Topological Noetherianity of polynomial functors

Abstract

We prove that any finite-degree polynomial functor over an infinite field is topologically Noetherian. This theorem is motivated by the recent resolution, by Ananyan-Hochster, of Stillman’s conjecture; and a recent Noetherianity proof by Derksen-Eggermont-Snowden for the space of cubics. Via work by Erman-Sam-Snowden, our theorem implies Stillman’s conjecture and indeed boundedness of a wider class of invariants of ideals in polynomial rings with a fixed number of generators of prescribed degrees.

1. Introduction

This paper is motivated by two recent developments in “asymptotic commutative algebra”. First, in 4, Ananyan-Hochster prove a conjecture due to Stillman 23, Problem 3.14, to the effect that the projective dimension of an ideal in a polynomial ring generated by a fixed number of homogeneous polynomials of prescribed degrees can be bounded independently of the number of variables. Second, in 9, Derksen-Eggermont-Snowden prove that the inverse limit over $n$ of the space of cubic polynomials in $n$ variables is topologically Noetherian up to linear coordinate transformations. These two theorems show striking similarities in content, and in 16, Erman-Sam-Snowden show that topological Noetherianity of a suitable space of tuples of homogeneous polynomials, together with Stillman’s conjecture, implies a generalisation of Stillman’s conjecture to other ideal invariants. In addition to being similar in content, the two questions have similar histories—e.g., both were first established for tuples of quadrics 315—but since 4 the Noetherianity problem has been lagging behind. The goal of this paper is to make it catch up.

1.1. Polynomial functors

Let $K$ be an infinite field and let $\mathbf{Vec}$ be the category of finite-dimensional vector spaces over $K$. We consider a covariant polynomial functor $P:\mathbf{Vec}\to \mathbf{Vec}$ of finite degree $d$. This means that for all $V,W$ the map $P:\operatorname {Hom}_\mathbf{Vec}(V,W) \to \operatorname {Hom}_\mathbf{Vec}(P(V),P(W))$ is polynomial of degree at most $d$, with equality for at least some choice of $V$ and $W$. The uniform upper bound $d$ rules out examples like $V \mapsto \bigwedge ^0 V \oplus \bigwedge ^1 V \oplus \bigwedge ^2 V \oplus \ldots$.

Then $P$ splits as a direct sum $P=P_0 \oplus P_1 \oplus \cdots \oplus P_d$ where

$$\begin{equation*} P_e(V):=\{p \in P(V) \mid P(t\cdot 1_V) p=t^e p \text{ for all } t\in K\}; \end{equation*}$$

see, e.g., 18. For each $e$ the map $\operatorname {Hom}(V,W) \to \operatorname {Hom}(P_e(V),P_e(W))$ is homogeneous of degree $e$, and we have $P_d(V) \neq 0$ for all $V$ of sufficiently large dimension.

1.2. Topological spaces over a category

The polynomial functor $P$ will also be interpreted as a functor to the category $\mathbf{Top}$ of topological spaces. Here we equip the finite-dimensional vector space $P(V)=P_0(V) \oplus \cdots \oplus P_d(V)$ with the Zariski-topology.

A general functor $X$ from a category $\mathbf{C}$ to $\mathbf{Top}$ is called a $\mathbf{C}$-topological space, or $\mathbf{C}$-space for short. A second $\mathbf{C}$-space $Y$ is called a $\mathbf{C}$-subspace of $X$ if for each object $A$ of $\mathbf{C}$ the space $Y(A)$ is a subset of $X(A)$ with the induced topology, and if, moreover, for a morphism $f:A \to B$ the continuous map $Y(f):Y(A) \to Y(B)$ is the restriction of $X(f)$ to $Y(A)$. We then write $Y \subseteq X$. The $\mathbf{C}$-subspace $Y$ is called closed if $Y(A)$ is closed in $X(A)$ for each $A$; we then also call $Y$ a closed $\mathbf{C}$-subset of $X$. The $\mathbf{C}$-space $X$ is called Noetherian if every descending chain of closed $\mathbf{C}$-subspaces stabilises.

Furthermore, a $\mathbf{C}$-continuous map from a $\mathbf{C}$-space $X$ to a $\mathbf{C}$-space $X'$ consists of a continuous map $\varphi _A:X(A) \to X'(A)$ for each object $A$, in such a way that for any morphism $f:A \to B$ we have $\varphi _B \circ X(f)=X'(f) \circ \varphi _A$. A $\mathbf{C}$-homeomorphism has the natural meaning.

1.3. The main theorem

We will establish the following fundamental theorem.

Theorem 1.

Let $\mathbf{Vec}$ be the category of finite-dimensional vector spaces over an infinite field $K$, and let $P:\mathbf{Vec}\to \mathbf{Vec}$ be a finite-degree polynomial functor. Then $P$ is Noetherian as a $\mathbf{Vec}$-topological space.

Remark 2.

The restriction to infinite $K$ is crucial for the set-up and the proofs below—e.g., it is used in the decomposition of a polynomial functor into homogeneous parts and, implicitly, to argue that if an algebraic group $\mathrm{GL}_n(K)$ preserves an ideal, then so does its Lie algebra. In future work, we will pursue versions of Theorem 1 over $\mathbb{Z}$ and possibly over finite fields.

Theorem 1 will be useful in different contexts where finiteness results are sought. In the remainder of this section we discuss several such consequences; since an earlier version of this paper appeared on the arXiv, several other ramifications have appeared, e.g., in 16.

