. Definition 1. Let f, g ∈ ℂ[x, y] be polynomials defining plane curves in A^{2}(ℂ). Suppose the origin is an isolated point of the intersection of these curves. Define the intersection multiplicity of f and g at the origin to be

χ(f, g) = length (\frac{ℂ[x, y]_{(x,y)}}{(f, g)} ) .

Here length(ℂ[x, y]_{(x,y)})/(f, g) is the vector space dimension of ℂ[x, y]_{(x,y)} over ℂ.

More generally, an arbitrary module M over a ring R is said to have finite length if the length l of a chain of distinct submodules 0 = M_{0} ⊊ ⋯ ⊊ M_{l} = M is bounded; if M has finite length, the length of M is the maximum value of l over all such chains of submodules.

For example, if f = y − x^{2} and g = y, then

\frac{ℂ[x, y]_{(x,y)}}{(f, g)} ≅ \frac{ℂ[x, y]}{(y − x^{2}, y)} ≅ \frac{ℂ[x]}{(x^{2})},

and the elements 1 and x form a basis for this ring as a ℂ-vector space, so the intersection multiplicity of y = x^{2} and y = 0 is, once again, 2.

More generally, it should be possible to define an intersection multiplicity for a pair of varieties in A^{n}(ℂ) meeting in an isolated point. Let R be the ring of local functions at the point of intersection, let m be the unique maximal ideal of R, and let I and J be the ideals of R corresponding to the two varieties. The requirement that the point of intersection be an isolated point forces the ideal I + J to be m-primary, which is equivalent to requiring that m be the only prime ideal containing I + J, that some power of m be contained in I + J, or that R/(I + J) have finite length. Since R/(I + J) has finite length, it would seem natural to define the intersection multiplicity to be this length; unfortunately, this turns out not to be a satisfactory definition.

The definition that works is due to Serre ([25]), and requires that we define functors Tor_{k}^{R} which give an R-module Tor_{k}^{R}(M, N) for any two R-modules M and N. In particular, the functor Tor_{0}^{R} is just the tensor product, so that Tor_{0}^{R}(M, N) ≅ M ⊗_{R} N. The intersection multiplicity of the varieties defined by the ideals I and J is defined to be the alternating sum

χ (\frac{R}{I}, \frac{R}{J}) = ∑_{k≥0}(−1)^{k} length (Tor_{k}^{R} (\frac{R}{I}, \frac{R}{J})) .

The fact that I + J is m-primary assures that these lengths are finite, and the fact that R is a localization of a polynomial ring implies that the Tor_{k}’s are eventually zero. Therefore this sum is well-defined. It turns out that all of the Tor_{k}’s for positive k are zero in the case where I and J both define plane curves, so in that case the intersection multiplicity is equal to the first term of the alternating sum, which is the length of

Tor_{0}^{R} (\frac{R}{I}, \frac{R}{J}) ≅ \frac{R}{I} ⊗_{R} \frac{R}{J} ≅ \frac{R}{I + J} .

This is just Definition 1.

As is the case with plane curves, this definition of the intersection multiplicity tells how many points of intersection should be expected nearby the given point when the two varieties are perturbed slightly. However, Serre’s definition also makes sense, more generally, when the ring of local functions on A^{n}(ℂ) is replaced by an arbitrary regular local ring, and when R/I and R/J are replaced by finitely generated Rmodules M and N. In such a situation, define the intersection multiplicity of M and N to be

χ(M, N) = ∑_{k≥0}(−1)^{k} length(Tor_{k}^{R}(M, N)).

Let d and e be the Krull dimensions of M and N and let n be the dimension of R. Thinking of d and e as corresponding to the dimensions of two intersecting varieties, and n as corresponding to the dimension of the ambient space, geometric intuition would suggest the following conjectures, originally due to Serre.

i. If length(M ⊗_{R} N) is finite, then d + e ≤ n; if the varieties corresponding to M and N are too “large”, then we would expect their intersection to be larger than an isolated point.

ii. If d + e < n, then χ(M, N) = 0; the varieties corresponding to M and N are too “small”, and we would expect them to cease intersecting completely if they were perturbed slightly.

iii. If d + e = n, then χ(M, N) > 0. In this case we’d expect the corresponding

varieties to continue to intersect after being moved slightly. Serre proved the first statement, but proved the other two only for equicharacter-

istic regular local rings ([25]). The second has been established for arbitrary regular local rings by P. Roberts ([20]) and, independently, by H. Gillet and C. Soulé ([8]). The third statement is still open in this generality, although O. Gabber has shown at least that χ(M, N) ≥ 0 (see [23] or [1]).

Serre’s definition of the intersection multiplicity can also be generalized to the case where more than two varieties intersect; to do so requires defining the functors Tor_{k} for more than two modules. We present the definition in this generality below. This definition of intersection multiplicity proves again to be equivalent, in the case of complex varieties, to a definition that depends on counting the number of points of intersection after the varieties are moved slightly.

In this thesis we investigate functions

f_{k} = length(Tor_{k}(M_{1}/I_{1}^{a_{1}}M_{1}, …, M_{n}/I_{n}^{a_{n}}M_{n})),

where R is an arbitrary local ring (not necessarily regular); M_{i} is a finitely generated R-module and I_{i} an ideal of R, for 1 ≤ i ≤ n; and I_{1}+⋯+I_{n}+Ann M_{1}+⋯+Ann M_{n} is primary to the maximal ideal of R.

Special cases of these functions have already been found to be 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 I_{1} and I_{2} using only properties of the function

f_{0}(a_{1}, a_{2}) = length(R/(I_{1}^{a_{1}} + I_{2}^{a_{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 particular, they are shown to have rational generating functions in the following cases:

i. n = 2, M_{1} = M_{2} = R, k ≥ 2; ii. n = 2, M_{1} = M_{2} = R, k is arbitrary, R is regular, and

⨁_{a_{1},a_{2}≥0}(I_{1}^{a_{1}} ∩ I_{2}^{a_{2}})t_{1}^{a_{1}}t_{2}^{a_{2}} ⊂ R[t_{1}, t_{2}]

is a finitely generated R-algebra; iii. M_{1}, …, M_{n} = R is a polynomial or power series ring, I_{1}, …, I_{n} are principal

monomial ideals, and k = 0; more generally, a weaker result is proved in the case when I_{1}, …, I_{n} are monomial but not necessarily principal.

1.2 Basic Definitions

All rings are commutative, and local rings are in addition assumed to be Noetherian. Definition 2. Given n modules M_{1}, …, M_{n} over a ring R, define the R-module Tor_{k}^{R}(M_{1}, …, M_{n}) by

i. choosing a projective resolution

⋯ → P_{i,3} → P_{i,2} → P_{i,1} → P_{i,0} → M_{i} → 0

for each M_{i}, ii. tensoring together all of these projective resolutions (with the modules M_{i} re-

moved) and taking the total complex,

⋯ → ⨁_{\begin{matrix}(i_{1},... ,i_{n}) \\ ∑_{j} i_{j}=2 \end{matrix}} P_{1,i_{1}} ⊗⋯⊗P_{n,i_{n}} → ⨁_{\begin{matrix}(i_{1},... ,i_{n}) \\ ∑_{j} i_{j}=1 \end{matrix}} P_{1,i_{1}} ⊗⋯⊗P_{n,i_{n}} → P_{1,0} ⊗⋯⊗P_{n,0} → 0,

and

iii. taking the kth homology of this complex.

This defines a functor, covariant in each of the n variables, which specializes for n = 2 to the usual Tor. It shares most of the basic properties with the usual 2-variable Tor; for example,

i. A short exact sequence in any of the variables gives the usual long exact sequence.

ii. Any element of R that kills one of M_{i} also kills Tor_{k}(M_{1}, …, M_{n}).

iii. Tor_{0}^{R}(M_{1}, …, M_{n}) = M_{1} ⊗_{R} ⋯ ⊗_{R} M_{n}.

iv. If all but one of the M_{i} is flat, then Tor_{k}(M_{1}, …, M_{n}) = 0 for all k > 0.

v. If the M_{i} are all finitely generated over R, so is Tor_{i}(M_{1}, …, M_{n}).

Definition 3. Let R be a Noetherian ring, n a positive integer, M_{1}, …, M_{n} finitely generated R-modules, and I_{1}, … I_{n} ideals of R. Let J = I_{1} + ⋯ + I_{n} + Ann_{R}(M_{1}) + ⋯+Ann_{R}(M_{n}), and assume that R/J has finite length. For k a non-negative integer, define

f_{k}(a_{1}, …, a_{n}) = length(Tor_{k}(M_{1}/I_{1}^{a_{1}}M_{1}, …, M_{n}/I_{n}^{a_{n}}M_{n})).

Note that the modules Tor_{k}(M_{1}/I_{1}^{a_{1}}M_{1}, …, M_{n}/I_{n}^{a_{n}}M_{n})) do have finite length, since they are finitely generated modules that are killed by the ideal J′ = I_{1}^{a_{1}} + ⋯ + I_{n}^{a_{n}} + Ann M_{1} + ⋯ + Ann M_{n}, and hence are finitely generated modules over R/J′, which has finite length. (This is because J′ has the same radical as the ideal J above, and R/J is assumed to be finite length.)

In this thesis, the ring R will always be local, and the modules M_{1}, …, M_{n} will almost always be equal to R. 1.3 Examples

i. If n = 1, then f_{0} is the usual Hilbert function, and is eventually a polynomial of degree d = dim R.

ii. In [3] and [4], W. Brown studies f_{0} in the case n = 2 and M = R; he calls this function the Hilbert function of I_{1} and I_{2}. In [3], he gives sufficient conditions for f_{0} to be eventually polynomial (in other words, for there to exist (b_{1}, b_{2}) such that f_{0}(a_{1}, a_{2}) agrees with a polynomial for all (a_{1}, a_{2}) ≥ (b_{1}, b_{2})). In [4], he considers the case where f_{0} is eventually a polynomial in a_{1}, a_{2}, and min(a_{1}, a_{2}). Both papers also provide several examples of such functions.

iii. If M_{1} = ⋯ = M_{N} = M, I_{1}, …, I_{n} are all principal ideals, and R has characteristic p > 0, then the function HK_{M,I} : a ↦ f_{0}(p^{a}, …, p^{a}) is called the Hilbert-Kunz function on M of the ideal I = I_{1} + ⋯ + I_{n}. The more standard definition is in terms of bracket powers,

I^{[p^{e}]} = ({i^{p^{e}} i ∈ I}).

With this notation, HK_{M,I}(a) = length(M/I^{[p^{e}]}M). Because R has characteristic p, I^{[p^{e}]} = I_{1}^{p^{e}} + ⋯ + I_{n}^{p^{e}}, which shows that HK_{M,I} is dependent only on I and not on the choice of I_{1}, …, I_{n}.

