Generalizations of Stillman’s conjecture via twisted commutative algebras

Generalizations of Stillman’s conjecture via
twisted commutative algebra

Daniel Erman Department of Mathematics, University of Wisconsin, Madison, WI derman@math.wisc.edu http://math.wisc.edu/~derman/ Steven V Sam Department of Mathematics, University of Wisconsin, Madison, WI svs@math.wisc.edu http://math.wisc.edu/~svs/  and  Andrew Snowden Department of Mathematics, University of Michigan, Ann Arbor, MI asnowden@umich.edu http://www-personal.umich.edu/~asnowden/
April 24, 2018
Abstract.

Combining recent results on noetherianity of twisted commutative algebras by Draisma and the resolution of Stillman’s conjecture by Ananyan–Hochster, we prove a broad generalization of Stillman’s conjecture. Our theorem yields an array of boundedness results in commutative algebra that only depend on the degrees of the generators of an ideal, and not the number of variables in the ambient polynomial ring.

2010 Mathematics Subject Classification:
14F05, 13D02.
DE was partially supported by NSF DMS-1302057 and NSF DMS-1601619. SS was partially supported by NSF DMS-1500069. AS was supported by NSF DMS-1303082 and DMS-1453893 and a Sloan Fellowship.

1. Introduction

The introduction of noetherianity conditions in commutative algebra streamlined and generalized boundedness results in invariant theory. We revisit this theme: using Draisma’s recent noetherianity result for twisted commutative algebras, and the resolution of Stillman’s conjecture by Ananyan–Hochster, we prove a boundedness result for a large class of ideal invariants. This can be seen as a far reaching generalization of Stillman’s conjecture.

As preparation for this result, we also establish a number of technical statements about certain topological spaces with group actions relevant to the study of ideal invariants.

1.1. Ideal invariants

Fix a field . An ideal invariant (over ) is a rule that associates a quantity to every homogeneous ideal in every standard-graded polynomial ring , such that only depends on the pair up to isomorphism. There are countless examples of ideal invariants: degree, projective dimension, regularity, the Betti number, etc.

Let be a tuple of positive integers. We say that an ideal is type if it is generated by where is homogeneous of degree . We say that an ideal invariant is bounded in degree if there exists such that for every type ideal we have or . We say that is degreewise bounded if it is bounded in degree for all . The main point of this definition is that the bound is independent of the number of variables.

There are two “niceness” conditions we require on our ideal invariants. We say that is cone-stable if for all , that is, adjoining a new variable does not affect the invariant. We say that is weakly upper semi-continuous if the following holds: given a polynomial ring , a variety over , and a homogeneous ideal sheaf of such that is -flat, the map is upper semi-continuous as varies over the geometric points of ; that is, for each , the locus of geometric points is Zariski-closed. (Here denotes the fiber of at , which is an ideal of by the flatness assumption.)

Stillman’s conjecture is exactly the statement that the ideal invariant “projective dimension” (which is cone-stable and weakly upper semi-continuous) is degreewise bounded. We prove the following generalization:

Theorem 1.1.

Any ideal invariant that is cone-stable and weakly upper semi-continuous is degreewise bounded.

1.2. The space of ideals

We introduce a topological space that parametrizes isomorphism classes of type ideals in the infinite polynomial ring . We construct as a quotient of an infinite dimensional variety that parametrizes generating sets of type ideals. Theorem 1.1 is deduced from two results about .

Theorem 1.2.

The space is noetherian.

Theorem 1.3.

The space admits a finite stratification such that the universal quotient ring is flat over each stratum see §3.6 for the precise meaning of this.

The primary input into the proof of Theorem 1.2 is Draisma’s theorem [draisma] on noetherianity of polynomial representations, while the primary input into the proof of Theorem 1.3 is the resolution of Stillman’s conjecture by Ananyan–Hochster [ananyan-hochster].