1.4. Equivariant Noetherinity of limits

The polynomial functor $P$ gives rise to a topological space $P_\infty :=\lim _{\leftarrow n} P(K^n)$, the projective limit along the linear maps $P(\pi _n): P(K^{n+1}) \to P(K^n)$ where $\pi _n$ is the projection $K^{n+1} \to K^n$ forgetting the last coordinate. By functoriality, each $P(K^n)$ is acted upon by the general linear group $\mathrm{GL}_n$, the map $P(\pi _n)$ is $\mathrm{GL}_n$-equivariant if we embed $\mathrm{GL}_n$ into $\mathrm{GL}_{n+1}$ via $g \mapsto \operatorname {diag}(g,1)$, and hence $P_\infty$ is acted upon by the direct limit $\mathrm{GL}_\infty :=\bigcup _n \mathrm{GL}_n$, the group of all invertible $\mathbb{N}\times \mathbb{N}$-matrices which in all but finitely many entries equal the infinite identity matrix.

Given a closed $\mathbf{Vec}$-subset $X$ of $P$, the inverse limit $X_\infty :=\lim _{\leftarrow n} X(K^n)$ is a closed, $\mathrm{GL}_\infty$-stable subset of $P_\infty$, and using embeddings $K^n \to K^{n+1}$ appending a zero coordinate, one finds that $X_\infty$ surjects onto each $X(K^n)$. Conversely, given a closed, $\mathrm{GL}_\infty$-stable subset $Y$ of $P_\infty$, then for any finite-dimensional vector space $V$ and any linear isomorphism $\varphi :K^n \to V$, we set $X(V):=P(\varphi )(Y_n)$, where $Y_n$ is the image of $Y$ in $P(K^n)$. One can check that $V \mapsto X(V)$ is a closed $\mathbf{Vec}$-subset of $P$ and that this construction $Y \mapsto X$ is inverse to the construction $X \mapsto X_\infty$ above. Thus the theorem is equivalent to the following corollary.

Corollary 3.

Let $\mathbf{Vec}$ be the category of finite-dimensional vector spaces over an infinite field $K$, let $P:\mathbf{Vec}\to \mathbf{Vec}$ be a finite-degree polynomial functor, and equip $P_\infty :=\lim _{\leftarrow n} P(K^n)$ with the inverse-limit topology of the Zariski topologies on the $P(K^n)$. Then $P_\infty$ is $\mathrm{GL}_\infty$-Noetherian, i.e., every chain $P_\infty \supseteq Y_1 \supseteq Y_2 \supseteq \ldots$ of $\mathrm{GL}_\infty$-stable closed subsets of $P_\infty$ is eventually constant. Equivalently, every $\mathrm{GL}_\infty$-stable closed subset $Y$ of $P_\infty$ is the set of common zeros of finitely many $\mathrm{GL}_\infty$-orbits of polynomial equations.

Example 4.

The paper 9 concerns the case where $P=S^3$, the third symmetric power. In this case, $P_\infty$ is the space of infinite cubics $\sum _{1 \leq i \leq j \leq k} c_{ijk} x_i x_j x_k$, and $\mathrm{GL}_\infty$ acts by linear transformations that affect only finitely many of the variables $x_i$. $\clubsuit$

Remark 5.

The proofs below could have been formulated directly in this infinite-dimensional setting, rather than the finite-dimensional, functorial setting. However, one of the key techniques, namely, shifting a polynomial functor by a constant vector space, is best expressed in the functorial language. Moreover, the functorial language allows us to stay in the more familiar realm of finite-dimensional algebraic geometry.

1.5. Generalisations of Stillman’s conjecture

In 16, Erman, Sam and Snowden use the following special case of Theorem 1.

Corollary 6.

Let $K$ be an infinite field, fix natural numbers $d_1,\ldots ,d_k$, and consider the polynomial functor $P:V \mapsto S^{d_1} V \oplus \cdots \oplus S^{d_k} V$. Then $P$ is a Noetherian $\mathbf{Vec}$-topological space, and hence its limit $P_\infty$ is $\mathrm{GL}_\infty$-Noetherian.

Let $\mu$ be a function that associates a number $\mu (I) \in \mathbb{Z}\cup \{\infty \}$ to any homogeneous ideal $I$ in a symmetric algebra $S V$ on $V \in \mathbf{Vec}$, in such a way that $\mu (\langle S(\varphi )I \rangle )=\mu (I)$ for any injective linear map $\varphi :V \to W$ with induced homomorphism $S(\varphi ):S V \to S W$, and such that $\mu$ is upper semicontinuous in flat families. In 16 the following is proved.

Theorem 7 (16).

Corollary 6 implies that for any ideal invariant $\mu$ with the properties above there exists a number $N$ such that for all $V \in \mathbf{Vec}$, any ideal $I \subseteq S V$ generated by $k$ homogeneous polynomials of degrees $d_1,\ldots ,d_k$ either has $\mu (I) \leq N$ or $\mu (I)=\infty$.

The crucial point here is that $N$ does not depend on $\dim V$. Stillman’s conjecture 23, Problem 3.14 is this statement for $\mu$ equal to the projective dimension, and it is used in the proof of the generalisation just stated. However, in a follow-up paper 17, the same authors give two new proofs of Stillman’s conjecture, one of which uses Corollary 6. An algorithmic variant of this latter proof is presented in 13.

