Initial ideals, Veronese subrings, and rates of algebras

David Eisenbud ^{∗} Dept. of Math., Brandeis Univ., Waltham MA 02254 eisenbud@math.brandeis.edu

Alyson Reeves Dept. of Math., Brandeis Univ., Waltham MA 02254 reeves@math.brandeis.edu

Burt Totaro Dept. of Math., Univ. of Chicago, Chicago IL 60637 totaro@math.uchicago.edu

Abstract

Let S be a polynomial ring over an infinite field and let I be a homogeneous ideal of S. Let T_{d} be a polynomial ring whose variables correspond to the monomials of degree d in S. We study the initial ideals of the ideals V_{d}(I) ⊂ T_{d} that define the Veronese subrings of S/I. In suitable orders, they are easily deduced from the initial ideal of I. We show that in(V_{d}(I)) is generated in degree ≤ max(⌈reg(I)/d⌉, 2), where reg(I) is the regularity of the ideal I. (In other words, the d^{th} Veronese subring of any commutative graded ring S/I has a Gröbner basis of degree ≤ max(⌈reg(I)/d⌉, 2).) We also give bounds on the regularity of I in terms of the degrees of the generators of in(I) and some combinatorial data. This implies a version of Backelin’s Theorem that high Veronese subrings of any ring are homogeneous Koszul algebras in the sense of Priddy [Pr70].

We also give a general obstruction for a homogeneous ideal I ⊂ S to have an initial ideal in(I) that is generated by quadrics, beyond the obvious requirement that I itself should be generated by quadrics, and the stronger statement that S/I is Koszul. We use the obstruction to show that in certain dimensions, a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics.

For the application to Backelin’s Theorem, we require a result of Backelin whose proof has never appeared. We give a simple proof of a sharpened version, bounding the rate of growth of the degrees of generators for syzygies of any multihomogeneous module over a polynomial ring modulo an ideal generated by monomials, following a method of Bruns and Herzog.

Notation: Throughout this paper we write S = k[x_{1}, …, x_{r}] for the graded polynomial ring in r variables over an infinite field k. We will generally deal with a monomial order > on S. We always suppose x_{1} > ⋯ > x_{r}; by the initial term in_{>}(p) of a polynomial p we mean the term with the largest monomial. Similarly, by the initial ideal in_{>}(I) of I, we mean the ideal generated by the initial terms of all polynomials in I:

∗The authors are grateful to the NSF for support during the preparation of this paper. The second and third authors were supported by NSF Postdoctoral Fellowships.

in_{>}(I) = ⟨in_{>}(p) | p ∈ I⟩.

We write T_{d} for the polynomial ring over k whose variables correspond to the monomials of degree d in S; there is a natural map ϕ_{d} : T_{d} → S sending each variable of T_{d} to the corresponding monomial in S. If I ⊂ S is an ideal, we write V_{d}(I) for the preimage of I in T_{d}, and A_{(d)} for T_{d}/V_{d}(I).

Furthermore, we make the following definitions:

Definition: For a homogeneous ideal I ⊂ S, the minimal generators of I are the homogeneous elements of I not in (x_{1}, …, x_{r})I. Let δ(I) be the maximum of the degrees of minimal generators of I, and let Δ(I) be the minimum, over all choices of variables and of monomial orderings of S of the maximum of the degrees of minimal generators of the initial ideal of I.

All ideals and rings that appear will be graded.

1 Introduction and motivating results

Given a homogeneous ideal I ⊂ S it is a matter of both computational and theoretical interest to know how low the degree of in_{>}(I) can be made by choosing variables and monomial order on S in an appropriate way. In particular, one may ask which ideals I admit quadratic initial ideals; that is, for which I are there choices of variables and order such that in_{>}(I) can be generated by monomials of degree 2?

One of the theoretical reasons for interest in this question is that if I admits a quadratic initial ideal then, by a result of Fröberg and a deformation argument noticed by Kempf and others, A := S/I is a (homogeneous) Koszul algebra in the sense of Priddy [Pr70]; that is, the residue field k of A admits an A-free resolution whose maps are given by matrices of linear forms. Using complex arguments about a lattice of ideals derived from a presentation of A as a quotient of a free noncommutative algebra, Backelin [Ba86] proved that for any graded ring A as above the d^{th} Veronese subring

A_{(d)} := ⨁_{i=1}^{∞} A_{di},

is Koszul for all sufficiently large d. Our work started from a request by George Kempf for a simpler proof. In this paper we shall prove the stronger result that A_{(d)} admits a quadratic initial ideal for all sufficiently large d.

To make these results more quantitative, we define a measure of the rate of growth of the degrees of the syzygies in a minimal free resolution:

Definition: Let A = k ⊕ A_{1} ⊕ A_{2} ⊕ ⋯ be a graded ring. For any finitely generated graded A-module M, set t_{i}^{A}(M) = max{j | T or_{i}^{A}(k, M)_{j} ≠ 0}, where T or_{i}^{A}(k, M)_{j} denotes the j^{th} graded piece of T or_{i}^{A}(k, M). The rate of A is defined by Backelin [Ba86] to be

rate(A) = sup{(t_{i}^{A}(k) − 1)/(i − 1)|i ≥ 2}.

For example, A is Koszul iff rate(A) = 1. It turns out that the rate of any graded algebra is finite (see for example [An86]) and Backelin actually proves

Theorem 1 ([Ba86]): rate(A_{(d)}) ≤ max(1, rate(A)/d).

One should compare this with the rather trivial result (Proposition 5 below) that if a homogeneous ideal I can be generated by forms of degree m, then the ideal V_{d}(I) defining A_{(d)} can be generated by forms of degree ≤ max(2, ⌈m/d⌉). In the notation introduced above, δ(V_{d}(I)) ≤ max(2, ⌈δ(I)/d⌉). A similar result with Δ (the minimum, over all choices of variables and of monomial orderings >, of the maximum degree of a minimal generator of in_{>}(I)) would lead to a bound on the rate by virtue of Proposition 3. Unfortunately, as we show in Example 3 below, it is not true that if some initial ideal of I can be generated by monomials of degree m then V_{d}(I) admits an initial ideal generated by forms of degree ≤ max(2, ⌈m/d⌉). But there is a replacement for m that makes such a formula true, and this is the main result of this paper.