Monsky has shown ([16]) that this function has the form

e_{HK}(M, I)p^{da} + O(p^{a(d−1)})

where d = dim M and e_{HK}(M, I) is a positive real constant. The constant e_{HK}(R, m) may be written e_{HK}(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}(R) is rational. However, many results have been proved in particular cases; see [16], [5], [17], [11], [19], [18], [6], and [24]. 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^{∗} = J^{∗} if and only if e_{HK}(R, I) = e_{HK}(R, J). This statement is true when R is a complete local domain, and in somewhat more generality. (See [14], thm. 8-17, or see the more thorough discussion in [13])

iv. In [7], M. 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. In this thesis, we similarly find that results for k > 0, in addition to being interesting in their own right, have applications to the case k = 0.

CHAPTER II

Quasipolynomial Functions and ℕ^{n}-graded modules

Given the functions f_{k} : ℕ^{n} → ℕ defined previously, there are corresponding generating functions

F_{k}(x_{1}, …, x_{n}) = ∑_{α=(a_{1},... ,a_{n})∈ℕ^{n}} f_{k}(α)x^{α}.

Later we will prove that, under certain conditions, these generating functions are rational. In fact, we will prove that the functions f_{k} are quasipolynomial, a somewhat stronger condition. The first section of this chapter discusses the basic properties of quasipolynomial functions.

Given a ℕ^{n}-graded module M having finite length in each graded piece, there is a natural notion of a Hilbert function ℕ^{n} → ℕ given by

(a_{1}, …, a_{n}) ↦ length(M_{(a_{1},... ,a_{n})}).

For n = 1 this is the usual Hilbert function. The second part of this chapter is devoted to showing that certain ℕ^{n}-graded algebraic objects (including, for example, finitely generated modules over finitely generated algebras over fields) have Hilbert functions which are quasipolynomial functions. This fact will be used in proofs of the results of the following chapter.

2.1 Quasipolynomial Functions

Most of the results of this section are known, though not perhaps in the form we need them; see, for example, [15].

Most of the functions considered here are defined on ℕ^{n}. However, it will be convenient to identify such functions with functions defined on all of ℤ^{n} that are zero for elements of ℤ^{n} not in ℕ^{n}. Definition 4. Given linearly independent vectors α_{1}, …, α_{e} ∈ ℕ^{n}, say that a function f : ℕ^{n} → ℤ is periodic with respect to α_{1}, …, α_{e} if, for any i, the function α ↦ f(α + γ) − f(α) is identically zero on ℕ^{n}, for γ any element of ℕα_{1} + ⋯ + ℕα_{e}. Definition 5. Given β ∈ ℕ^{n} and linearly independent vectors α_{1}, …, α_{e} ∈ ℕ^{n}, the cone with vertex β generated by α_{1}, …, α_{e} is

C_{β,α_{1},... ,α_{e}} = β + ℝ_{≥0}α_{1} + ⋯ + ℝ_{≥0}α_{e} ⊂ ℝ_{≥0}^{n}.

Definition 6. Let α_{1}, …, α_{e} ∈ ℕ^{n} be linearly independent vectors in ℝ^{n}, and let β be another element of ℕ^{n}. Let C = C_{β,α_{1},... ,α_{e}} be the cone with vertex β generated by α_{1}, …, α_{e}. Call f : ℕ^{n} → ℤ simple quasipolynomial of polynomial degree d on β, α_{1}, …, α_{e} if f(x) = 0 for x ∉ C and

f(x) = ∑_{α∈ℕ^{n}} c_{α}(x)x^{α}

for x ∈ C, where c_{α} is periodic with respect to α_{1}, …, α_{e}, is identically zero for all α = (a_{1}, …, a_{n}) such that ∑ a_{i} > d, and is not identically zero for at least one α satisfying ∑ a_{i} = d.

Note that polynomials of degree d are examples of simple quasipolynomials of diagonal degree d. We use the term “polynomial degree” and not just “degree” to distinguish it from another notion of degree.

Definition 7. The cumulative degree of a simple quasipolynomial of polynomial degree d on β, α_{1}, …, α_{e} is d + 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 8. A function f : ℕ^{n} → ℤ is simple quasipolynomial of cumulative degree 0 if it is nonzero on at most a single element of ℕ^{n}.

Lemma 9. Let F (x_{1}, …, x_{n}) be the generating function of f : ℕ^{n} → ℤ, fix β ∈ ℕ^{n}, and fix a linearly independent set of vectors α_{1}, …, α_{e} ∈ ℕ^{n}. Suppose

F (x_{1}, …, x_{n}) = \frac{1}{∏_{j=1}^{e}(1 − x^{α_{j}})^{d_{j}}}

with ∑_{j=1}^{e} d_{j} ≤ d. Then f is the simple quasipolynomial function of cumulative degree d given by

f(a_{1}, …, a_{n}) = \begin{cases}∏j \binom{d_{j}−1+m_{j}}{m_{j}} & if (a_{1}, …, a_{n}) = ∑_{j} m_{j}α_{j}, \\ 0 & otherwise. \end{cases}

Proof. Given α ∈ ℕ^{n}, there is at most one way of writing α as an ℕ-linear combination of α_{1}, …, α_{e}. This means that the coefficient of x^{α} in F is nonzero if and only if α can be written as the ℕ-linear combination of α_{1}, …, α_{e}. If α is such an exponent, write α = ∑ m_{j}α_{j}. Then the coefficient of x^{α} in F is the product of the coefficients of x^{m_{j}α_{j}} in 1/(1 − x^{α_{j}}), and it is easy to verify that these coefficients are just the binomial coefficients given in the description of f above.

Definition 10. Given a polynomial P (x_{1}, …, x_{n}), define the support of P, Supp P, to be the set of all α ∈ ℕ^{n} such that the coefficient of x^{α} in P is nonzero.

Definition 11. Given linearly independent vectors α_{1}, …, α_{e} ∈ ℕ^{n} and another vector β ∈ ℕ^{n}, let

Π_{β,α_{1},... ,α_{e}} = {α ∈ ℕ^{n} α = β + ∑_{i} m_{i}α_{i}, 0 ≤ m_{i} < 1} .

Given any α ∈ C_{β,α_{1},... ,α_{e}} ∩ ℕ^{n}, there is a unique representative of α modulo α_{1}, …, α_{e} in Π_{β,α_{1},... ,α_{e}}; in the following, we use \overline{α} to denote that unique representative. Lemma 12. Let F (x_{1}, …, x_{n}) be the generating function of f : ℕ^{n} → ℤ, fix β ∈ ℕ^{n}, and fix a linearly independent set of vectors α_{1}, …, α_{e} ∈ ℕ^{n}. Suppose

F (x_{1}, …, x_{n}) = \frac{P}{∏_{j=1}^{e}(1 − x^{α_{j}})^{d_{j}}}

with ∑_{j=1}^{e} d_{j} ≤ d and Supp P ⊂ Π_{β,α−1,... ,α_{e}}. Write P = ∑_{α} c_{α}x^{α}. Then f is the simple quasipolynomial function of cumulative degree d given by

f(α) = \{∏_{j} c\overline{α}\binom{d_{j}−1+m_{j}}{m_{j}} if α = \overline{α} + ∑_{j} m_{j}α_{j}.

Proof. The generating function F can be written as the sum of the fractions

\frac{c_{α}x^{α}}{∏_{j=1}^{e}(1 − x^{α_{j}})^{d_{j}}} .

The result then follows immediately from the previous lemma.

Lemma 13. Fix β ∈ ℕ^{n}, and fix a linearly independent set of vectors α_{1}, …, α_{e} ∈ ℕ^{n}. Then functions of the form

F (x_{1}, …, x_{n}) = \frac{P}{∏_{j=1}^{e}(1 − x^{α_{j}})^{d_{j}}}

with Supp P ⊂ Π = Π_{β,α_{1},... ,α_{e}}, and ∑_{j=1}^{e} d_{j} ≤ d, form a basis for the generating functions of the simple quasipolynomial functions of cumulative degree at most d on β, α_{1}, …, α_{e}.

Proof. This is an immediate consequence of the previous lemma and of the fact that the functions

(a_{1}, …, a_{e}) ↦ ∏_{j} \binom{d_{j} − 1 + a_{j}}{a_{j}},

with ∑(d_{j} − 1) at most d, form a basis for the polynomials of degree at most d.

Theorem 14. Let f_{1} be simple quasipolynomial on β, α_{1}, …, α_{e} of degree d. Pick β′ ∈ ℕ^{n} and linearly independent α′_{1}, …, α′_{e′} such that

C′ = C_{β′,α′_{1},... ,α′_{e′}} ⊂ C = C_{β,α_{1},... ,α_{e}}.

Define a new function f′ such that f′(α) = α for α ∈ C′ and f′(α) = 0 for α ∉ C′. Then there exist positive integers m_{i} such that f′ is simple quasipolynomial of degree at most d on β′, m_{1}α′_{1}, …, m_{e′}α′_{e′}.

Proof. For each α′_{i}, choose m_{i} such that m_{i}α′_{i} ∈ ℕα_{1} + ⋯ ℕα_{e}. Since f is simple quasipolynomial, we know

f(x) = ∑_{α∈ℕ^{n}} c_{α}(x)x^{α}

for x ∈ C, where c_{α} is periodic with respect to α_{1}, …, α_{e}, is identically zero for all α = (a_{1}, …, a_{n}) such that ∑ a_{i} > d. Since the c_{α}’s are periodic with respect to α_{1}, …, α_{e}, they are also periodic with respect to m_{1}α′_{1}, …, m_{e′}α_{e′}. Therefore f′ is also simple quasipolynomial.

Theorem 15. If f_{1} and f_{2} are both simple quasipolynomial functions on β, α, …, α_{e}, of cumulative degree d_{1} and d_{2} respectively, then f_{1} + f_{2} is also simple quasipolynomial on β, α_{1}, …, α_{e}, of degree at most max(d_{1}, d_{2}).

Proof. The proof is immediate from the definition of a simple quasipolynomial function.

Theorem 16. If f : ℕ^{n} → ℕ is polynomial of degree d, then

f^{∗}(a) = ∑_{a_{1}+···+a_{n}=a} f(a_{1}, …, a_{n})

is polynomial of degree d + n − 1. Proof. The set of generating functions of polynomials of degree at most d has a basis consisting of functions of the form

(x_{1}, …, x_{n}) ↦ \frac{1}{∏_{i=1}^{n}(1 − x_{i})^{d_{i}}}

where d_{i} ≥ 1 and ∑_{i}(d_{i} − 1) ≤ d. Given a function f with generating function F, the generating function of f^{∗} is F (x, …, x). Under this transformation the basis element above becomes

\frac{1}{(1 − x)^{∑ d_{i}}} ,

which is clearly the generating function of a polynomial of degree at most d + n − 1.