1.6. Twisted commutative algebras

For $K=\mathbb{C}$, the algebra of polynomial functions on $P_\infty$, i.e., the direct limit of the symmetric algebras on $P_n(K^n)^*$, is a twisted commutative algebra in one of its incarnations 3132. In this context, Theorem 1 says the following.

Corollary 8.

Any finitely generated twisted commutative algebra over $\mathbb{C}$ is topologically Noetherian.

1.7. Functors from $\mathbf{Vec}^\ell$

Theorem 1 has the following generalisation to functors that take several distinct vector spaces as input.

Corollary 9.

Let $K$ be an infinite field, let $\mathbf{Vec}$ be the category of finite-dimensional vector spaces over $K$, let $\ell$ be a positive integer, and let $P$ be a functor from $\mathbf{Vec}^\ell$ to $\mathbf{Vec}$ such that for any $\underline{V}, \underline{W}\!\!\in \!\! \mathbf{Vec}^\ell$ the map $\operatorname {Hom}_{\mathbf{Vec}^\ell }(\underline{V},\underline{W})\!\! \to \!\! \operatorname {Hom}_\mathbf{Vec}(P(\underline{V}),P(\underline{W}))$ is polynomial of uniformly bounded degree. Then $P$ is a Noetherian $\mathbf{Vec}^\ell$-topological space.

Note that the group of automorphisms of $(V_1,\ldots ,V_\ell )$ is $\prod _i \mathrm{GL}(V_i)$, which when the $V_i$ are all equal contains a diagonal copy of $\mathrm{GL}(V)$. This suggests that Theorem 1 for $V \mapsto P(V,\ldots ,V)$ is in fact stronger than this corollary, as we prove now.

Proof of Corollary 9 from Theorem 1.

Let $X_1 \supseteq X_2 \supseteq \cdots$ be a chain of closed $\mathbf{Vec}^\ell$-subsets of $P$. Let $Q:\mathbf{Vec}\to \mathbf{Vec}$ be the functor that sends $V$ to $P(V,\ldots ,V)$, and set $Y_n(V):=X_n(V,\ldots ,V)$, a closed $\mathbf{Vec}$-subset of the polynomial functor $Q$. By Theorem 1, the sequence $(Y_n)_n$ is constant for $n$ at least some $n_0$. We claim that so is $(X_n)_n$. Indeed, let $\underline{V}=(V_1,\ldots ,V_\ell ) \in \mathbf{Vec}^\ell$. Choose a $V \in \mathbf{Vec}$ with $\dim V \geq \dim V_i$ for each $i$, and choose surjections $\pi _i:V \to V_i$ and injections $\iota _i:V_i \to V$ such that $\pi _i \circ \iota _i=1_{V_i}$. Then for $n \geq n_0$ we have

$$\begin{align*} X_n(V_1,\ldots ,V_\ell ) &= P(\pi _1,\ldots ,\pi _\ell ) P(\iota _1,\ldots ,\iota _\ell ) X_n(V_1,\ldots ,V_\ell ) \\ &\subseteq P(\pi _1,\ldots ,\pi _\ell ) X_n(V,\ldots ,V) \\ &= P(\pi _1,\ldots ,\pi _\ell ) X_{n+1}(V,\ldots ,V) \subseteq X_{n+1}(V_1,\ldots ,V_\ell ), \end{align*}$$

as desired.

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1.8. Slice rank

Taking $P(V_1,\ldots ,V_\ell )=V_1 \otimes \cdots \otimes V_\ell$, a tensor in $P(\underline{V})$ is said to have slice rank $1$ if it is nonzero and of the form $v \otimes A$ for a $v$ in one of the $V_i$ and an $A \in \bigotimes _{j \neq i} V_j$. A tensor has slice rank at most $k$ if it is a sum of at most $k$ tensors of slice rank $1$; in this sum the slice index $i$ may vary through $\{1,\ldots ,d\}$. Being of slice rank at most a fixed number $k$ is a Zariski-closed condition (see Tao and Sawin’s blog post 33) and preserved under tensor products of linear maps. Corollary 9 implies the following.

Corollary 10.

Let $\mathbf{Vec}$ be the category of finite-dimensional vector spaces over an infinite field $K$. For fixed $\ell$ and $k$, there exists a tuple $\underline{W}\in \mathbf{Vec}^\ell$ such that for all $\underline{V}\in \mathbf{Vec}^\ell$ a tensor $T \in V_1 \otimes \cdots \otimes V_\ell$ has slice rank at most $k$ if and only if for all $\ell$-tuples of linear maps $\varphi _i:V_i \to W_i$ the tensor $(\varphi _1 \otimes \cdots \otimes \varphi _\ell )T$ has slice rank at most $k$.

Equivalently, in the space of $\ell$-way $\infty \times \infty \times \cdots \times \infty$-tensors, the variety of slice rank at most $k$ tensors is defined by finitely many $\mathrm{GL}_\infty ^\ell$-orbits (and even finitely many $\mathrm{GL}_\infty$-orbits, acting diagonally) of polynomial equations. A more in-depth study of the algebraic geometry of slice rank is a forthcoming work with Oosterhof.