We now sketch the proof of Theorem 1.1. Suppose is a cone-stable weakly upper semi-continuous ideal invariant. By cone-stability, defines a function on . Let be the locus where . These loci form a descending chain. By weak upper semi-continuity and Theorem 1.3, is closed in . By Theorem 1.2, is noetherian, and so stabilizes for each . Thus stabilizes, which shows that is bounded.

Remark 1.4.

As should be clear, our proof of Theorem 1.1 is a consequence of Stillman’s conjecture. A direct proof of Stillman’s conjecture using [draisma] appears in [ess-stillman]. See also Remark 3.13 for a comparison of the stratification in Theorem 1.3 and the one appearing in [ess-stillman, Theorem 5.13]. ∎

Surprisingly, one can go even further. In §5.2, we define a space which parametrizes isomorphism classes of all finitely generated homogeneous ideals of generated in degrees , with no restriction on the number of generators.

Theorem 1.5.

The space is noetherian.

Unfortunately, the analog of Theorem 1.3 fails for (it fails already for ). Nonetheless, one can deduce a certain analog of Theorem 1.1 from Theorem 1.5 (see Proposition 5.2).

1.3. Topological comparison results

To define the space , we realize it as the quotient space of another space , which parametrizes type ideals together with a choice of generating set of type . The space can also be described as the direct sum of the symmetric powers . Each summand is the space of homogeneous degree polynomials in infinitely many variables. A key property that we use is that these spaces (and finite direct sums of them) are noetherian under the change of basis action given by . We deduce this from a recent result of Draisma [draisma]. However, Draisma’s results is not directly applicable: the space we describe here is the direct limit of the space of homogeneous polynomials in variables with , but Draisma’s result applies to the corresponding inverse limit spaces.

Hence in §2, we prove some topological comparison results that show that noetherianity of direct limit spaces and noetherianity of inverse limit spaces are equivalent to one another under suitable conditions which include the situation above. Namely, we work in the more general context of finite length polynomial functors and consider the corresponding spaces together with the change of basis action of . There are a number of natural topologies that one might consider, and the key principle is that it does not matter which one is chosen as long as one works equivariantly.

1.4. Outline

In §2, we collect the results we need about certain infinite dimensional varieties. In §3, we prove the main theorems of the paper. In §4, we give examples of degreewise bounded invariants. Finally, §5 has some further comments and generalizations.

Acknowledgments

We thank Brian Lehmann and Claudiu Raicu for conversations inspiring §4.1, Giulio Caviglia for observations that simplified the proof of Theorem 1.3, and Bhargav Bhatt and Mel Hochster for additional useful conversations. We thank BIRS for hosting us during the workshop “Free Resolutions, Representations, and Asymptotic Algebra”, as some of the ideas in this paper were refined during that meeting.

2. Varieties defined by polynomial functors

2.1. Setup

For the purposes of this paper, a polynomial functor over a field is an endofunctor of the category of vector spaces that is a subquotient of a direct sum of tensor power functors. When has characteristic 0, every polynomial functor is a direct sum of Schur functors. The category of polynomial functors is abelian, and comes equipped with a tensor product defined by

This tensor product is equipped with the symmetry that interchanges and .

Fix a finitely generated algebra object in the category of polynomial functors. Examples of such algebra objects are easy to come by: if is a finite length polynomial functor then is such an algebra object. In fact, these are the only examples relevant to this paper. Let , a finitely generated -algebra, and let .

The standard inclusion , given by , induces an inclusion and thus a projection . We let be the direct limit of the , and let be the inverse limit of the ’s in the category of schemes, which is simply the affine scheme . We let be the topological space underlying the scheme , which is the inverse limit of the spaces .