Recall that the Castelnuovo-Mumford regularity of I is defined as follows: Definition: For I ⊂ S, the regularity of I is defined as

reg(I) = max{t_{i}^{S}(I) − i|i ≥ 0}.

Since t_{0}^{S}(I) = δ(I) ≤ reg(I), the regularity is ≥ the maximal degree of the generators of I. One may think of the regularity as a more stable measure of the size of the generators of I. Our main result is that we may replace the degree of the generators of I by the regularity and get a bound on the degrees of the initial ideals of Veronese powers:

Theorem 2

Δ(V_{d}(I)) ≤ max(2, ⌈reg(I)/d⌉).

In particular, if d ≥ reg(I)/2 then Δ(V_{d}(I)) = 2.

In section 5 we explain how to generalize this result to Segre products of Veronese embeddings.

To deduce a version of Backelin’s Theorem, one needs a result strengthening the theorem of Fröberg mentioned above. Such a result was stated without proof by Backelin ([Ba86, pp98ff]):

Proposition 3 If A = S/I with I a homogeneous ideal, then rate(A) ≤ Δ(I) − 1. In particular, if Δ(I) = 2 then A is Koszul.

We will give a simple proof of this proposition (and something more general) in section 4 following ideas of Bruns, Herzog, and Vetter [BrHeVe].

Unfortunately the converse of this result is not true: in particular, the algebra A may be Koszul without I admitting a quadratic initial ideal. In section 6, we formulate another obstruction for an ideal I ⊂ S to have an quadratic initial ideal.

An easily stated part of Theorem 19 is that if I admits a quadratic initial ideal then I contains far more quadrics of low rank than would a generic subspace of quadrics. We may make this quantitative as follows:

Corollary 4 If I ⊂ S admits a quadratic initial ideal, and dim(S/I) = n, then I contains an m-dimensional space of quadrics of rank ≤ 2(n + m) − 1 for every m ≤ codim(I).

The obstruction shows that in certain dimensions, a generic complete intersection of quadrics has no initial ideal in(I) which is generated by quadrics, even though every complete intersection of quadrics is a Koszul algebra. There seems no reason to believe that this obstruction, even with the Koszul condition, is enough to guarantee that an ideal admits a quadratic initial ideal, so we pose as a problem the question raised at the beginning of the introduction :

Find necessary and sufficient conditions for an ideal to admit a quadratic initial ideal.

A noncommutative analogue of some of the results on initial ideals, in which T_{d} is replaced by a free noncommutative algebra mapping onto S, is given in [Ei].

2 Initial ideals for Veronese subrings

As above, let

T_{d} = k[{z_{m}}] where m is a monomial of S of degree d,

and let ϕ_{d} : T_{d} → S be the map sending z_{m} to m. If J ⊂ S is a homogeneous ideal, let V_{d}(J) denote the preimage of J in T_{d}. It is easy to see that V_{d}(J) is generated by the kernel of ϕ_{d} and, for each generator g of J in degree e, the preimages of the elements of degree nd in (x_{1}, …, x_{r})^{nd−e}g, where nd is the smallest multiple of n that is ≥ e. These elements have degree n in T_{d}. Since ker(ϕ_{d}) is generated by forms of degree 2 it follows that V_{d}(J) is generated by forms of degree ≤ max(⌈δ(J)/d⌉, 2).

This gives a proof of the following well-known Proposition.

Proposition 5

δ(V_{d}(I)) ≤ max(2, ⌈δ(I)/d⌉).

In particular, if d ≥ δ(I)/2 then V_{d}(I) is generated by quadrics.

This Proposition is mentioned by Mumford in [Mu70] (in a slightly different form), though it is surely much older.

We extend the given monomial order on S to a monomial order on T_{d} as follows: If a, b are monomials in T_{d}, then a > b if ϕ(a) > ϕ(b) or ϕ(a) = ϕ(b) but a is bigger than b in the reverse lexicographic order: that is, given two monomials in T_{d} of the same degree having the same image in S we order the factors of each in decreasing order, and take as larger the monomial with the smaller factor in the last place where the two differ. Here the order of the variables z_{m} is defined to be the same as the order in S on the monomials m. We first compute the initial ideal of ker(ϕ_{d}).

Proposition 6 With notation as above, in(ker(ϕ_{d})) ⊂ T_{d} is generated by quadratic forms for every d.

Remark: Barcanescu and Manolache [BaMa82] proved that the Veronese rings are Koszul – which is a corollary of Proposition 6.

Proof: The ideal ker(ϕ_{d}) is generated by quadratic forms, each a difference of two monomials that go to the same monomial under ϕ_{d}. Let J be the monomial ideal generated by the initial terms of these quadratic elements of ker(ϕ_{d}). We have J ⊂ in(ker(ϕ_{d})), and we wish to prove equality. We will show that distinct monomials of T_{d} not in J map by ϕ_{d} to distinct monomials of S. It will follow that, for each e,

dimS_{de} ≥ dim(T_{d}/J)_{e}
≥ dim(T_{d}/(in(ker(ϕ_{d}))))_{e}
= dim(T_{d}/(ker(ϕ_{d})))_{e}
= dim(S_{de}).

Thus dim(T_{d}/J)_{e} = dim(T_{d}/(in(ker(ϕ_{d}))))_{e}, and J = in(ker(ϕ_{d})) as desired.

Call the monomials not in J “standard”, and say that a product of monomials of degree d in S is standard if its factors correspond to the factors of a standard monomial in T_{d}. Since J ⊂ in(ker(ϕ_{d})), any monomial of S_{de} may be written as a standard product of e monomials of degree d. We must show that if m ∈ S is a monomial of degree de, then there is a unique way of writing m as a standard product m_{1} ⋯ m_{e} of monomials of degree d. We claim that this unique product is obtained by writing out the de factors of m in decreasing order

m = x_{1} ⋯ x_{1}x_{2} ⋯ x_{2} ⋯ x_{r} ⋯ x_{r},

and taking m_{1} to be the product of the first d factors, m_{2} to be the product of the next d factors, and so on.