Thus f^{∗} is a polynomial of degree at most d + n − 1; it only remains to prove that f^{∗} has at least degree d + n − 1. To do this, it suffices to show that f_{d}^{∗} has degree d + n − 1, where f_{d} is the degree d part of f. We will prove this by bounding f_{d}^{∗} below by a polynomial of degree d + n − 1.

If f_{d}(a_{1}, …, a_{n}) < 0 for some (a_{1}, …, a_{n}) ∈ ℕ^{n}, then the polynomial function

a ↦ f(aa_{1}, …, aa_{n}) = f_{d}(aa_{1}, …, aa_{n}) + lower order terms

has negative leading term, and must eventually take on negative values. This is contrary to our hypotheses, so such (a_{1}, …, a_{n}) must not exist, and f_{d} : ℕ^{n} → ℕ must take on only nonnegative values. Furthermore, we can choose (a_{1}, …, a_{n}) ∈ ℝ_{≥0}^{n} such that ∑ a_{i} = 1 and f_{d}(a_{1}, …, a_{n}) > 0; if f_{d} were zero for every such (a_{1}, …, a_{n}) then it would be identically zero. In fact, the subset of

A = {(a_{1}, …, a_{n}) ∈ ℝ_{≥0}^{n} ∑_{i} a_{i} = 1}

on which f_{d} is zero is a proper algebraic subset of A. Therefore there is a closed (n − 1)-dimensional disk Δ ⊂ A of radius δ around the chosen point (a_{1}, …, a_{n}) such that f_{d}(β) ≥ c for c a fixed, positive real number and β any element of Δ.

Let a be any positive integer, and let α be any element of Δ. Then f_{d}(aα) = a^{d}f_{d}(α) ≥ ca^{d}. Therefore f_{d} is bounded below by a^{d} on the set aΔ = {aα α ∈ Δ}. By the same reasoning as that used in Lemma 50, the number of points in ℕ^{n} ∩ a^{d} is greater than or equal to the volume of (a − b)Δ for a fixed constant b ∈ ℝ. This volume is a polynomial V (a) of degree n − 1 in a. Therefore

f_{d}^{∗}(a) = ∑ ; ∑ a_{i}=a f_{d}(a_{1}, …, a_{n})
≥ ∑_{(a_{1},... ,a_{n})∈ℕ^{n}∩(a−b)Δ}f_{d}(a_{1}, …, a_{n})
≥ ∑_{(a_{1},... ,a_{n})∈ℕ^{n}∩(a−b)Δ}a^{d−1}c
≥ V (a)a^{d}c

Therefore f_{d}^{∗}(a) is bounded below by the product V (a)a^{d}c, which is a polynomial of degree d + n − 1.

Theorem 17. Given f : ℕ^{n} → ℕ simple quasipolynomial of cumulative degree d, define a new function f^{∗} : ℕ → ℕ by

f^{∗}(a) = ∑_{a_{1}+···+a_{n}=a} f(a_{1}, …, a_{n}).

Then f^{∗} is simple quasipolynomial of cumulative degree d.

Proof. Write F for the generating function associated with f, and F^{∗} for the generating function associated with f^{∗}. Then F^{∗} = F (x, …, x). By Theorem 13, F (x, …, x) is clearly the generating function of a simple quasipolynomial function if F (x, …, x) is. It is also clear that f has degree at most d. The fact that the degree of f is not less than d follows from the same fact for polynomials.

Definition 18. A function f : ℕ^{n} → ℤ is quasipolynomial of degree d if it can be written as the sum of simple quasipolynomial functions of cumulative degree at most d, and if it cannot be written as the sum of simple quasipolynomial functions of cumulative degree less than d.

Sometimes we will say that a function f : ℕ^{n} → ℤ is quasipolynomial on α_{1}, …, α_{e} ∈ ℕ^{n}; by this we mean that f can be written as the sum of simple quasipolynomial functions on subsets of α_{1}, …, α_{e}.

It might be more accurate to call such functions eventually quasipolynomial; this would make our usage agree, for example, with the definition given in 4.4 of [26].

It should be obvious from the definition that simple quasipolynomial functions of cumulative degree d are also quasipolynomial functions of degree d.

Theorem 19. If f_{1} : ℕ^{n} → ℤ and f_{2} : ℕ^{n} → ℤ are quasipolynomial, of degrees d_{1} and d_{2} respectively, then f_{1} + f_{2} is quasipolynomial of degree at most max(d_{1}, d_{2}). Also, if d_{1} ≠ d_{2}, then the degree of f_{1} + f_{2} is exactly max(d_{1}, d_{2}).

Proof. Immediate from the definition.

The following theorems provide a few examples of quasipolynomial functions.

Lemma 20. If f(α) is zero for all but finitely many values of α, then f is quasipolynomial of degree 0.

Proof. The proof follows immediately from the definition of a simple quasipolynomial of degree 0.

Theorem 21. A function f : ℕ → ℤ is quasipolynomial on 1 (in other words, is the sum of simple quasipolynomial functions on {β, 1}, for various β’s) of degree d iff it is eventually polynomial of degree d − 1.

Proof. A simple quasipolynomial function f(α) on β, 1 is a function that is zero for α < β, and that agrees for α ≥ β 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 − 1 is simple quasipolynomial of degree d. The difference between f and f′ is then nonzero for only finitely many values of ℕ, and is accounted for by the previous lemma.

Theorem 22. If a function f : ℕ → ℤ is quasipolynomial on e_{1}, …, e_{n} of degree d, where e_{i} is the vector in ℕ^{n} that is one in the ith position but zero elsewhere, then there exists α′ ∈ ℕ^{n} such that f(α) agrees with a polynomial of degree d − n for all α ≥ α′. (Here we take a polynomial of negative degree to be identically zero.)

Proof. A simple quasipolynomial function on a proper subset of e_{1}, …, e_{n} is easily seen to be eventually zero, whereas a simple quasipolynomial function on e_{1}, …, e_{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; if f is eventually polynomial but has nonpolynomial behavior on, for example, a finite set of lines extending parallel to one of the coordinate axes, then f is not quasipolynomial.

Theorem 23. The product of two quasipolynomial functions in distinct sets of variables is quasipolynomial of degree equal to the sum of the degrees of the two quasipolynomial functions. Proof. Let f(x_{1}, …, x_{m}) and g(y_{1}, …, y_{n}) be the two functions. Let f = f_{1} + ⋯ f_{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 = ∑_{i,j} f_{i}g_{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 (x_{1}, …, x_{m}) and G(y_{1}, …, y_{n}) for the generating functions of f and g, respectively. The generating function of fg is just F G and, by theorem 13, F G is the generating function of a simple quasipolynomial function of the correct degree.

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 24. Let F (x_{1}, …, x_{n}) be the generating function of f : ℕ^{n} → ℤ. Assume

F (x_{1}, …, x_{n}) = \frac{P (x_{1}, …, x_{n})}{∏_{j=1}^{e}(1 − x^{α_{j}})^{d_{j}}}

with α_{1}, …, α_{d} ∈ ℕ^{n} vectors that are linearly independent in ℝ^{n} and P a polynomial. Then f is quasipolynomial of cumulative degree at most ∑_{j=1}^{e} d_{j}.

Proof. The function F is the sum of fractions of the form

\frac{x^{γ}}{∏_{j=1}^{e}(1 − x^{α_{j}})^{d_{j}}}

Each of these is the generating function of a simple quasipolynomial function of degree d, by Theorem 13. Lemma 25. Let F (x) be a rational function of the form

F (x) = \frac{P (x)}{∏_{j=1}^{e}(1 − x^{α_{j}})^{d_{j}}} .

If the α_{j}’s are linearly dependent, then F (x) can be rewritten in the form

F (x) = ∑_{i} \frac{P_{i}(x)}{∏_{j=1}^{e_{i}}(1 − x^{α_{i,j}})^{d_{i,j}}} ,

such that each e_{i} is strictly less than e. Proof. Choose a linear dependence relation among the α_{j}’s. By clearing denominators and renumbering the α_{j}’s if necessary, this relation can be written

m_{1}α_{1} + ⋯ + m_{k}α_{k} = m_{k+1}α_{k+1} + ⋯ + m_{e}α_{e}

where each m_{k} is a non-negative integer. Write γ for the element of ℕ^{n} that both sides of this equation are equal to. Let d = 1+∑_{i=1}^{k}(d_{i}−1), and let d′ = 1+∑_{i=k+1}^{e}(d_{i}−1). Since

(1 − x^{mα_{i}}) = (1 − x^{α_{i}})(1 + x^{α_{1}} + x^{2α_{1}} + ⋯ + x^{(m−1)α_{1}}),

F can be rewritten in the form

\frac{P′(x)}{∏_{j=1}^{e}(1 − x^{m_{j}α_{j}})^{d_{j}}} ,

with P′ a new polynomial. Next write

F (x) = \frac{P′(x)(1 − x^{γ})^{d+d′}}{(1 − x^{γ})^{d+d′} ∏_{j=1}^{e}(1 − x^{m_{j}α_{j}})^{d_{j}}} ,  (2.1)

and note that

(1 − x^{γ}) = (1 − x^{∑_{i=1}^{k}(m_{i}α_{i})}) = (1 − x^{m_{1}α_{1}})x^{∑_{i=2}^{k}(m_{i}α_{i})}
+ (1 − x^{m_{2}α_{2}})x^{∑_{i=3}^{k}(m_{i}α_{i})}
+ ⋯
+ (1 − x^{m_{k}α_{k}}),

which shows that (1 − x^{γ}) is contained in the ideal generated by (1 − x^{m_{1}}), …, (1 − x^{m_{k}}). Similarly, (1 − x^{γ}) is also contained in the ideal generated by (1 − x^{m_{k+1}}), …, (1 − x^{m_{e}}). This fact, together with our choice of d and d′, allows us to rewrite (1 − x^{γ})^{d+d′} as the sum of terms each of which is contains one of both (1 − x^{m_{i}})^{d_{i}} for 1 ≤ i ≤ k and (1 − x^{m_{i}})^{d_{i}} for k < i ≤ e as factors. Finally we are able to write F in the desired form by expanding the numerator of Equation 2.1, breaking up the fraction into a sum of fractions, and canceling these factors.

Theorem 26. Let F (x_{1}, …, x_{n}) be the generating function of f : ℕ^{n} → ℤ. Then f is quasipolynomial iff F can be written in the form

F (x) = \frac{P (x)}{∏_{j=1}^{e}(1 − x^{α_{j}})^{m_{j}}} .

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 24. 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 Theorem 13, and the sum of such functions can clearly be written in the form required.