The standard projection , given by , induces a surjection and thus a closed immersion . We let be the inverse limit of the ’s, regarded as a topological ring, and we let be the ind-scheme defined by the directed system . Let be the direct limit of the sets . We define the ind-topology on to be the direct limit topology, and let denote the resulting topological space. The set is canonically identified with the set of open prime ideals in the topological ring . In this way, is a subset of , and one can give it the subspace topology. We call this the Zariski topology, and denote the resulting space by . The Zariski topology is, in many situations, the “correct” topology, since its closed sets are zero loci of equations, but it is often easier to check that a set is closed in the ind-topology, since this can be checked in each separately. Every Zariski closed set is ind-closed, but the converse is not true in general; see [anderson] for an example.

By definition, maps to . However, there is also a canonical map in the opposite direction. Indeed, for each the standard projection induces a ring homomorphism . These are compatible, and thus define a map . The composition is the identity, and so the map is injective while the map is surjective. These surjections are compatible as varies, and thus define a map . By the same reasoning, we get a map .

Recall that a topological space equipped with an action of a group is topologically -noetherian if every descending chain of -stable closed subsets stabilizes. Draisma proved that, over any field , is topologically -noetherian [draisma, Theorem 1], where here, and in what follows, denotes the algebraic group . The goal of this section is to transfer Draisma’s result to the spaces and and some related spaces.

2.2. Comparison of and

We begin by comparing to the Zariski topology on .

Proposition 2.1.

We have a bijection

given by . In particular, is -noetherian.

We refer to elements of the image of the map as finite polynomials.

Lemma 2.2.

Suppose is a closed, -stable ideal. Then every element of is the limit of finite polynomials that also belong to .

Proof.

Any continuous endomorphism of acts on , and any -stable subspace of (such as ) is automatically stable by these additional operators. Let , and let be its image under . The composite map is the action of a continuous endomorphism , and so it maps into itself. We thus see that . Since is the limit of the ’s, the result follows. ∎

Lemma 2.3.

If is a -stable closed subset, then every point of is a limit of points of .

Proof.

Let be the ideal defining . As in the previous proof, any endomorphism of acts on and carries to itself. Let be the map induced by the projection . We thus see . Let and put . The map factors as , and so for all . Since is the limit of the , the result follows. ∎

Proof of Proposition 2.1.

Define a map in the opposite direction by sending a -stable Zariski closed set of to its closure in . We claim that the two maps are inverse.

Start with . Let be the closure of in and let . Clearly . Let be a point of . We must show that every infinite polynomial that vanishes on also vanishes on . If is some infinite polynomial vanishing on , then by Lemma 2.2, it is a limit of finite polynomials vanishing on . Each vanishes on , and hence so does the limiting polynomial . So .

Now consider . Let be the closure of in . Then clearly . Let be a point of . Lemma 2.3 shows that every point in is a limit of points from and hence we obtain the opposite containment as well.

We have thus shown that defines a bijection as in the statement of the proposition. Draisma’s theorem shows that the source of this bijection satisfies the descending chain condition. Thus the target does as well, which shows that is -noetherian. ∎

2.3. Comparison of the ind- and Zariski topologies

The Zariski and ind-topologies on an ind-scheme are typically very different. We show that this distinction disappears in our situation, where we focus on -stable subsets.

Proposition 2.4.

A -stable subset of is Zariski closed if and only if it is ind-closed. In particular, is -noetherian.

Lemma 2.5.

Let be a -stable radical ideal of . Then its image in is radical, for any .

Proof.

Let be the image of in . We claim that . Since the composite is the identity, it is clear that any element of is contained in . Conversely, the projection map is obtained by applying the projection map , which can be realized as the limit of elements of , and so maps into itself. We thus see that any element of belongs to , which shows .

Now suppose and . Since is a radical ideal and , we have . Thus , and so is a radical ideal of . ∎

Proof of Proposition 2.4.