1.9. Related work

Theorem 1 fits in a trend at the interface between representation theory, algebraic geometry, commutative algebra, and applications, which studies algebraic structures over some base category and aims to establish stabilisation results. Recent examples, in addition to those referenced above, include the theory of modules over the category $\mathbf{FI}$ of finite sets with injective maps 7; Gröbner techniques 29 for modules over more general combinatorial categories that, among other things, led to a resolution of the Artinian conjecture 24 and to a resolution of a conjecture by Rauh-Sullivant 26 on iterated toric fibre products 14; and finiteness results for secant varieties of Segre and Segre-Veronese embeddings; see 21252728 and the notion of inheritance in 20. The current paper, while logically independent of these results, was very much influenced by the categorical viewpoint developed in these papers.

2. Proof of the main theorem

2.1. Overview of the proof

The proof of Theorem 1 is a double induction. The outer induction is on the polynomial functor $P$ via a (“lexicographic”) partial order $\prec$ on the class of polynomial functors introduced in §2.2. Using classical work by Friedlander-Suslin, we prove that this is a well-founded order. Any degree-zero polynomial functor, i.e., a constant vector space $U$ independent of the input $V$, is smaller than all polynomial functors of positive degree, and Hilbert’s basis theorem yields the base case of the induction.

So when we want to prove the theorem for $P$, we may assume that it holds for all polynomial functors smaller than $P$. We then show that every closed $\mathbf{Vec}$-subset $X$ of $P$ is Noetherian, by the inner induction on the smallest degree of a nonzero equation $f \in K[P(U)]$ vanishing on $X(U)$ for some $U \in \mathbf{Vec}$. Roughly, this works as follows (see the next paragraph for subtleties): fix an irreducible component $R$ of the highest-degree part $P_d$ of $P$, and find a direction $r_0 \in R(U)$ such that the directional derivative $h:={\partial {f}}/{\partial r_0}$ is not identically zero. Let $Y$ be the largest $\mathbf{Vec}$-closed subset of $P$ on which $h$ vanishes identically. Since $h$ has a lower degree than $f$, $Y$ is Noetherian by the inner induction hypothesis. On the other hand, set $P'(V):=P(U \oplus V)$ and $Q'(V):=P'(V)/R(V)$, so that $Q' \prec P$; this is discussed in §2.4. In §2.9 we show that the complement $Z$ of $Y$ has a closed embedding into a basic open subset of $Q'$, so $Z$ Noetherian by the outer induction hypothesis. Hence $X$, the union of two Noetherian spaces, is Noetherian.

There are four subtleties: First, $f$ may not depend on the coordinates on $R(U)$, so that ${\partial {f}}/{\partial r_0}=0$ for all $r_0 \in R(U)$. We therefore need to look for $f$ in the ideal of $X$ that is nonzero even after modding out the ideal of the projection of $X$ in $Q:=P/R$; see §2.5. Second, in positive characteristic, directional derivatives (linearisations) do not necessarily behave well; we replace these by additive polynomials in §2.6. Third, the closed embedding is in fact a Zariski homeomorphism to a closed subset; see §2.7 for the relevant lemma. Fourth, it is not quite $Z$ that embeds into a basic open subset of $Q'$—$Z$ is not functorial in $V$—but rather the locus $Z'(V)$ in $X'(V):=X(U \oplus V)$ where $h$ (which involves only coordinates on the constant vector space $U$) is nonzero. The closed embedding is then just the restriction of the projection $P'(V) \to Q'(V)$ along $R(V)$. Smearing around $Z'(V)$ by $\mathrm{GL}(U \oplus V)$, we obtain $Z(U \oplus V)$, and in §2.9 we show that this is good enough.

Example 11.

As a running example to illustrate the proof, we assume that $\operatorname {char}K \neq 2$, $P(V)=V \otimes V = Q(V) \oplus R(V)$ where $Q(V)$ is the space of symmetric two-tensors (matrices) and $R(V)$ is the space of skew-symmetric tensors. Let $(x_{ij})_{ij}$ be the standard coordinates on $P(K^n)$, let $(y_{ij})_{i \leq j}$ be the standard coordinates on $Q(K^n)$ extended to $P(K^n)$ by declaring them zero on $R(K^n)$, and let $(z_{ij})_{i<j}$ be the standard coordinates on $R(K^n)$, similarly extended to $P(K^n)$.

We then have

$$\begin{equation*} x_{ij}=\begin{cases} y_{ij}+z_{ij} & \text{if }i<j;\\ y_{ji}-z_{ji} & \text{if $j<i$; and} \\ y_{ii} & \text{if }i=j. \end{cases} \end{equation*}$$

Take $X(V)=\{v \otimes w \mid v,w \in V\}$, the variety of rank $1$ tensors. Then $X(K^1)$ is the entire space $P(K^1)$, but $X(K^2) \subsetneq P(K^2)$, so we may take $U=K^2$ and for $f$ the determinant

$$\begin{equation*} f=x_{11}x_{22}-x_{12}x_{21}=y_{11}y_{22}-(y_{12}+z_{12})(y_{12}-z_{12}) =y_{11}y_{22}-y_{12}^2+z_{12}^2. \end{equation*}$$

We take $r_0:=e_1 \otimes e_2 - e_2 \otimes e_1$ and find

$$\begin{equation*} h=\partial f / \partial r_0=\partial f / \partial z_{12} = 2 z_{12}. \end{equation*}$$