Theorem 27. Let f : ℕ^{n} → ℕ be quasipolynomial of degree d. Then

f^{∗}(a) = ∑_{a_{1}+···+a_{n}=a} f(a_{1}, …, a_{n})

is also quasipolynomial of degree d.

Proof. If f is quasipolynomial, and F (x_{1}, …, x_{n}) is its generating function, then the generating function of f^{∗} is F (x, …, x). The fact that f is quasipolynomial then follows immediately from Theorem 26. 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.

There must exist β, α_{1}, …, α_{e} such that f restricted to C_{β,α_{1},... ,α_{e}} is simple quasipolynomial of cumulative degree d. If not, then we could cover ℕ^{n} by cones in such a way that on each cone f was simple quasipolynomial (using Theorem 14) 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_{β,α_{1},... ,α_{e}}. Then f′ ≤ f, so f′^{∗} ≤ f^{∗}. By Theorem 17, f′^{∗} has degree d; it follows immediately from the definition of simple quasipolynomial that f^{∗} must have degree at least d.

Theorem 28. If f, g : ℕ^{n} → ℕ are quasipolynomial and f ≤ g, then deg f ≤ deg g.

Proof. The previous theorem allows us to reduce to the case n = 1. A quasipolynomial function of degree 1 is eventually equal to a polynomial with periodic coefficients, and the result is trivial for such functions.

Theorem 29. Let f : ℕ^{n} → ℤ, g : ℕ^{n} → ℤ, and β ∈ ℕ^{n} be such that f(α + β) − f(α) = g(α). If g is quasipolynomial, then so is f. In addition, if f : ℕ^{n} → ℕ, so all the values of f are nonnegative, then the cumulative 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(α + β) − f(α) = g(α) is equivalent to the relation F = G/(1 − x^{β}). 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 β. Then

F^{∗} = \frac{G^{∗}}{1 − x^{b}} ,

where F^{∗} and G^{∗} are the generating functions associated with, respectively, f^{∗} and g^{∗}. Thus by the previous theorem we can assume without loss of generality that n = 1.

It is not hard to see that in the case n = 1 a quasipolynomial function g is agrees eventually with a polynomial with periodic coefficients. Such a function is eventually bounded below by a function with generating function of the form

\frac{c}{(1 − 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

\frac{c}{(1 − x^{N})^{d}(1 − x^{b})},

and it follows that f has degree d + 1.

2.2 ℕ^{n}-graded algebras.

We now apply the results of the previous section to the problem of determining the Hilbert functions of certain ℕ^{n}-graded algebraic objects.

Theorem 30. Let S be an ℕ^{n}-graded algebra such that S_{0} is Artin and S is finitely generated as an algebra over S_{0}. Let M be a finitely generated, ℕ^{n}-graded S-module of dimension d. Then the Hilbert function f(α) = length(M_{α}) is quasipolynomial of degree d.

Proof. The proof will be by induction on d. If d = 0 then M has finite length, and length(M_{α}) is zero for all but finitely many α, 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 ∈ Spec(S).

We can map an ℕ^{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_{0}[s_{1}, …, s_{e}], with deg(s_{i}) = α_{i} ∈ ℕ^{n}.

Since d = dim M > 0, there must be some s_{i} not contained in P; otherwise M would have finite length. That s_{i} must be a non-zerodivisor on M. This means that the sequence

0 → M →^{s_{i}} M → \frac{M}{s_{i}M} → 0

is exact, so

length(M_{α}) − length(M_{α−α_{i}}) = length ((\frac{M}{s_{i}M} )_{α})
f(α) − f(α − α_{i}) = f_{M/s_{i}M}(α).

Since s_{i} is a non-zerodivisor on M, M/s_{i}M has dimension d − 1. Therefore, by the induction assumption, M/s_{i}M is quasipolynomial of degree d − 2, and f_{M} is quasipolynomial of degree d − 1.

It should be clear from the proof that if α_{1}, …, α_{e} are generators of S, then the generating function of the Hilbert function can be written as a rational function with denominator ∏_{i}(1 − x^{α_{i}}).

A few examples might be helpful.

i. If M is a finitely generated module over an ℕ-graded algebra S that is generated by elements of degree 1, then the Hilbert function f_{M} is quasipolynomial on {1}, and hence is eventually polynomial, as expected.

ii. If M is a finitely generated module over an ℕ^{2}-graded algebra S, and if every algebra-generator of S over S_{0} has degree (0, 1) or (1, 0), then S is quasipolynomial of degree d = dim S on (0, 1) and (1, 0). By Theorem 22, there exists a β such that, for all α ≥ β, f_{S}(α) is equal to a polynomial of degree at most d − 2. Thus we obtain Theorem 2 of [2], that Hilbert functions of bigraded modules are eventually polynomial of degree at most (dim M) − 2. More results about these functions can be found in [27].

iii. More generally, let M be a finitely generated module over a ℕ^{n}-graded algebra S, and assume every algebra-generator of S over S_{0} has degree e_{i}, where e_{i} is the vector in ℕ^{n} that is one in the ith position but zero elsewhere. Again, there is a β such that f_{M}(α) agrees with a polynomial of degree at most d − n for all α ≥ β. (See [12], where it is also shown that all the highest degree monomials of this polynomial have nonnegative coefficients.)

iv. If M = S = k[s_{1}, …, s_{e}], and if the degree of s_{i} is α_{i}, then the generating function of the Hilbert function f_{S} is

\frac{1}{∏_{i}(1 − x^{s_{1}})}

and f_{S} is quasipolynomial of degree d. It is not hard to see that any quasipolynomial function of the form

\frac{P (x)}{∏_{i}(1 − x^{α_{i}})}

such that P (x) has nonnegative coefficients can be written as the sum of translations 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 ℕ^{n}-graded modules.

Theorem 31. Let S be an ℕ^{n}-graded finite algebra over a local ring R, and let T be an ℕ^{n}-graded, finitely generated, S-algebra. Assume that S_{0} = T_{0} = R. Let N and M be finitely-generated ℕ^{n}-graded algebras over T and S, respectively, with N ⊃ M, and suppose that for every element of N there is a power of m_{R} multiplying that element into M. Then f_{N/M} : α → length((N/M)_{α}) is quasipolynomial of degree at most dim N = dim(T/(Ann_{T} N)).

Proof. We can filter N/M by N/T M and T M/M. The quotient N/T M is a T - module, not just an S-module. Also, N/T M is a finitely generated T -module, and since every element of N is multiplied into M (and hence into T M) by a power of m, it follows that there exists an n such that m^{n} kills every element of N/T M. Therefore N/T M can be thought of as a module over T/m^{n}T. This module now satisfies the conditions of the previous theorems, so the Hilbert function of N/T M is quasipolynomial. Since Ann_{T} N/T M ⊃ Ann T, this function is in addition quasipolynomial of degree at most dim N.

It remains only to show that T M/M has a quasipolynomial Hilbert function of degree at most dim N. So assume N = T M. Suppose that T is generated over S by r generators, so T = S[t_{1}, …, t_{r}]. Suppose that r is at least 2, and that the theorem has already been proved for smaller values of r. Apply the theorem with S[t_{1}] replacing S and S[t_{1}]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[t_{1}]M by r − 1 elements, so by assumption the Hilbert function of N/S[t_{1}]M is quasipolynomial. Similarly, apply the theorem with S and M as before, but T replaced by S[t_{1}] and N by S[t_{1}]M, and the conclusion is that S[t_{1}]M/M also has a quasipolynomial Hilbert function. The Hilbert function of N/M is the sum of the Hilbert functions of N/S[t_{1}]M and S[t_{1}]M/M, and sum of two quasipolynomial functions is quasipolynomial, so the result follows by induction if only we can deal with the case r = 1.

So assume N = S[t]M (where t = t_{1}). Construct the S-module

N′ = ⨁_{i=0}^{∞} \frac{t^{i+1}M + t^{i}M + ⋯ + tM + M}{t^{i}M + t^{i−1}M + ⋯ + tM + M} .

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

\frac{M}{M} ⊂ \frac{tM + M}{M} ⊂ \frac{t^{2}M + tM + M}{M} ⊂ ⋯,

and note that

⋃_{i=0}^{∞} \frac{t^{i}M + t^{i−1}M + ⋯ + M}{M} = \frac{N}{M} ,

thanks to our assumption that N = T M = S[t]M.

Note also that t(t^{i}M + ⋯ + M) ⊂ (t^{i+1}M + ⋯ + M), so multiplication by t induces a well-defined map

\frac{t^{i}M + ⋯ + M}{t^{i−1}M + ⋯ + M} → \frac{t^{i+1}M + ⋯ M}{t^{i}M + ⋯ + M} ,

and this in turn induces a well-defined map N′ → N′. Thus multiplication by t makes sense on N′, making N′ a T -module as well as an S-module.

In addition, N′ is generated over T by any set of generators for M, and in particular is finitely generated because M is. Any element of N′ is killed by a power of m_{R} because N′ is a direct sum of subquotients of N/M, which has the property that any element is killed by a power of m_{R}. Since N′ is a T -module, it must also be the case that any element of N′ is killed by a power of m_{R}T. Because N′ is finitely generated, it follows that N′ is killed by some fixed power of m_{R}T, hence is a finitely generated module over T/(m_{R}T)^{j} for some j. The fact that N′, and hence N/M, has a quasipolynomial Hilbert function, follows now from theorem 6. Note also that Ann_{T} N′ ⊃ Ann_{T} N, so dim N′ ≤ dim N, and N/M has a quasipolynomial Hilbert function of degree at most dim N.

CHAPTER III

Results about Tor_{k}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}})

This chapter applies the results of the previous chapter to prove theorems that describe the functions

f_{k}(a_{1}, …, a_{n}) = length(Tor_{k}(M_{1}/I_{1}^{a_{1}}M_{1}, …, M_{n}/I_{n}^{a_{n}}M_{n}))

in the case n ≥ 2, M_{1} = ⋯ = M_{n} = R. 3.1 Tor_{k}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}) for k ≥ 2

Let I be any ideal of a ring R. The short exact sequence 0 → I → R → R/I → 0 gives a long exact sequence

⋯ → Tor_{k}(I, M) → 0 → Tor_{k}(R/I, M) →
→ Tor_{k−1}(I, M) → 0 → Tor_{k−1}(R/I, M) → ⋯
⋯ → Tor_{0}(I, M) → M →  M/IM  → 0,

(3.1)

which yields an isomorphism Tor_{k}(R/I, M) ≅ Tor_{k−1}(I, M) for i ≥ 2 and an injection Tor_{1}(R/I, M) → Tor_{0}(I, M).