Every Zariski closed set is ind-closed (even without -stability), so it suffices to show that if is -stable and ind-closed then it is Zariski closed. Let , a Zariski closed subset of . Let be the unique radical ideal such that . Then is -stable. Let be the smallest -stable radical ideal of that contains . Its zero locus is the intersection of the -translates of the inverse image of in . Hence, if , the image of in defines , since . But this image is radical, by Lemma 2.5, so . If , the image of in is a closed subset of . Again, is radical by Lemma 2.5, so . Now let . Then , so and we see that is Zariski-closed. ∎

2.4. Some variants

Let , where denotes the dual space to . Of course, is isomorphic to as a -algebra, but the action of is different. Let and be defined analogously to before. Given an action of or on some object, we define the conjugate action as the precomposition with the automorphism .

Proposition 2.6.

We have isomorphisms of ind-schemes and that are -equivariant for the conjugate action on the source and the standard action on the target.

Proof.

Let be the standard basis for and the dual basis for . We have a linear isomorphism taking to . This is -equivariant using the conjugate action on the source and the standard action on the target. The thus induce the requisite isomorphisms and . ∎

A subset of or is stable under the standard -action if and only if it is stable under the conjugate -action. Thus, by Proposition 2.4 and its corollary, we find:

Corollary 2.7.

A -stable subset of is ind-closed if and only if it is Zariski closed.

Corollary 2.8.

The spaces , , and are topologically -noetherian.

Example 2.9.

Let be a finite length polynomial functor and let . Let and let . Also let be the restricted dual of , where the limit is taken with respect to the standard inclusions of into . Then we have canonical identifications

where here and are the usual linear duals of the spaces and . ∎

Remark 2.10.

The moral of this section is that all limit topological spaces one can sensibly form from are essentially equivalent when working equivariantly. This heuristic does not hold in some similar situations; see [eggsnow, §4] for examples. ∎

3. Theorems about

3.1. Notation

Let be the polynomial ring and let be the infinite polynomial ring . For a tuple of elements of , we let be the ideal of that they generate, and let be the ideal of that they generate. Let and .

For an integer , let , regarded as an affine scheme. For a degree tuple , we let

and we let be the ind-scheme defined by the system . This fits into the variant setup of the previous section, since is the scheme from Example 2.9 with . Let be the sheaf of algebras on . A -point of corresponds to a tuple of elements of . The family of ideals assembles to an ideal sheaf of (meaning that the image of the fiber in the fiber is ). We let be the quotient sheaf; its fiber at is .

3.2. The space

Let be the set of isomorphism classes of type ideals in , where we say that ideals and are isomorphic if there exists an isomorphism of graded rings with . Let be the set of closed points in and let be the map taking a tuple to the class of the ideal that it generates. The map is surjective and -invariant. We give the induced topology, using the Zariski topology on . Thus a subset of is closed if and only if is Zariski closed in . Since is -stable, it is Zariski closed if and only if it is ind-closed (Proposition 2.4), so the ind-topology on induces the same topology on .

Remark 3.1.

If , then two tuples generate the same ideal in if and only if they differ by an element of , and generate isomorphic ideals if and only if they differ by an element of . We thus see that is the quotient of by the group . For general , it is more complicated to describe directly. ∎

We define as the set of isomorphism classes of ideals in given the topology induced by the surjection . There are natural maps and .

Proposition 3.2.

The space is the direct limit of the spaces .

Proof.

It is clear that the set is the direct limit of the sets . If is a subset of , then the following are equivalent:

  • is closed;

  • is closed in (by definition of the topology on );

  • is closed for all (by Corollary 2.7);

  • is closed for all (it equals );

  • is closed for all (by definition of the topology on );

  • is closed in the direct limit topology.

Thus the topology on is the direct limit topology. ∎

Theorem (Theorem 1.2).

The space is noetherian.

Proof.

Suppose is a descending chain of closed subsets in . Then is a descending chain of -stable Zariski closed subsets of , and thus stabilizes by Corollary 2.8. It follows that stabilizes, and so is noetherian. ∎

Remark 3.3.