In this case, for $n \geq 2$, $Y(K^n)$ is the subvariety of $X(K^n)$ on which the $\mathrm{GL}_n$-orbit of $z_{12}$ vanishes identically, i.e., $Y(V)$ is the set of rank $1$ tensors in $Q(V)$. This is a coincidence; in the general setting of the proof, $Y(V)$ does not embed into $Q(V)$, but it always has a lower-degree polynomial vanishing on it. We discuss the complement $Z(V)$ in Example 26. $\clubsuit$

2.2. A well-founded order on polynomial functors

We will prove Theorem 1 by induction on the polynomial functor, along a partial order that we introduce now. Define a relation $\prec$ on polynomial functors of finite degree by $Q \prec P$ if $Q \not \cong P$ and moreover if $e$ is the highest degree with $Q_e \not \cong P_e$, then $Q_e$ is a homomorphic image of $P_e$; this is a partial order on (isomorphism classes of) polynomial functors.

Lemma 12.

The relation $\prec$ is a well-founded order on polynomial functors of finite degree.

Proof.

It suffices to prove that this order is well-founded when restricted to polynomial functors of degree at most a fixed number $d$. By 18, Lemma 3.4, if $V$ is any vector space of dimension at least $d$, then the map $P \mapsto P(V)$ is an equivalence of abelian categories from polynomial functors of degree at most $d$ and finite-dimensional polynomial $\mathrm{GL}(V)$-representations of degree at most $d$. Hence $Q \prec P$ implies that the sequence $(\dim Q_e(V))_{e=1}^d$ is strictly smaller than the sequence $(\dim P_e(V))_{e=1}^d$ in the lexicographic order where position $e$ is more significant than position $e-1$. Since this lexicographic order is a well-order, $\prec$ is well-founded.

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2.3. $\mathbf{Vec}$-varieties and their ideals

Write $K[P]$ for the contravariant functor from $\mathbf{Vec}$ to $K$-algebras that assigns to $V$ the coordinate ring $K[P(V)]$. A closed $\mathbf{Vec}$-subset $X$ of $P$ will be called a $\mathbf{Vec}$-variety in $P$, and denoted $X \subseteq P$. Its ideal is a contravariant functor that sends $V$ to the ideal of $X(V)$ inside $K[P(V)]$.

Using scalar multiples of the identity $V \to V$ and the fact that $K$ is infinite, one finds that the ideal of $X(V)$ is homogeneous with respect to the $\mathbb{Z}_{\geq 0}$-grading that assigns to the coordinates on $K[P_e(V)]$ the degree $e$. The degree function $\deg$ on $K[P(V)]$ and its quotients by homogeneous ideals is defined relative to this grading.

Example 13.

In our running Example 11, $f,h$ have degrees $4,2$, respectively. $\clubsuit$

2.4. The shift functor

Fixing a $U \in \mathbf{Vec}$, we let $\mathrm{Sh}_U:\mathbf{Vec}\to \mathbf{Vec}$ be the shift functor that sends $V \mapsto U \oplus V$ and the homomorphism $\varphi :V \to W$ to the homomorphism $\mathrm{Sh}_U(\varphi ):=\mathrm{id}_U \oplus \varphi : \mathrm{Sh}_U(V) \to \mathrm{Sh}_U(W)$.

Lemma 14.

For any polynomial functor $P$ of degree $d$, $P \circ \mathrm{Sh}_U$ is a polynomial functor of degree $d$ whose degree-$d$ homogeneous part is canonically isomorphic to that of $P$.

Proof.

Set $P':=P \circ \mathrm{Sh}_U$. For $V,W \in \mathbf{Vec}$ the map

$$\begin{equation*} \operatorname {Hom}(V,W) \to \operatorname {Hom}(P'(V),P'(W)) \end{equation*}$$

given by $\varphi \mapsto P'(\varphi )$ is the composition of the affine-linear map $\varphi \mapsto 1_U \oplus \varphi$ and the polynomial map $\psi \mapsto P(\psi )$ of degree at most $d$, so $P'$ is a polynomial functor of degree at most $d$.

For $V \in \mathbf{Vec}$ let $\iota _V$ be the embedding $V \to U \oplus V,\ v \mapsto (0,v)$ and let $\pi _V$ be the projection $U \oplus V \to V,\ (u,v) \mapsto v$. These give rise to morphisms of polynomial functors $\alpha :P \!\to \! P'$ and $\beta :P' \!\to \! P$ given by $\alpha (V) \!:=\! P(\iota _V):P(V) \to P(U \oplus {} V)$ and $\beta (V):=P(\pi _V)$. Straightforward computations show that $\alpha$ and $\beta$ map each homogeneous part $P_e$ into $P'_e$ and vice versa, and that $\beta \circ \alpha$ is the identity. Conversely, for $q$ in the highest-degree part $P_d'(V)$ we have $P(1_U \oplus t 1_V)q=t^d q$ for all $t \in K$. The coefficient of $t^d$ in the left-hand side equals $P(0 \oplus 1_V)q$, so we have $P(0 \oplus 1_V)q=q$ and therefore

$$\begin{equation*} \alpha (V) \beta (V) q = P(\iota _V) P(\pi _V) q = P(0 \oplus 1_V) q = q, \end{equation*}$$

which proves that $\beta :P'_d(V) \to P_d(V)$ is indeed a linear isomorphism.

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Example 15.