If I_{1} and I_{2} are two ideals of R, then Tor_{k}(R/I_{1}, R/I_{2}) ≅ Tor_{k−1}(I_{1}, I_{2}) for i ≥ 3, by two applications of the above isomorphisms. Since direct sums commute with Tor, there is also an isomorphism

⨁_{a_{1},a_{2}≥0} Tor_{k}^{R}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}) ≅ Tor_{k}^{R}(⨁_{a_{1}≥0} I_{1}^{a_{1}}, ⨁_{a_{2}≥0} I_{2}^{a_{2}})

for k ≥ 3. Let t_{1} and t_{2} be indeterminates; then we can replace ⨁_{a_{i}≥0} I_{i}^{a_{i}} by the algebra R[I_{i}t] = ⨁_{a_{i}≥0} I_{i}^{a_{i}}t^{a_{i}}. This allows us to impose the structure of an algebra on ⨁_{1_{1},a_{2}≥0} Tor_{k}^{R}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}).

Lemma 32. Let I_{1} and I_{2} be ideals of a ring R. Let S_{i} = R[I_{i}t_{i}] = ⨁_{a_{i}≥0} I_{i}^{a_{i}}t_{i}^{a_{i}}. Then the R-modules

⨁_{a_{1},a_{2}≥0} Tor_{k}^{R}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}})

can be given the structure of ℕ^{2}-graded, finitely generated S_{1} ⊗_{R} S_{2}-modules, for k ≥ 2.

Proof. For k > 2, we know that this direct sum is isomorphic to Tor_{k−2}^{R}(S_{1}, S_{2}). Let r_{j,1}, …, r_{j,m_{i}} generate I_{i}, and let T_{1} = R[x_{1,1}, …, x_{1,m_{1}}] and T_{2} = R[x_{2,1}, …, x_{2,m_{2}}] be polynomial rings, and map T_{i} onto S_{i} by mapping x_{i,j} onto r_{i,j}. This makes S_{i} into a T_{i}-module. Choose a resolution of S_{i} by finitely generated projective T_{i}-modules,

⋯ → P_{i,1} → P_{i,0} → S_{i}.

Since T_{i} is a free R-module, this resolution is also a resolution for S_{i} over R. Therefore Tor_{k}^{R}(S_{1}, S_{2}) is the homology of the total complex

⋯ → P_{1,1} ⊗_{R} P_{2,0} ⊕ P_{1,0} ⊗_{R} P_{2,1} → P_{1,0} ⊗_{R} P_{1,0} → 0.

The R-modules in this complex are easily seen to also be finitely generated T_{1} ⊗_{R} T_{2}- modules, and the maps T_{1} ⊗ T_{2}-maps. Therefore the homology modules Tor_{k}^{R}(S_{1}, S_{2}) are also finitely generated modules over T_{1} ⊗T_{2}. In addition, note that these modules must be killed by the kernels of the maps T_{i} → S_{i}; thus, each Tor_{k}^{R}(S_{1}, S_{2}) is also a finitely generated S_{1} ⊗ S_{2}-module.

For k = 2, apply direct sums to the end of the long exact sequence of Equation 3.1 above to get

0 → ⨁_{a_{1},a_{2}≥0} Tor_{2}^{R}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}) → R[I_{1}t_{1}] ⊗_{R} R[I_{2}t_{2}] → R[I_{1}t_{1}] ⊗_{R} R[t_{2}].

The map R[I_{1}t_{2}] ⊗_{R} R[I_{2}t_{2}] → R − I_{1}t_{1} ⊗_{R} R[t_{2}] is clearly a map of S_{1} ⊗_{R} S_{2}-modules, so the kernel is an S_{1} ⊗_{R} S_{2}-module.

Theorem 33. Let I_{1} and I_{2} be ideals of a local ring (R, m) such that I_{1} + I_{2} is m-primary. Then for k ≥ 2 the function

f_{k}(a_{1}, a_{2}) = length(Tor_{k}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}))

is quasipolynomial of degree at most 2d.

Proof. Fix k ≥ 2. Then ⨁_{a_{1},a_{2}} Tor_{k}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}) is isomorphic to (or, in the case of k = 2, is at least a submodule of) Tor_{k−2}(S_{1}, S_{2}). Since I + J is m-primary, I_{1}^{a_{1}} + I_{2}^{a_{2}} is also m-primary, so some power of m is contained in I_{1}^{a_{1}} + I_{2}^{a_{2}} and some power of m kills Tor_{k}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}). Therefore every graded piece of Tor_{k−2}(S_{1}, S_{2}) is killed by a power of m. By the previous lemma, Tor_{k−2}(S_{1}, S_{2}) is a finitely generated S_{1} ⊗ S_{2}- module. It follows that every element of Tor_{k−2}(S_{1}, S_{2}) is also killed by a power of m(S_{1} ⊗ S_{2}) and, in fact, that a fixed power m^{N}(S_{1} ⊗ S_{2}) kills every element of the module. Therefore Tor_{k−2}(S_{1}, S_{2}) is actually a module over (R/m^{N}) ⊗_{R} (S_{1} ⊗ S_{2}).

We now know that the module ⨁_{a_{1},a_{2}} Tor_{k}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}) is isomorphic, by an isomorphism that preserves the grading, to a finitely generated, bigraded module over (R/m^{N})⊗_{R} (S_{1} ⊗S_{2}). Therefore f_{k} is the Hilbert function of a finitely generated bigraded module over the bigraded ring (R/m^{N}) ⊗_{R} (S_{1} ⊗ S_{2}). Theorem 30, from the previous chapter, says that this Hilbert function is quasipolynomial of degree equal to the dimension of the ring. So it remains only to calculate the dimension of (R/m^{N})⊗_{R}(S_{1}⊗S_{2}). The ideal m(S_{1}⊗S_{2}) is nilpotent, so the ring (R/m)⊗_{R}(S_{1}⊗S_{2}) has the same dimension, and

(R/m) ⊗_{R} (S_{1} ⊗ S_{2}) ≅ ((R/m) ⊗_{R} S_{1}) ⊗_{R/m} ((R/m) ⊗_{R} S_{2}).

The dimension of (R/m) ⊗_{R} S_{i} = (R/m) ⊗_{R} R[I_{i}t] is the analytic spread of I_{i}, which is at most d. The claimed result follows.

The modules whose Hilbert functions we calculate above have generators all of whose multidegrees are either (0, 1) or (0, 1). By remarks in the previous chapter, such functions are eventually polynomial of degree at most 2 less than the dimension of the module. Therefore Corollary 34. Let I_{1} and I_{2} be ideals of a local ring (R, m) such that I_{1} + I_{2} is mprimary. Then there exists a (b_{1}, b_{2}) ∈ ℕ^{2} such that, for k ≥ 2 and (a_{1}, a_{2}) ≥ (b_{1}, b_{2}), the function

f_{k}(a_{1}, a_{2}) = length(Tor_{k}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}))

is polynomial of degree at most 2d − 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 I and J 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. 3.2 Tor_{1}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{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 35. Let I_{1} and I_{2} be ideals of a local ring (R, m) such that I_{1} + I_{2} is m-primary. Then the function

f_{1}(a_{1}, a_{2}) = length(Tor_{1}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}})) is quasipolynomial of degree at most dim R + 2.

However, all we know now is this: Theorem 36. Let I_{1} and I_{2} be ideals of a local ring (R, m) such that I_{1} + I_{2} is m-primary, and such that the R-algebra

T = ⨁_{a_{1},a_{2}}(I_{1}^{a_{1}} ∩ I_{2}^{a_{2}})x_{1}^{a_{1}}x_{2}^{a_{2}} ⊂ R[x_{1}, x_{2}]

is finitely generated over R. Then the function

f(a_{1}, a_{2}) = length(Tor_{1}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}))

is quasipolynomial of degree at most dim R + 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 = R[I_{1}x_{1}, I_{2}x_{2}] = ⨁_{a_{1},a_{2}} I_{1}^{a_{1}}I_{2}^{a_{2}}x_{1}^{a_{1}}x_{2}^{a_{2}}.

Since S ⊂ T, T is also an S-algebra. Now observe that

T/S = ⨁_{a_{1},a_{2}} \frac{I_{1}^{a_{1}} ∩ I_{2}^{a_{2}}}{I_{1}^{a_{1}}I_{2}^{a_{2}}} x_{1}^{a_{1}}x_{2}^{a_{2}}

and recall that T or_{1}^{R}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}) = (I_{1}^{a_{1}} ∩ I_{2}^{a_{2}})/I^{r}J^{s}. Apply Theorem 31, with N = T, M = 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 mR and by (I^{a}∩J^{b})s^{a}t^{b}, for all a, b satisfying a + b ≥ 1. Clearly M is maximal, with T/M ≅ R/m_{R}. Also, dim T = height M. Let P be minimal a minimal prime such that dim T/P = dim T, and let p = P ∩ R. Apply the dimension formula to get

dim T + 0 = height M + tr. deg_{R/mR}T/MT ≤ height(m_{R}(R/p)) + tr. deg_{R/p}T
≤ dim R + 2

Definition 37. Given a pair (I_{1}, I_{2}) of ideals of the local ring (R, m), call the algebra ⨁_{a_{1},a_{2}}(I_{1}^{a_{1}} ∩ I_{2}^{a_{2}})x_{1}^{a_{1}}x_{2}^{a_{2}} the intersection algebra of I_{1} and I_{2}. If this algebra is finite, we will say that I_{1} and I_{2} have finite intersection algebra.

The previous theorem can now be restated as follows. Corollary 38. If I_{1} and I_{2}, ideals of the local ring (R, m), have finite intersection algebra, then the function

f(a_{1}, a_{2}) = length(Tor_{1}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}))

is quasipolynomial of degree at most dim R + 2. 3.3 Tor_{0}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}})

Throughout this section, (R, m) is a regular local ring of dimension d.

Let I_{1} and I_{2} be ideals of R such that I_{1} + I_{2} is m-primary, and let f_{χ}(a_{1}, a_{2}) be the Euler characteristic

f_{χ}(a_{1}, a_{2}) = χ(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}) = ∑_{k}(−1)^{k} length(Tor_{k}^{R}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}})).

This sum is well-defined because the Tor’s are all zero for k > dim R, and because the condition that I_{1} + I_{2} be m-primary forces all of the Tor_{k}’s to have finite length. The function χ is biadditive (so, for example, χ(M ⊕ N, P) = χ(M, P) + χ(N, P)) because each of the Tor_{k}’s are. Also, χ(M, N) is zero whenever M and N are such that dim(M) + dim(N) < d (see [8] or [20]). And if length(M ⊗ N) < ∞, as is the case when M = R/I^{a_{1}} and N = R/I^{a_{2}}, then dim(M) + dim(N) ≤ d (see [25]). Theorem 39. For large a_{1} and a_{2}, f_{χ}(a_{1}, a_{2}) is either identically zero, or is a eventually a polynomial of degree d.