First we prove the claim for a standard product m_{1}m_{2} with just two factors. Suppose the sequences of indices of the two factors are

i_{1} = (i_{11} ≤ ⋯ ≤ i_{1d}), i_{2} = (i_{21} ≤ ⋯ ≤ i_{2d})

and m_{1} > m_{2}. We must show that i_{1d} ≤ i_{21}. The product of the monomials m′_{1} and m′_{2} obtained from m_{1} and m_{2} by interchanging the factors x_{i_{1d}} and x_{i_{21}} represents the same element of S. The difference of these products represents a quadratic element of ker(ϕ_{d}) . If i_{1d} > i_{21} then x_{i_{1d}} < x_{i_{21}} and thus m′_{2} < m_{2}. Consequently the leading term of this quadratic element of ker(ϕ_{d}) would correspond to m_{1}m_{2}, and the product m_{1}m_{2} would not be standard.

Now suppose that m_{1}·· · ··m_{e} is a standard product of degree de, with e arbitrary, and m_{1} ≥ ⋯ ≥ m_{e}. Let i_{j1} ≤ ⋯ ≤ α_{jd} be the indices of the variables in m_{j}. If i_{jd} > i_{j+1,1} for some j then the product m_{j}m_{j+1} would not be standard, contradicting the standardness of the entire product.

Note: Even when r = 3 and d = 2, the initial terms of the minimal system of generators for the ideal ker(ϕ_{2}) do not generate in_{>}(ker(ϕ_{2})) under all possible orders >. In this case, the ideal is minimally generated by the 2 × 2 minors of the symmetric 3 × 3 matrix with entries z_{ij}, i ≤ j. There are 29 different initial ideals (depending upon the order chosen), 23 of which are generated entirely in degree 2. The other 6 each require an additional generator of degree three.

It would be interesting to characterize the orders > for which the ideal in_{>}(ker(ϕ_{d})) is generated by elements of degree 2.

Following the proof of Proposition 6 we say a monomial of T_{d} is standard if it is not in in(ker(ϕ_{d})). Let σ : S → T_{d} be the k-linear map that takes each monomial to its unique standard representative. Because of the way we have defined the order on T_{d} we have in(σ(p)) = σ(in(p)). Also, σ takes J into V_{d}(J). As a consequence we have:

Lemma 7 If J is a homogeneous ideal of S, then in(V_{d}(J)) is the ideal K generated by in(ker(ϕ_{d})) and the monomials σ(m) for m ∈ in(J) ∩ (im(ϕ_{d})).

Proof: For each degree e it is clear that dim_{k}(T_{d}/K)_{e} ≤ dim_{k}(S/J)_{de}, so it is enough to show that K ⊂ in(V_{d}(J)). Let p ∈ J be a form with initial term m. Clearly σ(p) ∈ V_{d}(J), and in(σ(p)) = σ(m).

Definition: If m ∈ S is a monomial, then following Eliahou and Kervaire [ElKe90] we write max(m) for the largest index i such that x_{i} divides m. We call a monomial ideal I (combinatorially) stable if for every monomial m ∈ I and j < max(m), the monomial (x_{j}/x_{max(m)})m ∈ I.

Theorem 8 If J is a homogeneous ideal of S such that in(J) is combinatorially stable, then in(V_{d}(J)) is generated by in(ker(ϕ_{d})) and the monomials σ(m) where m runs over the minimal generators of in(J) ∩ im(ϕ_{d}). Thus if δ(in(J)) ≤ u, then δ(in(V_{d}(J))) ≤ max(2, ⌈u/d⌉).

Proof: Let n ∈ in(J) ∩ im(ϕ_{d}). By Lemma 7 it suffices to show that σ(n) is divisible by some σ(m), where m is a minimal generator of in(J) ∩ im(ϕ_{d}). Let m′ be an element of in(J) ∩ im(ϕ_{d}) of minimal degree among those dividing n, and write n = x_{i_{1}} · · · · · x_{i_{sd}} with i_{1} ≤ ⋯ ≤ i_{sd}. Say deg m′ = de. Since in(J) is combinatorially stable, it follows that m := x_{i_{1}} · · · · · x_{i_{de}} ∈ in(J), and since m has degree de, we have m ∈ im(ϕ_{d}) as well. If m were not a minimal generator of in(J) ∩ im(ϕ_{d}), then some proper divisor of it would be in in(J) ∩ (im(ϕ_{d})) and would divide n, contradicting our choice of m′. As σ(m) divides σ(n), we are done.

Example 1: The hypothesis of stability cannot be dropped. For example, if r = 3 and J = (x_{1}x_{2}x_{3}), and we take the lexicographic or reverse lexicographic order on S, then the initial ideal in(V_{d}(J)) (defined using the order on T_{d} we associate to the given order on S) requires a cubic generator for all d.

To obtain Theorem 2 as given in the introduction, we need to recall the notion of Castelnuovo-Mumford regularity.

Definition: For I ⊂ S, the regularity of I is defined as

reg(I) = max{t_{i}^{S}(I) − i|i ≥ 0}.

Note: t_{0}^{S}(I) = δ(I) ≤ reg(I).

Bayer and Stillman [BaSt87] give the following criterion for an ideal to be mregular, assuming (as we do throughout this paper) that the field k is infinite.

Theorem 9 (BaSt87) Let I ⊂ S be an ideal generated in degrees ≤ e. The following conditions are equivalent:

1. I is e-regular,

2. (a) For some j ≥ 0 and for some linear forms h_{1}, …, h_{j} ∈ S_{1} we have

((I, h_{1}, …, h_{i−1}) : h_{i})_{e} = (I, h_{1}, …, h_{i−1})_{e}

for i = 1, …, j, and

(b)

(I, h_{1}, …, h_{j})_{e} = S_{e}.

3. Conditions 2a) and 2b) hold for some j ≥ 0 and for generic linear forms h_{1}, …, h_{j} ∈ S_{1}.

Note: Let g be generically chosen in the Borel group B, the subgroup of Gl(r) consisting of the upper triangular matrices. Then ⟨gx_{r}, …, gx_{r−j+1}⟩ is a generic linear subspace for I. Since gx_{i} is a generic linear form with respect to (I, gx_{r}, …, gx_{i+1}), x_{i} is a generic linear form with respect to (g^{−1}I, x_{r}, …, x_{i+1}). If I is Borel-fixed, then g^{−1}I = I, and hence we can replace h_{1}, …, h_{j} by x_{r}, …, x_{r−j+1} in the statement of Theorem 9 in this case.

Proposition 10 Let I ⊂ S be a Borel-fixed monomial ideal generated in degrees ≤ e. Then I is e-regular if and only if I_{e} is combinatorially stable.