If $P(V)=S^d V$, then $(P \circ \mathrm{Sh}_U)(V)=S^d (U \oplus V) \cong \bigoplus _{e=0}^d S^{d-e} U \otimes S^e V$, so $P \circ \mathrm{Sh}_U$ equals $P$ plus a polynomial functor of degree $d-1$. $\clubsuit$

Example 16.

In our running Example 11, $P(V)=V \otimes V=Q(V) \oplus R(V)$, $U=K^2$ and

$$\begin{align*} P \circ \mathrm{Sh}_U(V)&=(U \oplus V) \otimes (U \oplus V)=U \otimes U + U \otimes V + V \otimes U + V \otimes V\\ &\cong K^4 + V^4 + Q(V) + R(V); \end{align*}$$

note that the degree-two part of $(P \circ \mathrm{Sh}_U)/R$ is $Q$, so that $(P \circ \mathrm{Sh}_U)/R \prec P$. $\clubsuit$

2.5. Splitting off a term of highest degree

Assume that $P$ is a polynomial functor of degree $d>0$. Let $R$ be any irreducible sub-polynomial functor of the highest-degree part $P_d$ of $P$, define $Q:=P/R$, and let $\pi :P \to Q$ be the natural projection. Then $K[Q]$ embeds into $K[P]$ via the pull-back of $\pi$. If $X$ is a $\mathbf{Vec}$-variety in $P$, then let $X_Q$ be the $\mathbf{Vec}$-variety in $Q$ defined by setting $X_Q(V)$ equal to the Zariski-closure in $Q(V)$ of $\pi (V)(X(V))$.

We will think of $X$ as a $\mathbf{Vec}$-variety over $X_Q$. Accordingly, we write $\mathcal{I}_X$ for the contravariant functor that assigns to $V$ the ideal of $X(V)$ in $K[\pi (V)^{-1}(X_Q(V))]$, the quotient of $K[P(V)]$ by the ideal in $K[P(V)]$ generated by the ideal of $X_Q(V)$ in $K[Q(V)]$.

In particular, we have $\mathcal{I}_X \!=\! 0$ if and only if for all $V$ we have $X(V) \!=\! \pi ^{-1}(X_Q(V))$. We write $\delta _X \in \{1,2,\ldots ,\infty \}$ for the minimal degree of a nonzero homogeneous polynomial $f \in \mathcal{I}_X(V)$ as $V$ runs over $\mathbf{Vec}$. Note that any polynomial in $K[P(V)]$ of degree zero is contained in $K[P_0(V)] \subseteq K[Q(V)]$; here we use that $d>0$. So if a degree-zero polynomial vanishes on $X(V)$, then it is an equation for $X_Q(V)$ and has already been modded out in the definition of $\mathcal{I}_X$. This explains why $\delta _X \geq 1$. Furthermore, note that $\delta _X=\infty$ if and only if $\mathcal{I}_X=0$.

Example 17.

In Examples 11,13, $\delta _X=\deg f=4$. $\clubsuit$

2.6. Additive polynomials as directional derivatives

For a finite-dimensional vector space $W$ over the infinite field $K$, we write $\mathrm{Add}(W)$ for the subset of $K[W]$ consisting of polynomials $f$ such that $f(v+w)=f(v)+f(w)$ for all $v,w \in W$. Then $\mathrm{Add}(W)$ is a $K$-subspace of $K[W]$, and equal to $W^*$ when $\operatorname {char}K=0$. In general, if we let $p$ be the characteristic exponent of $K$—so $p=1$ if $\operatorname {char}K=0$ and $p=\operatorname {char}K$ otherwise—and if we choose a basis $x_1,\ldots ,x_n$ of $W^*$, then $\mathrm{Add}(W)$ has as a basis the polynomials $x_i^{p^e}$ where $i$ runs through $\{1,\ldots ,n\}$ and $e$ through $\mathbb{Z}_{\geq 0}$ if $p>1$ and through $\{0\}$ if $p=1$. The span of these for fixed $e$ is denoted $\mathrm{Add}(W)_e$.

Lemma 18.

Let $W' \supseteq W$ be finite-dimensional vector spaces over the infinite field $K$, let $f$ be a polynomial on $W'$, and consider the expression $f(w'+tw)$, a polynomial function of the triple $(w',t,w) \in W' \times K \times W$. Then one of the following hold:

(1)

$f(w'+tw)$ is independent of $t$; this happens if and only if $f$ factors through the projection $\pi _{W'/W}:W' \to W'/W$; or

(2)

the nonzero part of $f(w'+tw)$ of lowest degree in $t$ is of the form $t^{p^{e}} h(w',w)$ for a unique $e$ (taken $0$ if $\operatorname {char}K=0$). Then for each fixed $w' \in W'$ the map $w \mapsto h(w',w)$ is in $\mathrm{Add}(W)_e$.

Proof.

For $w \in W$, let $D_w^{(r)}:K[W'] \to K[W']$ denote the $r$th Hasse directional derivative in the direction $w$. This linear map is defined in characteristic $0$ by $D_w^{(r)} g =\frac{1}{r!}(\frac{\partial }{\partial w})^r g$ and in arbitrary characteristic by realising that the latter expression actually has integer coefficients relative to any monomial basis of $K[W']$. Explicitly: let $x_1,\ldots ,x_{n-1}$ be a basis of $(W'/K w)$ and let $x_n \in (W')^*$ with $x_n(w)=1$. Then

$$\begin{equation*} D_w^{(r)} x_1^{a_1} \cdots x_n^{a_n}:=\binom{a_n}{r} x_1^{a_1} \cdots x_{n-1}^{a_{n-1}} x_n^{a_n-r}; \end{equation*}$$

in particular, if $r=a_n$, then this is nonzero as a polynomial over $K$, even when $a_n \cdots (a_n-r+1)$ is zero. A straightforward check shows that this is independent of the choice of basis, and that $D_{cw}^{(r)}=c^r D_w^{(r)}$.