Proof. Fix a_{1} and a_{2} and let P_{u,1}, …, P_{u,s_{u}} be the primes associated to I_{u}^{a_{u}}. For u = 1 and u = 2, choose filtrations of R/I_{u}^{a_{u}} by prime cyclic modules, and let p_{u,i} be the number of times that R/P_{u,i} occurs in the corresponding filtration. Then by biaddativity,

χ(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}) = ∑_{i≤s_{1},j≤s_{2}} p_{1,i}p_{2,j}χ(R/P_{1,i}, R/P_{2,j}).

Assume that the P_{u,i}’s are ordered so that P_{u,1}, …, P_{u,s′_{u}} have the least height of any of the P_{u,i}’s, and let e_{u} be this minimal height. Since each P_{u,i} contains I_{u}^{a_{u}}, we know that P_{1,i} + P_{2,j} is m-primary for any i and j. Therefore e_{1} + e_{2} ≥ d. This means that if i > s′_{1} or j > s′_{2} then height(P_{1,i}) + height(P_{2,j}) > e_{1} + e_{2} ≥ d, so χ(R/P_{1,i}, R/P_{2,j}) = 0. Therefore

χ(\frac{R}{I_{1}^{a_{1}}} , \frac{R}{I_{2}^{a_{2}}} ) = ∑_{i≤s′_{1},j≤s′_{2}} p_{1,i}p_{2,j}χ(\frac{R}{P_{1,i}} , \frac{R}{P_{2,j}}).

Note that p_{u,i} is independent of the choice of filtration; in fact, p_{u,i} is just the length of R_{P_{u,i}}/I_{u}^{a_{u}}R_{P_{u,i}}. Also, the ideals P_{u,1}, …, P_{u,s_{u}} are also the associated primes of minimal height for any power of I_{1}. So, for any a_{1} and a_{2},

f_{χ}(a_{1}, a_{2}) = χ(\frac{R}{I_{1}^{a_{1}}} , \frac{R}{I_{2}^{a_{2}}} )
= ∑_{i≤s′_{1},j≤s′_{2}} length(\frac{R_{P_{1,i}}}{I_{1}^{a_{1}}R_{P_{1,i}}}) length(\frac{R_{P_{2,i}}}{I_{2}^{a_{2}}R_{P_{2,i}}})χ(\frac{R}{P_{1,i}} , \frac{R}{Q_{2,j}}).

Since I_{u} is primary to the maximal ideal of R_{P_{u,i}}, the functions

a_{u} ↦ length(\frac{R_{P_{u,i}}}{I_{u}^{a_{u}}R_{P_{u,i}}})

are just Hilbert functions, and are eventually polynomial of degree e_{u}. Therefore, the function f_{χ}(a_{1}, a_{2}) is polynomial of degree d for sufficiently large a and b, if e+f = d, and is identically zero if e + f > d, since in that case all of the χ(R/P_{1,i}, R/P_{2,j})’s are zero.

Actually from the proof and the results of the previous chapter this function is quasipolynomial of degree at most d + 2, and is the sum of periodic-polynomials on subsets of {(1, 0), (0, 1)}. Here is the result in more detail.

Theorem 40. f_{χ}(a_{1}, a_{2}) is either identically zero, or quasipolynomial of degree d+2.

Proof. By the proof of the previous theorem, f_{χ} is the sum of terms of the form cf(x)g(y), where f(x) and g(y) are eventually polynomial and c is a non-negative integer. By Theorem 19, 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 + 2. By Theorem 21, f(x) and g(y) are each quasipolynomial of degree e + 1 and f + 1 (where e and f are as in the proof of the previous theorem). So, Theorem 23, cf(x)g(y) is quasipolynomial of degree e + f + 2 = d + 2, as long as c is nonzero.

Theorem 41. If I_{1} and I_{2} have finite intersection algebra, then the function

f_{0}(a_{1}, a_{2}) = length (\frac{R}{I_{1}^{a_{1}} + I_{2}^{a_{2}}} )

is quasipolynomial of degree dim R + 2.

Proof. By the previous theorems, we already know that f_{i} is quasipolynomial for i > 0, and we know that the alternating sum χ = f_{0}−f_{1}+f_{2}+⋯ is quasipolynomial. Since f_{0} = χ + f_{1} − f_{2} + ⋯, 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.

Both I_{1} and I_{2} are contained in m, so I_{1}^{a_{1}} + I_{2}^{a_{2}} ⊂ m^{min(a_{1},a_{2})}. Therefore f_{0} is bounded below by, for example, the function given by

f(a_{1}, a_{2}) = \begin{cases}length(R/m^{(a_{1}+a_{2})/3}) & if 1/2 ≤ a_{1}/a_{2} ≤ 2 \\ 0 & otherwise \end{cases}

Since length(R/m^{a_{1}}) is eventually polynomial of degree d, this function is quasipolynomial of degree d + 2. So f_{0} is bounded below by a quasipolynomial function of degree d + 2.

Since I_{1} + I_{2} is m-primary, there exists N such that m^{N} ⊂ I_{1} + I_{2}. It follows that m^{N(a_{1}+a_{2}−1)} ⊂ I_{1}^{a_{1}} + I_{2}^{a_{2}}, so

f_{0}(a_{1}, a_{2}) ≤ length(R/m^{N(a_{1}+a_{2})}).

Therefore f_{0}(a_{1}, a_{2}) can also be bounded above by a polynomial of degree d, and hence by a quasipolynomial function of degree d + 2.

The result now follows from Theorem 28.

CHAPTER IV

Monomial Ideals

This chapter focuses on the cases where k = 0, M_{1} = ⋯ = M_{n} = R = k[[x_{1}, …, x_{d}]], and the ideals I_{1}, …, I_{n} are monomial ideals. When n = 2 the results of the previous chapter apply, because the pair I_{1} and I_{2} have finite intersection algebra whenever I_{1} and I_{2} are both monomial ideals. The first section of this chapter presents a proof of this fact. The rest of the chapter is devoted to a different approach which gives more useful information even for arbitrary n. 4.1 The case n = 2

Let A = k[[x_{1}, …, x_{d}]], with x_{1}, …, x_{d} indeterminates. Let B be a sub-k-alebra of A generated as an algebra by monomials in x_{1}, …, x_{d}. Then B is also a sub-k-module of A, and is generated over k as a module by monomials. The set of exponents α ∈ ℕ^{d} of monomials x^{α} generating B as a module over k form a submonoid M of the additive monoid ℕ^{n}. It is easy to see that M is a finitely generated monoid iff B is a finitely generated R-algebra.

Now consider the situation at hand. Let I_{1}, I_{2} ⊂ R = k[[x_{1}, …, x_{d}]] be ideals generated by monomials. Let T be the intersection algebra of I_{1} and I_{2}. Consider the two algebras

T_{1} = R[I_{1}t_{1}, t_{2}] ⊂ R[t_{1}, t_{2}]
T_{2} = R[t_{1}, I_{2}t_{2}] ⊂ R[t_{1}, t_{2}].

If x^{β_{1}}, …, x^{β_{m}} generate I, then T_{1} is generated over R by t_{2}, x^{β_{1}}t_{1}, …, x^{β_{m}}t_{1}. The situation for T_{2} is the same, so both T_{1} and T_{2} are finitely generated R-algebras. The intersection of T_{1} and T_{2} is T. Let M be the submonoid of ℕ^{d} consisting of those vectors (a_{1}, …, a_{d}, b_{1}, b_{2}) which occur as the exponents of monomials x_{1}^{a_{1}} ⋯ x_{d}^{a_{d}}t_{1}^{b_{1}}t_{2}^{b_{2}} contained in T. Define monomials M_{1} and M_{2} that correspond in the same way to T_{1} and T_{2}. Then M_{1} and M_{2} are finitely generated, and M = M_{1} ∩ M_{2}. Therefore we need only show that Theorem 42. If M_{1} and M_{2} are finitely generated submonoids of ℕ^{m}, then so is M = M_{1} ∩ M_{2}. Proof. This is a direct result of a theorem from integer programming (see, e.g., section I.3 of ), which states that if M ⊂ ℕ^{n} is the set of solutions to a finite set of ℤ-linear equations, then M is a finitely generated monoid.

Let α_{i,1}, …, α_{i,e_{i}} ∈ ℕ^{m} generate M_{i}. Define a new monoid M′ ⊂ ℕ^{m+e_{1}+e_{2}} as the set of all (a_{1}, …, a_{m}, b_{1,1}, …, b_{1,e_{1}}, b_{2,1}, …, b_{2,e_{2}}) satisfying the 2m equations

(a_{1}, …, a_{m}) = e_{1,1}α_{1,1} + ⋯ + e_{1,e_{1}}α_{1,e_{1}}
(a_{1}, …, a_{m}) = e_{2,1}α_{2,1} + ⋯ + e_{2,e_{2}}α_{2,e_{2}}

Then M′ is finitely generated, and it is easy to see that M = M_{1} ∩M_{2} is the image of M′ under the map which projects ℕ^{m+e_{1}+e_{2}} onto the first m coordinates. Therefore M is also finitely generated.

Therefore we have Theorem 43. If I_{1}, I_{2} ⊂ k[[x_{1}, …, x_{d}]] are monomial ideals, then the length functions f_{k} are quasipolynomial for all k ≥ 0.

Proof. Now that we know I_{1} and I_{2} have finite intersection algebra, this is a consequence of the theorems of the previous chapter. 4.2 The case n ≥ 2

Throughout this section, I_{1}, …, I_{n} will be ideals of R = k[[x_{1}, …, x_{d}]] which are generated by monomials.

I take for granted the basic definitions and theorems from the theory of convex polytopes as in, for example, the first three chapters of [10].

The monomials of R correspond to elements of (ℤ_{≥0})^{d} ⊂ ℝ^{d}, under a map “log” which takes a monomial x^{α} to the integer vector α determined by its exponents. Thus log x^{α} = α. Similarly, monomial ideals correspond to subsets of ℝ^{d}; write log I for the discrete subset of ℝ^{d} corresponding to the exponents of the monomials contained in I.

If S is a subset of ℝ^{d}, aS will denote the subset {as s ∈ S}. Note that a log I is not equal to log I^{a}.

Let K_{i} be the convex hull of log I_{i}. Write P = ℝ_{≥0}^{d} = {(r_{1}, …, r_{d}) ∈ ℝ^{d} r_{i} ≥ 0}, let A_{i} be the closure of P \K_{i}. I will show that f_{0}(a_{1}, …, a_{n}) is closely approximated, when a_{1}, a_{2}, …, a_{n} are all sufficiently large, by the d-dimensional volume of the region ⋂_{i} a_{i}A_{i}. Along the way, I will show that the volume of ⋂_{i} a_{i}A_{i} is piecewise a degree d homogeneous polynomial in the a_{i}’s. The conclusion will be that f(a_{1}, …, a_{n}) is approximated by a forms of degree d in a_{1}, …, a_{n}, for large a_{1}, …, a_{n}.