Proof: By the note following the statement of Theorem 9, we may replace h_{1}, …, h_{j} by x_{r}, …, x_{r−j+1} in the statement of the theorem.

Suppose I is e-regular. Then 2b) of Theorem 9 implies that I includes all monomials in x_{1}, …, x_{r−j} of degree e. And 2a) of the same theorem implies that for every monomial m ∈ I of degree e with max(m) > r − j, I also contains x_{k}/x_{max(m)} · m for every k with 1 ≤ k ≤ max(m). Taken together, these two statements imply I_{e} is combinatorially stable.

Conversely, suppose I_{e} is combinatorially stable, and let j be the smallest integer such that I contains a power of x_{r−j}. It follows that x_{r−j}^{e} ∈ I_{e}, and by stability, (x_{1}, …, x_{r−j})^{e} ⊂ I, and hence 2b) holds. Let m ∈ ((I, x_{r}, …, x_{i+1}) : x_{i})_{e} for some r − j + 1 ≤ i ≤ r; since I is a monomial ideal, we can assume that m is a monomial in proving 2a). If m is divisible by x_{k} for some i + 1 ≤ k ≤ r, then it is clear that m ∈ (I, x_{r}, …, x_{i+1}). Thus we may assume m is not divisible by x_{i+1}, …, x_{r}. Since the monomial mx_{i} belongs to (I, x_{r}, …, x_{i+1}), it must belong to I. Since it has degree e + 1, there must be a monomial m′ ∈ I_{e} and an l such that mx_{i} = m′x_{l}. Clearly l ≤ i. Since m = (x_{l}/x_{i})m′, combinatorial stability of I_{e} implies that m ∈ I_{e}. Thus 2a) of Theorem 9 holds as well, and I is e-regular.

Proof of Theorem 3: If I is an ideal in generic coordinates which is e-regular, then by Theorem 9, in_{>}(I) is generated in degrees ≤ e, where < is the reverse lexicographic order. By the above proposition, in(I_{e}) with respect to reverse lexicographic order is combinatorially stable, and hence by Theorem 8 we have:

Δ(V_{d}(I)) ≤ δ(in_{>′}(V_{d}(gI)))
≤ max(2, ⌈reg(gI))/d⌉)
= max(2, ⌈reg(I)/d⌉)

(where g is a “general” choice of coordinates, > is reverse lexicographic order, and >′ is the induced order), proving Theorem 2.

3 Comments on the main theorem

We have proved that for any homogeneous ideal I ⊂ S, we have Δ(V_{d}(I)) ≤ max(2, ⌈reg(I)/d⌉). In particular, for suitable coordinates and order on T_{d}, the Veronese ideal V_{d}(I) has quadratic initial ideal for d ≥ reg(I)/2, and it follows that the Veronese subring T_{d}/V_{d}(I) ⊂ S/I is Koszul for d ≥ reg(I)/2. In this section we will estimate the regularity of I in order to bound Δ(V_{d}(I)) in terms of other invariants of I such as Δ(I). These results can probably be improved, but we will give an example to show that the most optimistic hopes are false.

Theorem 11 Let r be the number of generators of the polynomial ring S. For any homogeneous ideal I ⊂ S, Δ(V_{d}(I)) ≤ max(2, ⌈(rΔ(I) − r + 1)/d⌉). In particular, for suitable coordinates and order on T_{d}, the Veronese ideal V_{d}(I) has quadratic initial ideal for d ≥ (rΔ(I) − r + 1)/2.

Proof: The Taylor resolution [Ta60] gives an upper bound on reg(I), specifically:

reg(I) ≤ rΔ(I) − r + 1.

With Theorem 2, this gives the result. It is worth mentioning that, by Bayer and Stillman’s Theorem 9, there are actually upper and lower bounds relating reg(I) and Δ(I):

Δ(I) ≤ reg(I) ≤ rΔ(I) − r + 1.

The assumption d ≥ Δ(I)/2 is not enough to imply that V_{d}(I) has quadratic initial ideal, by the example at the end of this section. We do not know the best estimate for Δ(V_{d}(I)) in terms of Δ(I). The problem is combinatorial in the sense that it suffices to consider monomial ideals I.

We have been assuming that the field k is infinite. For arbitrary (in particular, finite) fields k, we have a slightly weaker version of Theorem 11: there is an order on T_{d} such that V_{d}(I) has quadratic initial ideal for d ≥ r⌈Δ(I)/2⌉. We omit the proof, which is not too difficult given a definition of the correct order. The ordering which yields this result is defined as follows:

Definition: For each monomial m in S of degree d, we produce a vector

ν(m) = (ν_{11}(m), ν_{12}(m), …, ν_{1r}(m), ν_{21}(m), …, ν_{2r}(m), ν_{31}(m) …),

where ν_{ij}(m) = \{ \begin{matrix}0 & if x_{j}^{i}|m \\ 1 & else \end{matrix} . The order on monomials in S of degree d is then

defined by m > n if ν(m) > ν(n) in lexicographic order. We define the order of the variables in T_{d} using the above order on S. Specifically, z_{m} > z_{n} if m > n in the order on S defined above. Given this ordering on the variables in T_{d}, let the order on the monomials in T_{d} be reverse lexicographic order.

For some monomial ideals I, we can improve the Taylor bound on the regularity of I. First, since one direction of the proof of Proposition 10 does not use the Borel-fixed hypothesis, we have:

Proposition 12 If in(I) is generated in degrees ≤ u and in(I)_{u} is combinatorially stable, then

reg(I) ≤ u.

Next we generalize the definition of combinatorial stability.

Definition: Let q be an integer. A monomial ideal I is q-combinatorially stable, if for every m ∈ I and for each j < max(m) there exists an integer s with 1 ≤ s ≤ q such that x_{j}^{s}/x_{max(m)}^{s}m ∈ I.

Proposition 13 Let I be an ideal in generic coordinates, and let e = δ(in_{>}(I)), where > is reverse lexicographic order. If I is q-combinatorially stable, then reg(I) ≤ e + (r − 1)(q − 1).