Taylor’s formula in arbitrary characteristic reads

$$\begin{equation*} f(w'+tw)=\sum _{r \geq 0} (D_w^{(r)} f)(w') t^r. \end{equation*}$$

If $D_w^{(r)} f=0$ for all $r>0$, then we see from the above that $f$ does not involve the variable $x_n$, i.e., $f$ factors through $W' /\langle w \rangle$. Similarly, if $D_w^{(r)}f=0$ for all $w \in W$ and all $r>0$, then $f$ factors through $W'/W$.

Suppose that there exist $r>0$ and $w \in W$ such that $D_w^{(r)}f \neq 0$; take such a pair $(r,w)$ with $r$ minimal. Then, in the coordinates above, $f$ contains a monomial $x_1^{a_1} \cdots x_n^{a_n}$ for which $\binom{a_n}{r}$ is nonzero in $K$, but $\binom{a_n}{r'}=0$ in $K$ for all $r'$ with $0<r'<r$. By Lucas’s theorem on binomial coefficients, $r$ is a power of $p$, $a_n$ is divisible by $r$, and $\binom{a_n}{r}=a_n/r$ in $K$. Since in fact $\binom{a_n'}{r'}=0$ holds for all $r'<r$ and all monomials $x_1^{a_1'} \cdots x_n^{a_n'}$ in $f$, $a_n'$ is a multiple of $r$, and hence $f=g(x_1,\ldots ,x_{n-1},x_n^r)$ for a unique polynomial $g(x_1,\ldots ,x_{n-1},y_n)$. Moreover, $D_w^{(r)} f=(\partial g/\partial y_n)(x_1,\ldots ,x_{n-1},x_n^r)$.

More generally, let $x_1,\ldots ,x_k$ be a basis of $(W'/W)^*$ and extend to a basis $x_1,\ldots ,x_n$ of $(W')^*$.

We then find, by minimality of $r$, that $f=g(x_1,\ldots ,x_k,x_{k+1}^r,\ldots ,x_n^r)$ for a unique polynomial $g(x_1,\ldots ,x_n,y_{k+1},\ldots ,y_n)$, and for all $u \in W$ we have

$$\begin{equation*} D_{u}^{(r)} f=\sum _{i=k+1}^n (x_i(u))^r (\partial g/\partial y_i)(x_1,\ldots ,x_k,x_{k+1}^r,\ldots ,x_n^r). \end{equation*}$$

Since $r=p^e$ for some $e$, the right-hand side is additive in $u$, which concludes the proof of the lemma.

■

For $W' \supseteq W$ and $f \in K[W']$ and $h \in K[W'] \otimes K[W]$ as in the second case of the lemma and for $w \in W$ we write $\partial _w f \in K[W']$ for the polynomial $w' \mapsto h(w',w)$, and call this the directional derivative of $f$ in the direction $w$. This polynomial has degree less than $\deg f$, and agrees with the usual directional derivative for $\operatorname {char}K=0$. Note that $\partial _w f$ depends on the choice of $W$ inside $W'$, not just on $w$: if $f(w'+tw)$ depends on $t$ but vanishes to a higher degree at $t=0$ for a specific $w \in W$ than it does for general $w \in W$, then we have $\partial _w f=0$.

Also in the first case of the lemma we write $\partial _w f:=0$ for all $w \in W$. We extend the notation to rational functions with nonzero denominator $h \in K[W'/W]$ by $\partial _w(f/h):=(\partial _w f)/h$. The following lemma is immediate from Lemma 18.

Lemma 19.

For $W' \supseteq W$, $f \in K[W']$, $e \in \mathbb{Z}_{\geq 0}$ as in Lemma 18, $h \in K[W'/W] \setminus \{0\}$, we have $\partial _{v+w} (f/h)= \partial _v (f/h) + \partial _w (f/h)$ and $\partial _{cw} (f/h)=c^{p^e} \partial _w (f/h)$ for $c \in K$ and $w \in W$. $\square$

Example 20.

Assume that $p>2$. Let $W'=K^3$ with standard coordinates $x,y,z$, let $W$ be the span of the first two standard basis vectors, and let $f=y^{p^2} z^2 + x^{2p} y^{p^2} z$. Then

$$\begin{align*} f((x,y,z)+t(a,b,0))&=(y+tb)^{p^2} z^2 + (x+ta)^{2p} (y+tb)^{p^2} z \\ &= f(x,y,z) + t^p (2 a^p x^p y^{p^2} z) + \cdots \end{align*}$$

where the remaining terms are divisible by $t^{2p}$. Hence $\partial _{(a,b,0)} f = 2 a^p x^p y^{p^2} z$. $\clubsuit$