An ideal invariant induces a function for each . An ideal invariant is cone-stable if and only if it is compatible with the transition maps . It follows that a cone-stable ideal invariant induces a function . ∎

3.3. Finiteness of initial ideals

For an ideal we let denote the generic initial ideal of under the revlex order. We note that for , essentially by definition. The proof of the following theorem crucially depends on the theorem of Ananyan–Hochster [ananyan-hochster] (Stillman’s conjecture).

Theorem 3.4.

Given there exist and such that for any and any type ideal of the ideal is generated by monomials of degree at most in the variables .

Proof.

By Stillman’s conjecture for [ananyan-hochster, Theorem C], there exists a bound such that for any type ideal of . By a theorem of Caviglia [ms, Theorem 2.4], this implies that there exists such that for any such (the proof of that result shows that only depends on and not on the underlying field). Both and are independent of .

Let be a type ideal of and put . After a general change of coordinates, we can assume that form a regular sequence on . Since the projective dimension of is at most , the Auslander–Buchsbaum Theorem [eisenbud, Theorem 19.9] implies that . By the Bayer–Stillman criterion [bayer-stillman, Theorem 2.4], the same form a regular sequence on the quotient of by the revlex initial ideal of . Thus, is definable in at most variables. Moreover [bayer-stillman, Corollary 2.5] implies that the revlex generic initial ideal is definable in degree at most . ∎

It is a consequence of [bayer-stillman, Lemma 2.2] that formation of revlex is cone-stable, and thus is well-defined for any finitely generated homogeneous ideal , and is a finitely generated monomial ideal of . We define a type revlex generic initial ideal as an ideal of of the form where is a type ideal. Theorem 3.4 yields:

Corollary 3.5.

There are only finitely many type revlex generic initial ideals.

Remark 3.6.

Corollary 3.5 fails for other term orders. See [snellman, Appendix A.2]. ∎

3.4. Hilbert numerators

For a homogeneous ideal of , the Hilbert series can be written as a rational function with . We call the Hilbert numerator of , and we denote it by . Terminology for varies in the literature: our usage follows [kreuzer-robbiano, p. 282], but it is also called the -polynomial [miller-sturmfels, Definition 8.21], and has other names elsewhere. Note that the Hilbert numerator is not necessarily the numerator of when written in lowest terms.

The advantage of the Hilbert numerator is that it is cone-stable: . We can thus define the Hilbert numerator of for any finitely generated homogeneous ideal of . We define a type Hilbert numerator to be a polynomial of the form for a type ideal of .

Theorem 3.7.

There are only finitely many type Hilbert numerators.

Proof.

The Hilbert series associated to an ideal coincides with that of . It follows that the Hilbert numerator associated to an ideal coincides with that of , and so the result follows from Corollary 3.5. ∎

3.5. Two partial orders

Given polynomials , define if for all . We use to mean that either or that , which is not the same as for all . Given series and in , define if for all . Note that , and similarly for .

Proposition 3.8.

Let . The following are equivalent:

  1. .

  2. can be expressed in the form for some and non-negative coefficients .

  3. There exists such that for all .

  4. There exists such that .

Proof.

(a) (b). After a change of coordinates, this is [polyaszego, Part 6, §6, Problem 49].

(b) (c). Write as in (b). Let . Then . Since for all in the sum, the series has non-negative coefficients. Since the are non-negative, it follows that has non-negative coefficients.

(c) (d). Obvious.

(d) (a). If then obviously (a) holds. Thus suppose . Let be such that , and let . The series is non-zero, has non-negative coefficients, and converges at . Thus its value at is positive. Since is also positive, we conclude that is positive, and so (a) holds. ∎

3.6. The flattening stratification

Let be the set of all type Hilbert numerators, where is a finite index set. Define a partial order on by if , that is, if for all . For , let be the locus in where the corresponding ideal class has Hilbert numerator . Let .