First I’ll prove the theorem about the volume of ⋂_{i} a_{i}A_{i}. But before doing this it would be useful to know what kind of object ⋂_{i} a_{i}A_{i} is. Lemma 44. If I is an ideal generated by finitely many monomials, then K = conv(log I) is the intersection of finitely many closed half-spaces. Proof. Let x^{α_{1}}, …, x^{α_{k}} generate an ideal I. Then log I is the union of the sets α_{i} + ℕ^{d}, for 1 ≤ i ≤ k. The convex hull in ℝ^{d} of one of these sets is α_{i} + P, so the convex hull K of I in ℝ^{d} is the convex hull of the sets α_{i} + P. Now, embed ℝ^{d} in P^{d}(ℝ) by the map

(r_{1}, …, r_{d}) ↦ (r_{1} : ⋯: r_{d} : 1),

and let \overline{K} and \overline{P} be the closures of K and P in this projective space. Let A be another copy of ℝ^{d} in P^{d}(ℝ) containing \overline{K}. Then α_{i} + \overline{P} is the convex hull in A of the point α_{i} and the points at infinity

(1 : 0 : ⋯: 0), (0 : 1 : 0 : ⋯: 0), …, (0 : ⋯: 0 : 1 : 0).

It follows that \overline{K} is the convex hull in A of the points \{α_{i}\} corresponding to the generators of I along with the above points at infinity. Since \overline{K} is the convex hull of a finite number of points, \overline{K} can be written as the intersection of finitely many half-spaces of A. Half-spaces in A correspond to half-spaces in ℝ^{d}, so the original set K can also be written as the intersection of finitely many half-spaces.

Thus each of our sets K_{i} can be written as the intersection of finitely many halfspaces.

Let H_{i,1}^{−}, …, H_{i,k_{i}}^{−} be the closed half-spaces whose intersection is K_{i}, and let H_{i,i}^{+}, …, H_{i,k_{i}}^{+} be the complementary closed half-spaces.

Note that A_{i} = ⋃_{j} H_{i,j}^{+}. We can also write

A_{i} = ⋃_{\begin{matrix}S⊂\{1,... ,k_{i}\} \\ S≠∅ \end{matrix}} A_{i,S}

where

A_{i,S} = ⋂_{j∈S} H_{i,j}^{+} ∩ ⋂_{j∉S} H_{i,j}^{−}.

The advantage of this decomposition is that we have written A_{i} as the union of sets with disjoint interiors: if S′ and S′′ are distinct non-empty subsets of {1..k_{i}}, then there must be some j contained in one but not the other; thus A_{i,S′} ∩ A_{i,S′′} is contained in the hyperplane H_{i,j}^{+} ∩ H_{i,j}^{−}.

Let T be the set of all pairs (i, j) ∈ ℤ^{2} with 1 ≤ i ≤ n and 1 ≤ j ≤ k_{j}. If S ⊂ T, write S_{i′} for the set of all (i, j) ∈ S with i = i′. Then

⋂_{i} a_{i}A_{i} = ⋃_{\begin{matrix}S⊂T \\ S_{i}≠∅ for any i \end{matrix}}( ⋂_{(i,j)∈S} H_{i,j}^{+} ∩ ⋂_{(i,j)∉S} H_{i,j}^{−}).

Proving that this union of intersections is actually equal to ⋂_{i} a_{i}A_{i} is routine. Once again, the sets that we are taking the union of have disjoint interiors. Also, each of these sets is contained in ⋂_{i} a_{i}A_{i}, hence is bounded, and each is the intersection of finitely many half-spaces, hence is a polytope.

So we have established that ⋂_{i} a_{i}A_{i} is the union of polytopes with disjoint interiors. To prove that the volume of ⋂_{i} a_{i}A_{i} is a piecewise polynomial in the a_{i}, it therefore suffices to prove the following theorem. Note that the H_{i,j}’s in the statement below do not correspond exactly with the similarly-named half-spaces above. Theorem 45. Let H_{1,1}^{+}, …, H_{1,k_{1}}^{+}, …, H_{n,1}^{+}, …, H_{n,k_{n}}^{+} be half-spaces, and let

Q_{a_{1},... ,a_{n}} = ⋂_{i,j} a_{i}H_{i,j}

be a family of polytopes indexed by a_{1}, …, a_{n}. Then the volume of Q_{a_{1},... ,a_{n}} is a continuous function of a_{1}, …, a_{n}. In addition, there exist a finite number of hyperplanes in ℝ_{≥0}^{n} such that, for a_{1}, …, a_{n} not contained in any of the hyperplanes, the volume of the polytope Q_{a_{1},... ,a_{n}} is given by a degree d form in a_{1}, …, a_{n}.

Proof. I believe this fact is actually well known, but have been unable to find a complete reference in the literature. Therefore I will present a sketch of the proof.

If (a_{1}, …, a_{n}) are fixed, then volume of the polytope Q_{a_{1},... ,a_{n}} can be calculated by triangulating Q_{a_{1},... ,a_{n}} and then calculating the volume of each simplex in the triangulation. It is also helpful to choose the triangulation so that each vertex of a simplex in the triangulation is also a vertex of the polytope. The existence of such a triangulation can be proved by induction on the dimension of the polytope: first triangulate the boundary of Q (which has lower dimension), then pick a fixed vertex of Q, and construct simplices each of which is a “pyramid” whose vertex is the chosen vertex and whose base is one of the simplices in the triangulation of the boundary.

If (a_{1}, …, a_{n}) are now allowed to vary, such a triangulation can be made to vary along with the polytope as long as the combinatorial type of the polytope does not change. The vertices of Q_{a_{1},... ,a_{n}} are the intersections of hyperplanes a_{i}H_{i,j}, and it can be shown that locally the location of one of these vertices is determined by solving a system of linear equations that depends linearly on the a_{i} and that the coordinates of the vertices are therefore locally linear functions of (a_{1}, …, a_{n}). The volume of a simplex can be written as a degree d form in the coordinates of its vertices. This means the volume of the simplices in the triangulation of the polytope, and hence the volume of the polytope itself, can be written as a degree d form in (a_{1}, …, a_{n}).

This only works for (a_{1}, …, a_{n}) constrained to a region where the combinatorial type of the polytope does not change. But it can be shown that the values of (a_{1}, …, a_{n}) for which the combinatorial type of Q_{a_{1},... ,a_{n}} changes are solutions to one of finitely many systems of linear equations, and hence are contained in the union of a finite set of hyperplanes.

From the description given in the theorem, it is clear that this volume is actually a quasipolynomial function of (a_{1}, …, a_{n}). The next goal is to show that f_{0} is well approximated by this volume for large enough (a_{1}, …, a_{n}). However, without even doing this we’ve already gained some information in the case where the ideals I_{i} are all principal.

Corollary 46. If I_{1}, …, I_{n} are each generated by a single monomial, then f_{0} has the form given in the above theorem, hence is quasipolynomial of degree d + 2.

Proof. Let I_{i} = (x^{α_{i}}). In this case A_{i} is just P \(α_{i} + P), and the volume of ∩_{i}A_{I} is easily seen to be exactly equal to the number of lattice points contained in ∩_{i}A_{i}, and to the length of R/(I_{1}^{a_{1}} + ⋯ + I_{n}^{a_{n}}). The result follows immediately from the previous theorem.

Of course, this result is exactly what we would expect given the results of the previous chapter. The following corollary is a slight generalization of a result from [6].

Corollary 47. Let I and J be monomial ideals in R = k[[x_{1}, …, x_{d}]], where k has characteristic p, and let S = R/I. Assume I + J is m_{R}-primary. Then the Hilbert-Kunz function (see chapter 1), HK_{S,J}(e), is eventually a polynomial of degree dim S in p^{e}.

Proof. Suppose I = (x^{α_{1}}, …, x^{α_{b}}), and J = (x^{α_{b+1}}, …, x^{α_{n}}). Let I_{i} = (α_{i}). Apply the previous corollary to I_{1}, …, I_{n}, and note that

HK_{S,I}(e) = f_{0}( \underbrace{1, …, 1}_{b} , \underbrace{p^{e}, …, p^{e}}_{n − b}).

Since f_{0} is piecewise polynomial, the point (1, …, 1, e, …, e) must eventually end up in one of the “pieces.”

The statement about the degree follows from Monsky’s result about the form of Hilbert-Kunz functions [16].

As noted in [6], the assumption about the characteristic of p is actually unnecessary if we generalize the definition of the Hilbert-Kunz function in a natural way.

The next lemmas will be used to bound f(a_{1}, …, a_{n}) using the volume of ∩a_{i}A_{i}. Lemma 48. If H is a half-space containing log I, then aH contains log I^{a}. Proof. We can write H = {x ∈ ℝ^{d} ϕ(x) ≥ c} for some linear function ϕ and real number c, and aH can then be described as the set of all x ∈ ℝ^{d} such that ϕ(x) ≥ ac. Let α ∈ log I^{m}. Then α = α_{1} + ⋯ + α_{a} + β, with α_{i} ∈ log I and β ∈ log R. Then

ϕ(α) = ϕ(α_{1} + ⋯ + α_{a} + β)
= ϕ(α_{1}) + ⋯ + ϕ(α_{a}) + ϕ(β)
≥ c + ⋯ + c + ϕ(β)
≥ ac.

For the last step we used the fact that ϕ(β) ≥ 0 for any β ∈ log R; establishing this fact will complete the proof. So let β be an arbitrary element of log R, and let α be an arbitrary element of log I. Then α + mβ ∈ log I for any m ∈ ℤ^{+}, so ϕ(α + mβ) = ϕ(α) + mϕ(β) ≥ c for any m. From this it follows that ϕ(β) ≥ 0.

It follows from the previous lemma that a_{i}A_{i} contains log(I_{i}^{a_{i}}). There is also an inclusion the other way. Lemma 49. If I is generated by h monomials, and if K is the convex hull of log(I), then log(I^{a}) ⊃ ℤ^{d} ∩ (a + h)K. Proof. Let x^{α_{1}}, …, x^{α_{h}} generate I, and let β be any element of (a + h)K. Then β can be written as a + h times an element of K. Since K is convex, this element can be written in the form ∑_{i} b_{i}α_{i}, where the b_{i}’s are non-negative real numbers such that ∑_{i} b_{i} ≥ 1. So β = ∑_{i}(a + h)b_{i}α_{i}. Write c_{i} for the greatest integer less than or equal to (a + h)b_{i}. Each c_{i} is non-negative, and

∑_{i=1}^{h} c_{i} ≥ ∑_{i=1}^{h}((a + h)b_{i} − 1) = (a + h)(∑_{i=1}^{h} b_{i}) − h ≥ (a + h) · 1 − h = a

Therefore

x^{∑_{i} c_{i}α_{i}} = ∏_{i}(x^{α_{i}})^{c_{i}}

is an element of I^{a}, and since each coordinate of the vector ∑_{i}(a + h)b_{i}α_{i} is at least as large as the corresponding coordinate of ∑_{i} c_{i}α_{i},

β = x^{∑_{i}(a+h)b_{i}α_{i}}

must also be contained in I^{a}.