Proof: Let t = e + (r − 1)(q − 1). By Proposition 12, we need only show that J := in(I)_{t} is combinatorially stable. Let m ∈ J. m = x_{1}^{b_{1}+c_{1}} ⋯ x_{r}^{b_{r}+c_{r}}, where l := x_{1}^{b_{1}} ⋯ x_{r}^{b_{r}} ∈ in(I)_{e}, and set n := x_{1}^{c_{1}} ⋯ x_{r}^{c_{r}}. We have ∑_{i=1}^{r} c_{i} = (r − 1)(q − 1). If max(m) = max(n), then (x_{i}/x_{max(m)})m ∈ I_{t} for all i with 1 ≤ i ≤ x_{max(m)} because n is divisible by x_{max(m)}. If max(m) > max(n), then max(m) = max(l), and either there exists some index k such that c_{k} ≥ q, or else c_{j} = q − 1 for all j = 1, …, n. In the first case, we can rewrite m as l′n′, where l′ = (x_{k}^{s}/x_{max(l)}^{s})l and n′ = (x_{max(l)}^{s}/x_{k}^{s})n, for some 1 ≤ s ≤ q. After doing so, max(m) = max(n) and we conclude as before. In the second case, the degree of x_{i} in x_{i}/x_{max(m)}m is b_{i} + c_{i} + 1, and c_{i} + 1 = q. We may rewrite (x_{i}/x_{max(m)})m as (x_{i}^{s}/x_{max(l)}^{s})ln′, where n′ = (x_{max(l)}^{s−1}/x_{i}^{s−1})n, and 1 ≤ s ≤ q is chosen so that (x_{i}^{s}/x_{max(l)}^{s})l ∈ I. Thus (x_{i}/x_{max(m)})m is in I_{t}.

If char k = 0, then every ideal I in generic coordinates is Borel-fixed and hence 1-combinatorially stable. In this case, the proposition above yields reg(I) ≤ e, and in fact, equality holds, as Bayer and Stillman proved in [BaSt87]. In char k = p, every ideal I in generic coordinates has a q-combinatorially stable initial ideal for some q that is a power of p ≤ δ(in(I)) [Pa], but even if q is chosen to be as small as possible, reg(I) can be strictly less than e + (r − 1)(q − 1). An example is the ideal I = {a^{6}, a^{2}b^{4}, a^{2}c^{4}, b^{8}, c^{8}}, which is 8-combinatorially stable (implying a bound of 22 on the regularity), but has regularity 16. Also, I has a quadratic initial ideal in the Veronese embedding of degree 5, which is strictly less than the degree of 7 given by Theorem 8.

As noted above, in characteristic 0 and generic coordinates, the regularity of I is equal to δ(in(I)), where the initial ideal is with respect to reverse lexicographic order. In characteristic p, we cannot replace reg(I) with δ(in(I)) in the statement of Theorem 2, as the following example illustrates.

Example 2: A Borel-fixed ideal I ⊂ k[a, b], with char k = 2, e := δ(I) = 6, such that the algebra T_{3}/V_{3}(I) is not Koszul. It follows that the initial ideal of the Veronese embedding of degree ⌈e/2⌉ = 3 is not generated in degree 2 under any order and any generators for the graded algebra T_{3}. In fact the ideal defined below has the same properties for k of characteristic 0, except that it is not Borel-fixed in characteristic 0.

Let I = (a^{6}, a^{2}b^{4}), and consider the embedding in degree 3 = ⌈6/2⌉. Let A = T_{3}/V_{3}(I). Thus, in the obvious coordinates y_{i} = a^{3−i}b^{i},

A = k[y_{0}, y_{1}, y_{2}, y_{3}]/(y_{0}^{2} = 0, y_{0}y_{2} = y_{1}^{2}, y_{0}y_{3} = y_{1}y_{2}, y_{1}y_{3} = y_{2}^{2} = 0).

The graded vector space T or_{3}^{A}(k, k) is not entirely in degree 3: it has dimension 26 in degree 3 and dimension 2 in degree 4. So A is not Koszul.

In fact, under the induced order used throughout this paper, in(V_{3}(I)) requires 2 cubic generators. However, in(V_{4}(I)) is generated in degree 2. The regularity of I is 9.

4 Resolution of multihomogeneous modules

Fundamental to the discussion of rates above is the estimate of the rate for a monomial ideal given (without proof) by Backelin in [Ba86]. The case of quadratic monomials follows at once from the more precise result of Fröberg in [Fr75]. Fröberg’s result was recently reexamined and reproved by Bruns, Herzog and Vetter [BrHeVe] using a different method. Jürgen Herzog has pointed out to us that their method actually proves the entire result claimed by Backelin, in a somewhat strengthened form, and we now present this argument.

Let S = k[x_{1}, …, x_{r}] be a polynomial ring over a field k. We will regard S as a Z^{r}-graded ring, graded by the monomials. Suppose that I is a monomial ideal of S, and set A := S/I; the ring A is again Z^{r}-graded. If M is a finitely generated Z^{r}-graded module over A, then M has a Z^{r}-graded minimal free resolution over A. The vector spaces Tor_{i}^{A}(k, M) are Z^{r}-graded.

For the purpose of bounding degrees it is convenient to turn these multigradings into single gradings. Rather than simply using the total degree, we get a more refined result by defining weights, as follows: Let w_{1}, …, w_{r} be non-negative real numbers. For any monomial m = x_{1}^{α_{1}} ⋯ x_{r}^{α_{r}} define the weight of m to be w(m) = ∑ w_{i}α_{i}. Generalizing the definition of t_{i}^{A}(M) used above we define t_{i}^{A}(w, M) to be the maximal weight, with respect to w, of a nonzero vector in T or_{i}^{A}(k, M).

We can estimate the t_{i}^{A}(w, M) as follows. Given an ordered set {g_{1}, …, g_{s}} of generators of M we get a filtration

Ag_{1} ⊂ Ag_{1} + Ag_{2} ⊂ ⋯ ⊂ Ag_{1} + ⋯ + Ag_{s} = M

of M with quotients the cyclic modules A/J_{i} where J_{i} = ((g_{1}, …, g_{i−1}) : g_{i}). The set of generators {g_{i}} also gives rise to a surjection ϕ of a free A-module A^{s} to M sending the i^{th} basis element to g_{i}. It is easy to show that the kernel of ϕ has a filtration whose successive quotients are the ideals J_{i} (see the proof of Theorem 15). Thus the weights of the generators of the J_{i} added to the weights of the g_{i} give a bound for t_{1}^{A}(w, M) (we get a bound and not an exact result because the set of generators for the first syzygy of M produced from sets of generators for the J_{i} may not be minimal). Moreover, if we have a method for bounding the weights of syzygies of the J_{i}, we may continue this process. The following Lemma provides what we require:

Lemma 14 Let S = k[x_{1}, …, x_{r}] be a polynomial ring over a field k. Suppose that I is an ideal of S, generated by monomials n_{1}, …, n_{t}, and set A := S/I.