Proposition 3.9.

For any , the set is closed in .

Proof.

By Corollary 2.7 and the definition of the topology on , it suffices to show that is closed in for all . Let be such that for any and we have if and only if ; this exists by Proposition 3.8 and the fact that is finite. Let and let be a point of . Then if and only if , which in turn is equivalent to . Since is the Hilbert series , we see

which is closed by the usual semi-continuity property of Hilbert series. ∎

Corollary 3.10.

Each is locally closed in .

Let be the closure of in , endowed with the reduced subscheme structure. Let be the complement of in , considered as an open subscheme of . The set of closed points in is exactly .

For simplicity, we have only defined as a topological space (as opposed to an ind-stack). To make sense of the flatness statement from Theorem 1.3, we thus pass to the cover , and interpret Theorem 1.3 as the following concrete statement:

Proposition 3.11.

The restriction of to is -flat.

Proof.

For we have . Thus the Hilbert series of the fiber of is constant on . Let be one of the graded pieces of . Then is a coherent sheaf on whose fiber at all closed points has the same dimension, say dimension . By semi-continuity of fiber dimension, the locus of (not necessarily closed) points where the fiber dimension is is the union of a closed set (where the dimension is ) and an open set (where the dimension is ). Yet this locus contains no closed points, and since is of finite type over a field, this implies that the fiber of has the same dimension on the non-closed points as well, which in turn implies that is locally free [eisenbud, Ex. 20.14(b)]. ∎

Corollary 3.12.

Let be a cone-stable weakly upper semi-continuous ideal invariant, and let be given. Then is a closed subset of .

Proof.

It suffices to show that is a closed subset of for all . We have . This is closed, since is flat over and is weakly upper semi-continuous. ∎

3.7. Proof of Theorem 1.1

Fix a cone-stable weakly upper semi-continuous ideal invariant , and let be given. Let be the locus defined by . Observe that is closed in by Corollary 3.12, and that the space is noetherian, being a subspace of the noetherian space (see Theorem 1.2). We thus see that the descending chain stabilizes. Since there are only finitely many , it follows that the chain stabilizes. Let be such that for all . We thus see that implies for all ; thus implies . We therefore find that or holds at all points in , and so is bounded in degree . ∎

Remark 3.13.

The stratification of differs from the stratification produced in [ess-stillman, Theorem 5.13] in two ways. First, each -point of corresponds to a tuple in a polynomial ring for some , whereas [ess-stillman] allows “infinite polynomials”, i.e., tuples which lie in the graded inverse limit ring. Second, since the Betti table determines the Hilbert numerator, the stratification in [ess-stillman] refines the above stratification of . ∎

4. Examples of ideal invariants

4.1. The number of linear subspaces in a variety

In this section we work over an algebraically closed field . A smooth cubic surface in contains exactly lines. An arbitrary cubic surface can contain fewer than lines or it can contain an infinite number of lines (e.g., if is reducible); but if contains a finite number of lines, then it contains at most lines [milne, Theorem 9.48]. In this section, we prove a sort of generalization of this.

Fix a non-negative integer . Let be the Grassmannian of codimension subspaces of . For a homogeneous ideal , let be the closed subscheme of whose -points are those families of subspaces of scheme-theoretically contained in . Thus is exactly the set of subspaces of of codimension contained in . We say that a point of is rigid if the Zariski tangent space to at vanishes. Define an ideal invariant as follows: if contains a non-rigid point; otherwise, . We note that if and only if is étale over .

Proposition 4.1.

The ideal invariant is degreewise bounded.

Lemma 4.2.

Let be a flat, proper map of schemes, with of finite type. Then the locus of points where is étale over is open.

Proof.

Since is algebraically, the locus where is étale is the intersection of the locus where is dimension and the locus where is reduced. These loci are open by [stacks, 0D4I and 0C0E]. ∎

Remark 4.3.