Write vol(A) for the d-dimensional volume of A ⊂ ℝ^{d}, and write #(A) for the number of points in A ∩ ℤ^{d}.

Lemma 50. Let A be a subset of ℝ^{d} such that both A and its boundary ∂A have finite volume. Then

vol(A) − vol(∂A)\sqrt{2} ≤ #(A) ≤ vol(A) + vol(∂A)\sqrt{2}

Proof. Given α ∈ ℤ^{d}, we can form a half-open unit cube α + [0, 1)^{d}. This has volume 1 and contains exactly the one integer point α. Form the set A_{0} made up exactly of those cubes α + [0, 1)^{d} which are completely contained within A. Note that A_{0} is contained in A, and that the volume of A_{0} is exactly equal to the number of integer points contained in A_{0}. Similarly, form a set A^{0} that is the union of those cubes α + [0, 1)^{d} which have nonzero intersection with A. Then A^{0} contains A, and again has the nice property that the number of integer points in A^{0} is equal to the volume of A^{0}.

Note also that every point of A\A_{0} lies within \sqrt{2} units of the boundary of S, and, similarly, that every point of A^{0}\A is within \sqrt{2} units of the boundary of A. The stated formula follows immediately.

Theorem 51. Fix (b_{1}, …, b_{n}) ∈ ℕ^{n}, a > 0. Then

lim_{m → ∞} \frac{length (\frac{R}{I_{1}^{mb_{1}} + ⋯ + I_{n}^{mb_{n}}} )}{vol(⋂_{i} mb_{i}A_{i})} = 1.

Proof. Note that log(∑_{i} I_{i}^{a_{i}}) = ⋂_{i} log(I^{a_{i}}). Therefore length(R/(∑_{i} I_{i}^{a_{i}})) is just the number of points contained in ℕ^{d} but not in any log(I_{i}^{a_{i}}). Let I be generated by h monomials. Then by Lemmas 48 and 49,

(⋂_{i} mb_{i}A_{i}) ∩ ℤ^{d} ⊂ ℕ^{d}\ log(∑_{i} I_{i}^{mb_{i}}) ⊂ (⋂_{i}(mb_{i} + h)A_{i}) ∩ ℤ^{d}.

Therefore

#(⋂_{i} mb_{i}A_{i}) ≤ length (\frac{R}{∑_{i} I_{i}^{mb_{i}}} ) ≤ #(⋂_{i}(mb_{i} + h)A_{i}).

Let b be the d−1-volume of the boundary of ⋂_{i} b_{i}A_{i}, making mb the d−1-volume of the boundary of ⋂_{i} mb_{i}A_{i}. Combine the inequalities above with the inequalities given by lemma 50:

vol(⋂_{i} mb_{i}A_{i}) − mb\sqrt{2}

≤ #(⋂_{i} mb_{i}A_{i})

≤ length (\frac{R}{∑_{i} I_{i}^{a_{i}}} )

≤ #(⋂_{i}(mb_{i} + h)A_{i})

≤ vol(⋂_{i} mb_{i}A_{i}) + (m + h)b\sqrt{2}.

Divide through by vol(⋂_{i} mb_{i}A_{i}) to get

\frac{vol(⋂_{i} mb_{i}A_{i}) − mb\sqrt{2}}{vol(⋂_{i} mb_{i}A_{i})} ≤ \frac{length (_{\frac{R}{∑_{i} I_{i}^{b_{i}}}} )}{vol(⋂_{i} mb_{i}A_{i})} ≤ \frac{vol(⋂(mb_{i} + h)A_{i}) + mb\sqrt{2}}{vol(⋂_{i} mb_{i}A_{i})}.

Recalling that vol(⋂_{i} mb_{i}A_{i}) is a degree d form in m and n, it is easy to see that the left- and right- hand sides of this inequality approach 1 as m and n approach infinity along the line n = ma. The desired result follows.

This theorem, together with Theorem 45 and the discussion proving it, shows, roughly speaking, that f_{0} agrees asymptotically with a piecewise degree d polynomial when I_{1}, …, I_{n} are monomial ideals. 4.3 Examples

In general, the functions f_{0} seem to be quite difficult to compute, but in the monomial case, at least. there are lots of very tractable examples. It is worth examing a few, if only to verify that the piecewise behavior really does occur and that f_{0} is not always eventually a polynomial. i. Let R = ℚ[[x_{1}, …, x_{d}], I_{1} = ⋯ = I_{n} = (x_{1}, …, x_{d}). Then

f_{0}(a_{1}, …, a_{n}) = \binom{n + min(a_{1}, …, a_{n}) − 1}{n}.

If we divide ℝ_{≥0}^{n} into the n subsets

A_{i} = {(a_{1}, …, a_{n}) ∈ ℕ^{n} a_{i} ≤ a_{j} for all j ≠ i} ,

each of which is the cone over a (d − 1)-dimensional simplex, then f_{0} is polynomial of degree d on each A_{i}.

ii. Fix a positive integer c and take R = ℚ[[x_{1}, x_{2}]], I_{1} = (x_{1}, x_{2})^{c} ∩ (x_{1}), and I_{2} = (x_{2}^{c}). Then it is not hard to calculate that

f_{0}(a_{1}, a_{2}) = \begin{cases}\frac{(c−1)^{2}}{2} a_{1}^{2} + ca_{1}a_{2} + \frac{c−1}{2}a_{1} & for a_{1} ≤ \frac{c}{c−1}a_{2} \\ c^{2}a_{1}a_{2} − \frac{c^{2}}{2} a_{2}^{2} + \frac{c}{2}a_{1} & for a_{1} ≥ \frac{c}{c−1}a_{2}. \end{cases}

APPENDICES

49

APPENDIX A

Open Questions

i. Is it possible to calculate the intersection multiplicity χ(I_{1}, I_{2}) knowing only the behavior of the function f_{0}? Or do there exist pairs of ideals (I_{1}, I_{2}) and (J_{1}, J_{2}) that have different intersection multiplicities (χ(I_{1}, I_{2}) ≠ χ(J_{1}, J_{2})) but such that the functions f_{0} are identical?

ii. Are the f_{i} always quasipolynomial when n = 2? The results of this thesis are compatible with the following conjecture:

Conjecture 52. For (R, m) of dimension d and I_{1}, I_{2} ⊂ R such that I_{1} + I_{2} is m-primary,

f_{k}(a_{1}, a_{2}) = length(Tor_{k}(R/I_{1}^{a_{1}}, R/I_{2}^{a_{2}}))

is quasipolynomial of degree d+n, if k = 0, is quasipolynomial of degree at most d + 2, if k = 1, and is quasipolynomial of degree 2d, if k ≥ 2.

In fact, from this dissertation it might appear that a more general conjecture would be true for arbitrary n; however, there are counterexamples among the Hilbert-Kunz functions. If

R = \frac{(ℤ/5ℤ)[[x_{1}, x_{2}, x_{3}, x_{4}]]}{(x_{1}^{4} + x_{2}^{4} + x_{3}^{4} + x_{4}^{4})},

then

HK_{R}(a) = \frac{168}{61}(5^{a})^{3} − \frac{107}{61}(3^{a}),

which is not quasipolynomial in 5^{a}. (See [11] for details.) For counterexamples to the more restricted conjecture above, ideals I_{1}, I_{2} not having finite intersection algebra would be particularly interesting to examine. Such examples can be constructed from examples existing in the literature of non-finitely generated symbolic power algebras. (See [22], [21], or [9]; the latter paper gives a family of examples based on monomial curves which may be easier to do calculations with.) In particular, given an ideal P ⊂ R such that the algebra

R ⊕ P^{(1)} ⊕ P^{(2)} ⊕ ⋯

is not finitely generated, it is possible to find a f ∈ R such that the ideals (f) and P do not have finite intersection algebra. To do this, choose f so that (P^{a} : f^{a}) = P^{(a)}, for all a.

Lemma 53. If R regular and P ⊂ R prime, then there exists f ∈ R such that (P^{a} : f^{a}) = P^{(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, gr_{P R_{P}} (R_{P}) = (R − P)^{−1}gr_{P} (R) is a domain. Let I be the kernel of the natural map gr_{P} (R) →(R − P)^{−1}gr_{P} (R). Then (R − P)^{−1}I = 0, and, since I is finitely generated, there exists f ∈ R − P such that fI = 0. (We are implicitly identifying elements in R with their images in gr_{P} (R) via the natural map R → R/P ⊂ gr_{P} (R)). This means that fg ∈ P^{n+1} for any element g of P^{n} which can be multiplied into P^{n+1} by an element of R − P. More generally, any element g ∈ P^{n} which can be multiplied into P^{n+m} by an element of R − P satisfies f^{m}g ∈ P^{n+m}. The result follows.

Lemma 54. If f and P are chosen as suggested above, then the intersection algebra ⨁_{a_{1},a_{2}≥0}(P^{a_{1}} ∩ (f^{a_{2}}))x_{1}^{a_{1}}x_{2}^{a_{2}} is not finitely generated.

Proof. We will prove the contrapositive. Assume that the intersection algebra is finitely generated. Note that (P^{a_{1}} : f^{a_{2}})f^{a_{2}} = P^{a_{1}} ∩ (f^{a_{2}}), so the algebra can be rewritten as ⨁(P^{a_{1}} : f^{a_{2}})x_{1}^{a_{1}}(x_{2}f)^{a_{2}}. Map this to the symbolic power algebra ⨁_{a} P^{(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.

iii. For the cases where we proved results f_{k} was quasipolynomial for k ≥ 1, we only found an upper bound on the degree of f_{k}. Is this upper bound actually an equality when f_{k} is not identically zero? If not, is it the best possible upper bound, or is there a smaller one?

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ABSTRACT

Length functions determined by killing powers of several ideals in a local ring

by J. Bruce Fields

Chair: Melvin Hochster

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 to the length of the quotient of R by sum of the ideals raised to the respective exponents. This quotient can also be obtained by taking the tensor product of the quotients of R by the various powers of the ideals. This thesis studies these functions as well as the functions obtained by replacing the tensor product by a higher Tor. These functions are shown to have rational generating functions under certain conditions.