If J = (m_{1}, …, m_{s}) ⊂ A is an ideal generated by the images m_{i} of monomials m′_{i} of S, then the quotient ((m_{1}, …, m_{s−1}) :_{A} m_{s}) is generated by the images in A of divisors of the monomials m′_{1}, …, m′_{s−1} and proper divisors of the monomials n_{1}, …, n_{t}.

Proof: The quotient is the image in A of ((n_{1}, …, n_{t}, m_{1}, …, m_{s−1}) :_{S} m_{s}) and is thus generated by divisors of the monomials n_{1}, …, n_{t}, m_{1}, …, m_{s−1}. The divisors of the n_{i} that are not proper go to zero in A.

Using Lemma 14 with the idea above we obtain:

Theorem 15 Let S = k[x_{1}, …, x_{r}] be a polynomial ring over a field k, and let w be a weight function on S as above. Suppose that I is an ideal of S, generated by monomials, and set A := S/I. Let M be a Z^{r}-graded A-module with Z^{r}-homogeneous generators {g_{1}, …, g_{s}}, of weights ≤ d, and set J_{i} = (Ag_{1} + … + Ag_{i−1} : _{A} g_{i}). If the J_{i} are generated by elements of weight ≤ e, and both these elements and the proper divisors of the generators of I have weights ≤ f, then for each integer i ≥ 1 we have

t_{i}^{A}(w, M) ≤ d + e + (i − 1)f.

Proof: We will inductively construct a (not necessarily minimal) free resolution

⋯ → F_{2} → F_{1} → F_{0} → M → 0

such that the generators of F_{0} have weights ≤ d, the generators of F_{1} have weights ≤ d+e, and for i ≥ 2 the weights of the generators of F_{i} are ≤ d+e+(i−1)f. Since the minimal (multigraded) free resolution is a summand of any free resolution, it follows that the weights of the i^{th} free module in the minimal resolution are also ≤ d + e + (i − 1)f, proving the desired inequality.

Let F_{0} be Z^{r}-graded free A-module with s generators whose degrees match those of the g_{i}, so that the surjection ϕ_{0} : F_{0} → M sending the i^{th} basis vector of F_{0} to g_{i} is multihomogeneous. The weights of the generators of F_{0} are ≤ d.

We will prove by induction on s that the module ker(ϕ_{0}) admits a filtration with successive quotients isomorphic, up to a shift in multidegree, to the ideals J_{i}, and that the weight of the generators of this kernel are ≤ d + e. If we define F′_{0} to be F_{0} modulo the first basis element, and define M′ by the short exact sequence

0 → Ag_{1} → M → M′ → 0

then by the snake lemma we get a short exact sequence

0 → J_{1} → ker(ϕ_{0}) → ker(ϕ′_{0}) → 0,

where ϕ′_{0} : F′_{0} → M′ is the induced map. By induction ker(ϕ′_{0}) has a filtration with quotients J_{2}, …, J_{s}, and generators of weights ≤ d + e. This gives the desired filtration of ker(ϕ_{0}). The weights of the generators of the copy of J_{1} in the kernel are the weights of the monomials generating the ideal J_{1} plus the weight of g_{1}, so they are also ≤ d + e, and we are done.

Using this filtration of ker(ϕ_{0}), we define a free module F_{1} whose generators have weights ≤ d + e and a map ϕ_{1} : F_{1} → F_{0} sending the generators of F_{1} to representatives h_{l} in ker(ϕ_{0}) of the generators of the successive quotients J_{i}.

We now repeat the argument, replacing M by ker(ϕ_{0}) and ϕ_{0} by ϕ_{1}. Lemma 14 applied to the ideals J_{i} implies that the argument works as before if we replace e by f: that is, ker(ϕ_{1}) has a filtration with successive quotients isomorphic (up to a shift in mult-degree) to ideals with generators of weight ≤ f. This allows us to construct F_{2} with generators of degrees ≤ d + e + f that map onto generators h_{i} of ker(ϕ_{0}) such that the ideals (Ah_{1} + … + Ag_{l−1} : _{A} g_{l}) have generators of weight ≤ f.

We may continue to repeat the argument, using the bound f from the second step on, and constructing the desired resolution.

In the special case of the resolution of the residue class field k, we may take J_{1} to be the maximal ideal, and we get Backelin’s result referred to above:

Corollary 16 Let S = k[x_{1}, …, x_{r}] be a polynomial ring over a field k. Suppose that I is an ideal of S, generated by monomials of degree ≤ f, and set A := S/I. We have

t_{i}^{A}(k) ≤ 1 + (i − 1)(f − 1).

Note that Theorem 15 does not prove much about the entries of the matrices in even a non-minimal resolution.

With the ring A as in the Theorem, it would be interesting to know whether there is a minimal free resolution (say of the residue class field ), with bases for the free modules occurring, such that the entries of the matrices representing the maps of the resolution with respect to the given bases all have low multidegree (or low weight). The rate bound given in the above theorem, together with the requirement that the resolution be minimal (so that each syzygy has weight at least 1) implies that the individual entries of the i^{th} syzygy matrix must have weights bounded by d+e+(i−1)f −i+1. We do not know whether there always exists a free resolution with bases — even non-minimal — such that the entries appearing in the matrices are all proper divisors of the generators of I, or even of the least common multiple of the generators of I.

5 Segre Products of Veronese embeddings

The proofs of Proposition 6 and Theorem 8 can be easily generalized to the Segre-Veronese case. Below are the definitions and statements we can make in this case.

Let

S := k[x_{11}, …, x_{1r_{1}}, …, x_{s1}, …, x_{sr_{s}}]

be the coordinate ring of P_{1}^{r} × ⋯ × P^{r_{s}}, where x_{i} = (x_{i1}, …, x_{ir_{i}}) are the homogeneous coordinates on P^{r_{i}}. And let

T := k[{z_{m}}, m a monomial of S of multi-degree (d_{1}, …, d_{s})]

be the coordinate ring of P^{N}.