We believe the ideal invariant is degreewise bounded. It is cone-stable, but not upper semi-continuous. By contrast, we could also consider the scheme structure on , setting when is finite and else. However, the invariant is upper semi-continuous, but not cone-stable. ∎

4.2. Invariants of singularities

Our ideal invariants are integer-valued. However, there are interesting ideal invariants that are not integer-valued. The methods used in the proof of Theorem 1.1 can sometimes be applied to these invariants. Here is an example:

Proposition 4.4.

Suppose is a field of characteristic , and fix . Let be the set of rational numbers occurring as the log-canonical threshold at the cone point of some type ideal. Then satisfies the descending chain condition.

Proof.

Cone-stability follows from the definition; for instance, using the analytic definition (as in [mustata-impanga, Theorem 1.2]), a rational function is integrable around the origin of if and only if the corresponding function is integrable around the origin of . Lower semi-continuity follows from [lazarsfeld, Example 9.5.41]. For each and each we say that lies in if and only if the log canonical threshold of at the cone point in is at most . Since is -invariant, and since is closed by semicontinuity, it follows that is closed. An infinite decreasing chain of values in would thus yield an infinite decreasing chain of -stable closed subsets in , contradicting Theorem 1.2. ∎

Remark 4.5.

A similar statement holds for -pure thresholds in positive characteristic. Here the semi-continuity follows from [mustata-yoshida, Theorem 5.1]. ∎

4.3. Previously known degreewise bounded invariants

Many ideal invariants have been previously shown to be degreewise bounded. We catalogue some here to hint at the ubiquity of the phenomenon.

Proposition 4.6.

The following invariants of an ideal are known to be degreewise bounded by previous results in the literature:

  1. The degree of .

  2. The maximal codimension of a minimal or associated prime of .

  3. The projective dimension of .

  4. The Castelnuovo–Mumford regularity of .

  5. The Betti number for any .

  6. The sum of the degrees of: the minimal primes of or the minimal primary components of .

  7. The th arithmetic degree of , as defined in [bayer-mumford, Definition 3.4].111Since the embedded primary components of an ideal are not uniquely defined, it would not make sense to talk about the degree of an embedded primary component. To remedy this, [bayer-mumford, §3] introduces a notion of the multiplicity of an embedded component of that depends only on the ideal and the corresponding associated prime. This leads to their definition of the th arithmetic degree of an ideal.

  8. The number of: minimal primes, embedded primes, or associated primes of .

  9. The degree of .

  10. The minimal such that the symbolic power belongs to for all .

  11. The minimal such that .

  12. The largest degree of a generator, or the number of generators, for: any associated prime of , the radical , or the symbolic power for any integer .

Proof.
  1. Refined Bezout’s Theorem [fulton, Theorem 12.3].

  1. The principal ideal theorem [eisenbud, Theorem 10.2].

  2. Stillman’s conjecture [ananyan-hochster, Theorem C].

  3. Stillman’s conjecture combined with Caviglia’s theorem [ms, Theorem 2.4].

  4. We can combine (3) and (4) with Boij–Söderberg theory [eisenbud-schreyer, Theorems 0.1, 0.2] to see that only finitely many Betti tables for are possible, and the statement follows.

  5. For either sum, one can apply Refined Bezout’s Theorem [fulton, Theorem 12.3].

  6. We use [bayer-mumford, Proposition 3.6].

  7. The number of minimal primes is bounded by (6), so it suffices to bound the number of embedded primes. By (2), we know that the codimension of an embedded prime of can take on only finitely many distinct values, so it suffices to bound the number of embedded primes of a given codimension. Since each embedded prime of codimension contributes at least to the th arithmetic degree of , we can then apply (7).

  8. Follows from (6).

  9. Follows from (2) plus [hochster-huneke, Theorem 1.1(c)].

  10. Follows from the Effective Nullstellensatz [brownawell, kollar, sombra].

  11. For an associated prime of , or for a minimal primary component of , this follows from [ananyan-hochster, Theorem D(b)]. Since is the intersection of the minimal primes of , it suffices to show that we can bound the number and degree of defining equations of in terms of the number and degree of defining equations of and . Using parts (3) and (4) above, we can bound the regularity and projective dimension of and . Using the exact sequence relating these ideals to , we can bound the regularity and projective dimension of as well. There are thus only a finite number of possible revlex gins of , yielding the desired bounds. This proves the statement for ; a similar argument works for symbolic powers. ∎

Remark 4.7.

Combining parts l and d answers  [peeva-stillman, Problem 3.9]. See also [cd, ravi] for related work on the degreewise boundedness of taking radicals. ∎

5. Additional comments

5.1. A converse theorem

Before Draisma’s paper [draisma] appeared, we had Theorem 1.1 in the form “if sums of symmetric power functors are topologically noetherian then ideal invariants are bounded.” The converse of this statement, that boundedness of ideal invariants implies the noetherianity of of the space , follows from an equivariant version of the Hilbert basis theorem that appears in [eqhilb, §1.2].

5.2. An improvement to Theorem 1.2

Let be the set of all isomorphism classes of finitely generated homogeneous ideals of that are generated in degrees at most (but with no condition on the number of generators). We have a surjective map

defined by sending to the ideal generated by the . Alternatively, we may view an element in as a finite rank map and the ideal is generated by the image. We can topologize as a quotient space of . The following statement greatly strengthens Theorem 1.2:

Theorem (Theorem 1.5).

The space is noetherian.

Proof.

The space carries a natural action of , and the map is equivariant if we have this group act trivially on . Consider the diagonal action of . The tensor product is a finite length representation for each . So by [draisma], is topologically -noetherian. Since the map is a quotient map, is also noetherian (since acts trivially on it). ∎

We give two consequences of the theorem.

Proposition 5.1.

Let be the set of all polynomials of the form with a homogeneous ideal finitely generated in degrees at most . Endow with the partial order from §3.5. Then satisfies the ascending chain condition.

We say that an ideal invariant is strongly upper semi-continuous if for any family of ideals over (with no flatness condition imposed), the function is upper semi-continuous. (Here denotes the ideal at , which is a homomorphic image of the fiber.) We say that is strongly degreewise bounded if for every there exists a such that for any homogeneous ideal finitely generated in degrees at most . By a straightforward adaptation of the proof of Theorem 1.1, we also obtain:

Proposition 5.2.

Any cone-stable strongly upper semi-continuous ideal invariant is strongly degreewise bounded.

Remark 5.3.

The analogue of Theorem 1.3 fails for : even would need a separate stratum for each integer consisting of the isomorphism class of the ideal . ∎

Remark 5.4.

Let be the set of isomorphism classes of all finitely generated ideals in . We claim that is not noetherian. Indeed, if it were then the set of all polynomials of the form , with any finitely generated homogeneous ideal, would satisfy the ascending chain condition. But it does not: indeed, is a Hilbert numerator for any , and these form an ascending chain. ∎

5.3. Boundedness of tca ideal invariants

Draisma’s theorem states that if is any finite length polynomial representation of then is topologically noetherian. In our proof of Theorem 1.2, we only applied this result with being a finite sum of symmetric powers. It is natural to wonder, therefore, if the remaining cases of Draisma’s theorem have implications for ideal invariants. We now give one possible answer to this question.

Recall that a twisted commutative algebra (tca) over is a commutative associative unital graded -algebra equipped with an action of the symmetric group on satisfying certain conditions, the most important being that multiplication is commutative up to a “twist” by ; see [expos] for the full definition. Let be the tca freely generated by indeterminates of degree one. Explicitly, is the tensor algebra on , equipped with the natural action of on the th tensor power.

We define a tca ideal invariant to be a rule associating to every ideal in any a quantity