Definition: ϕ : T ⟶ S by ϕ(z_{m}) = m = m_{1} ⋯ m_{s}, where m_{i} = x_{i1}^{α_{i1}} ⋯ x_{is}^{α_{is}}.

ker(ϕ) is generated by the quadratic binomials of the form z_{m}z_{n} − z_{m′}z_{n′}, where m · n = m′ · n′ in S. As in section 2, if a, b are monomials in T, a > b if ϕ(a) > ϕ(b), or ϕ(a) = ϕ(b) and a > b in reverse lexicographic order.

Proposition 17 In reverse lexicographic order, the initial terms of the binomials z_{m}z_{n} − z_{m′}z_{n′} generate in(ker(ϕ)).

Definition: The stabilization {I} of an ideal I is defined to be the ideal generated by

{(x_{j}/x_{max(m)})m |m ∈ I, j = 1, …, max(m)}.

Definition: Call a multi-homogeneous monomial ideal I combinatorially stable if it is combinatorially stable in each set of variables independently. That is, given

x_{1}^{α_{1}} ⋯ x_{s}^{α_{s}} ∈ I,

we must have

{x_{1}^{α_{1}}} · {x_{2}^{α_{2}}} · · · · · {x_{s}^{α_{s}}} ⊂ I,

where {x_{i}^{α_{i}}} is the set of all monomials necessary for an ideal in k[x_{i0}, …, x_{ir_{i}}] containing x_{i}^{α_{i}} to be combinatorially stable, and where the above product is the outer product, i.e. all possible products of elements taken one from each set.

Theorem 18 If I is a multi-homogeneous ideal whose initial ideal is combinatorially stable, then in(σ(I)) = σ(in(I)) (where the initial terms on the left are computed with respect to the induced order with ties broken by reverse lex). Thus if in(I) is generated in degrees ≤ (u_{1}, …, u_{s}) (u_{i} = maximum degree of any generator with respect to the i^{th} set of variables), then in(V (I)) is generated in degrees ≤ max(2, ⌈u_{1}/d_{1}⌉, …, ⌈u_{s}/d_{s}⌉).

6 Another obstruction to having a quadratic initial ideal

In this section, we formulate a general obstruction to the existence of a quadratic initial ideal for a given polynomial ideal, beyond the obvious requirement that the ideal must be generated by quadratic polynomials, and even beyond the stronger requirement that the quotient ring must be a Koszul algebra. We use the obstruction to show that in certain dimensions, the ideal of a generic complete intersection of quadrics has no quadratic initial ideal, although every complete intersection of quadrics is a Koszul algebra [BaFr85].

Note. In discussing the existence of a quadratic initial ideal for a homogeneous ideal I in a polynomial ring S = k[x_{1}, …, x_{r}], we are asking whether there exists a set of coordinates x′_{1}, …, x′_{r} (linear combinations of x_{1}, …, x_{r} ∈ S_{1}) and a monomial order with respect to which in(I) is generated by quadratic polynomials.

Theorem 19 Let I be a homogeneous ideal in a polynomial ring S. Consider the Krull dimensions r = dim(S), n = dim(S/I), e = r − n. (Thus, if n ≥ 1, n is one more than the dimension of the projective variety defined by I.) If there are coordinates and a monomial order such that the initial ideal in(I) has quadratic generators, then the ideal I contains e linearly independent quadratic elements of the form:

c_{1}x_{1}^{2} = x_{2}L_{1,2} + ⋯ + x_{e+n}L_{1,e+n}
⋮
c_{e}x_{e}^{2} = x_{e+1}L_{e,e+1} + ⋯ + x_{e+n}L_{e,e+n}

for some basis x_{1}, …, x_{e+n} for S_{1} and some c_{i} ∈ k and linear forms L_{ij} ∈ S_{1}.

In particular, I contains an m-dimensional space of quadrics of rank ≤ 2(n + m) − 1 for every m ≤ codim(I).

We recall that the rank of a quadratic form Q over a field k is the rank of a symmetric matrix representing the form.

Proof. We are given that there is a basis x_{1}, …, x_{e+n} for the vector space S_{1} and an ordering of the x-monomials, such that the resulting initial ideal in(I) is generated by in(I)_{2}. We can assume that x_{1} > ⋯ > x_{e+n} in the monomial ordering.

We observe that for i = 1, …, e, there must be at least i quadratic monomials x_{j}x_{k} with e − i + 1 ≤ j, k ≤ e + n which are not allowable. Otherwise the Hilbert series of S/I would be at least equal to the Hilbert series of an algebra k[x_{e−i+1}, … x_{e+n}]/(< i relations), so the dimension of S/I would be at least that of the latter ring, which is greater than n; this contradicts dim(S/I) = n.

Thus, for i = 1, …, e, there are i monomials x_{j}x_{k}, e − i + 1 ≤ j, k ≤ e + n, which are linear combinations of earlier monomials x_{l}x_{m}. No matter what monomial ordering we are using, at least one of l and m must be > e − i + 1 in this situation. So, for all 1 ≤ i ≤ e, R satisfies i independent relations of the form:

bx_{e−i+1}^{2} + (a_{e−i+2,1}x_{e−i+2}x_{1} + ⋯ + a_{e−i+2,e+n}x_{e−i+2}x_{e+n})
+(a_{e−i+3,1}x_{e−i+3}x_{1} + ⋯ + a_{e−i+3,e+n}x_{e−i+3}x_{e+n}) + ⋯
+(a_{e+n,1}x_{e+n}x_{1} + ⋯ + a_{e+n,e+n}x_{e+n}^{2}) = 0.

This implies the statement of the lemma.

Corollary 20 Let k be an infinite field, and let I be an ideal in S = k[x_{1}, …, x_{e+n}] generated by e generic quadratic forms defined over k. We assume that n ≥ 0. (If n ≥ 1, S/I is the homogeneous coordinate ring of an (n − 1)-dimensional complete intersection of quadrics in P^{e+n−1}.) If

n < \frac{(e − 1)(e − 2)}{6}

then generic complete intersection ideals I as above do not admit any quadratic initial ideal.

Proof. Any n-dimensional complete intersection of homogeneous quadrics in affine (e+n)-space can be described by a point of the Grassmannian of e-dimensional subspaces of S^{2}V, Gr(X_{e} ⊂ S^{2}V), where we let V = S_{1}, a vector space of dimension e + n over k; conversely, a nonempty open subset of this Grassmannian corresponds to complete intersections.

Those e-dimensional linear spaces of quadrics which generate an ideal which admits a quadratic initial ideal can, by Lemma 19, be written in the form:

c_{1}x_{1}^{2} = x_{2}L_{1,2} + ⋯ + x_{e+n}L_{1,e+n}
⋮  (∗)
c_{e}x_{e}^{2} = x_{e+1}L_{e,e+1} + ⋯ + x_{e+n}L_{e,e+n}

for some basis x_{1}, …, x_{e+n} for V and some c_{i} ∈ k and linear forms L_{ij} ∈ V. We want an upper bound for the dimension of the space of e-dimensional linear subspaces of S^{2}V which can be written in this form. If our bound is less than the dimension of the whole Grassmannian Gr(X_{e} ⊂ S^{2}V), then we will know that generic complete intersections of quadrics in this dimension do not have any quadratic initial ideal.

Our dimension estimate (which is not always sharp) is based on the following observation. The basis x_{1}, …, x_{e+n} for V is not important, only the flag ⟨x_{e+n}, …, x_{e+1}⟩ ⊂ ⟨x_{e+n}, …, x_{e}⟩ ⊂ ⋯ ⊂ ⟨x_{e+n}, …, x_{1}⟩ = V. That is, if a linear system of quadrics has the form (∗) for one basis x_{1}, …, x_{e+n} of V, then it has the same form for any basis which gives the same flag ⟨x_{e+1}, …, x_{e+n}⟩ ⊂ · · ·.

For any flag V_{n} ⊂ V_{n+1} ⊂ ⋯ ⊂ V_{e+n} = V, we consider an associated flag W_{a_{1}} ⊂ W_{a_{2}} ⊂ ⋯ ⊂ W_{a_{e}} = S^{2}V defined by

W_{a_{i}} = (V_{n+i−1} · V) + (V_{n+i} · V_{n+i}) ⊂ S^{2}V.

(That is, in terms of any basis x_{1}, …, x_{e+n} adapted to the flag V_{n} ⊂ ⋯, W_{a_{i}} is the space of quadrics of the form c_{e−i+1}x_{e−i+1}^{2} = x_{e−i+2}L_{e−i+2} + ⋯ + x_{e+n}L_{e+n}, L_{j} ∈ V.)

Note. We always use the notation V_{i}, W_{i}, etc. to denote i-dimensional vector spaces.

Then, also associated to any flag V_{n} ⊂ V_{n+1} ⊂ ⋯ ⊂ V_{e+n} = V, we can consider the space of flags X_{1} ⊂ ⋯ ⊂ X_{e} ⊂ V such that X_{i} ⊂ W_{a_{i}} for i = 1, …, e. Let Q_{n,e} be the space of flags V_{n} ⊂ V_{n+1} ⊂ ⋯ ⊂ V_{e+n} = V and X_{1} ⊂ ⋯ ⊂ X_{e} ⊂ S^{2}V such that X_{i} ⊂ W_{a_{i}} for i = 1, …, e. Then the image of the map

Q_{n,e} → Gr(X_{e} ⊂ S^{2}V)
(V_{i}, X_{i}) ↦ X_{e}

contains the set of complete intersections of e homogeneous quadrics in (e+n)-space which have a quadratic initial ideal.

Moreover, it is easy to compute the dimension of Q_{n,e}, which is an iterated projective bundle over the flag manifold Fl(V_{n} ⊂ V_{n+1} ⊂ ⋯ ⊂ V_{e+n} = V): given a flag V_{n} ⊂ ⋯ and hence the associated subspaces W_{a_{i}}, we first choose the line X_{1} ⊂ W_{a_{1}}, then a line X_{2}/X_{1} ⊂ W_{a_{2}}/X_{1}, and so on.

The dimension of the vector space W_{a_{i}}, for i = 1, …, e, is

a_{i} = (e + n) + (e + n − 1) + ⋯ + (e − i − 2) + 1
= (e + n)(e + n + 1)/2 − (e − i + 1)(e − i + 2)/2 + 1.

Using this, we compute the dimension of the variety Q_{n,e}:

dim Q_{n,e} = dim Fl(V_{n} ⊂ ⋯ ⊂ V_{e+n} = V) + ∑_{i=1}^{e} dim P(W_{a_{i}}/X_{i−1})

= en + \frac{e(e − 1)}{2} + ∑_{i=1}^{e}[\frac{(e + n)(e + n + 1)}{2} − \frac{(e − i + 1)(e − i + 2)}{2} + 1 − i]

= en + \frac{e(e − 1)}{2} + \frac{e(e + n)(e + n + 1)}{2} − (∑_{j=1}^{e} \frac{j(j + 1)}{2}) + e − \frac{e(e + 1)}{2}

= e(n + \frac{(e + n)(e + n + 1)}{2} − \frac{(e + 1)(e + 2)}{6}).

So

dim Q_{n,e} < dim Gr(X_{e} ⊂ S^{2}V)
⟺ e(n + (e + n)(e + n + 1)/2 − (e + 1)(e + 2)/6) < e((e + n)(e + n + 1)/2 − e)
⟺ n < (e + 1)(e + 2)/6 − e
⟺ n < (e − 1)(e − 2)/6

Thus, if n < (e − 1)(e − 2)/6, then the image of the map Q_{n,e} → Gr(X_{e} ⊂ S^{2}V) has dimension less than the dimension of Gr(X_{e} ⊂ S^{2}V). Thus for n < (e − 1)(e − 2)/6, the ideal generated by e generic quadratic forms in e + n variables has no quadratic initial ideal.

For example, the ideal generated by 3 generic quadratic forms in 3 variables has no quadratic initial ideal. Similarly for a generic complete intersection of 5 quadrics in P^{5}, or a generic complete intersection of 6 quadrics in P^{7}. (These last examples are smooth curves of genus 129.)

Explicitly, say over a field k of characteristic 0, the ideal

I = (x(x + y), y(y + z), z(z + x)) ⊂ k[x, y, z] = S

is a complete intersection of quadrics, and so S/I is a Koszul algebra [BaFr85], but one can check using Lemma 19 that it has no quadratic initial ideal, for any coordinates and any monomial order. (One has to check that no nonzero linear combination of the relations x(x + y), y(y + z), z(z + x) is the square of a linear form, which is easy.)

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