2.7. A closed embedding

Retaining the notation in the previous section, let $B$ be a basic open subset in $W'/W$ defined by the nonvanishing of some polynomial $h \in K[W'/W] \setminus \{0\}$ (we allow $h=1$, in which case $B=W'/W$). Let $Z$ be a Zariski-closed subset of $A:=\pi _{W'/W}^{-1}(B) \subseteq W'$ and let $J$ be the ideal of $Z$ inside $K[A]=K[W'][1/h]$. Fix a number $e \in \mathbb{Z}_{\geq 0}$, equal to $0$ if $\operatorname {char}K=0$, and let $J_e$ be the set of elements $k \in J$ such that $k(a+tw)=k(a)+t^{p^e}(\partial _w k)(a)$ for all $a \in A, t \in K, w \in W$ (so $k$ is affine-additive in $W$ with additive part of degree $p^e$). Note that via the pull-back $\pi _{W'/W}^*: K[B] \to K[A]$, $J_e$ is a $K[B]$-submodule of $K[A]$.

Lemma 21.

Assume that $K$ is algebraically closed and suppose that for each $a \in A$ the map $J_e \to \mathrm{Add}(W)_e, k \mapsto (w \mapsto (\partial _w k)(a))$ is surjective. Then $\pi _{W'/W}$ restricts to a Zariski-homeomorphism from $Z$ to a closed subset of $B$.

Proof.

Fix any tuple $x_1,\ldots ,x_n \in (W')^*$ whose restrictions to $W$ form a basis of $W^*$. Then the natural map $K[B][x_1,\ldots ,x_n] \to K[A]$ is an isomorphism by which we identify the two algebras. Under this identification, each element of $J_e$ can be written uniquely as $k_0 + \sum _{j=1}^n k_j x_j^{p^e}$ for suitable elements $k_0,k_1,\ldots ,k_n \in K[B]$. Let $k^{(1)},\ldots ,k^{(m)}$ be $K[B]$-module generators of $J_e$, and let $k^{(i)}_j \in K[B]$ be the coefficient of $x_j^{p^e}$ in $k^{(i)}$. Let $M \in K[B]^{m \times n}$ be the matrix whose $(i,j)$ entry equals $k^{(i)}_j$. The condition in the lemma says that $M(b)$ has rank $n$ for all $b \in B$.

Since $K$ is algebraically closed, the Nullstellensatz implies that $1$ lies in the ideal generated by the nonzero $n \times n$-subdeterminants $\Delta _1,\ldots ,\Delta _N \in K[B]$ of $M$: $1=\sum _{l=1}^N f_l \Delta _l$ for suitable $f_l \in K[B]$.

Fix a $j \in \{1,\ldots ,n\}$. For each $l \in \{1,\ldots ,N\}$ we can write $e_j^T=v_{jl}^T M$, where $e_j$ is the $j$th standard basis vector in $K[B]^n$ and $v_{jl}$ is a vector in $K(B)^m$ (supported only on the positions corresponding to $\Delta _l$) satisfying $\Delta _l v_j \in K[B]^m$; this is just Cramer’s rule. Now

$$\begin{equation*} e_j=1 e_j=\sum _{l=1}^N f_l \Delta _l e_j=\sum _{l=1}^N f_l (\Delta _l v_{jl}^T) M, \end{equation*}$$

and we conclude that $J_e$ contains an element of the form $k_{0,j} + x_j^{p^e}$ with $k_{0,j} \in K[B]$.

Consider the morphism $\varphi :A \to A$ dual to the homomorphism $K[A] \to K[A]$ that restricts to the identity on $K[B]$ and sends each $x_j$ to $x_j^{p^e}$. Since $K$ is algebraically closed, $\varphi$ is a homeomorphism in the Zariski-topology, and since $\varphi$ commutes with the projection $A \to B$, it suffices to show that this projection restricts to a closed embedding from $Z':=\varphi (Z)$ into $B$. Let $J'$ be the ideal of $Z'$. By the previous paragraph, $J'$ contains an element of the form $k_{0,j} + x_j$ with $k_{0,j} \in K[B]$ for each $j$. Hence the map $K[B] \to K[A]/J'$ is surjective, so $Z' \to B$ is a closed embedding, as desired.

■

Remark 22.

In characteristic $0$, the Zariski-homeomorphism from the lemma is in fact a closed embedding. In positive characteristic, it need not be.

2.8. Extending the field

Let $P:\mathbf{Vec}\to \mathbf{Vec}$ be a finite-degree polynomial functor over the infinite field $K$, let $L$ be an extension field of $K$, and denote by $\mathbf{Vec}_L$ the category of finite-dimensional vector spaces over $L$. We construct a polynomial functor $P_L:\mathbf{Vec}_L \to \mathbf{Vec}_L$ as follows. For every $U \in \mathbf{Vec}_L$ we fix a $V_U \in \mathbf{Vec}$ and an isomorphism $\psi _U: U \to L \otimes _K V_U$ of $L$-vector spaces.

At the level of objects, $P_L$ is defined by $P_L(U):=L \otimes _K P(V_U)$. To define $P_L$ on morphisms we proceed as follows. For $U,U' \in \mathbf{Vec}_L$ the polynomial map $P:\operatorname {Hom}_K(V_U,V_{U'}) \to \operatorname {Hom}_K(P(V_U),P(V_{U'}))$ extends uniquely to a polynomial map

$$\begin{equation*} P'_L:L \otimes _K \operatorname {Hom}_K(V_U,V_{U'}) \to L \otimes _K \operatorname {Hom}_K(P(V_U),P(V_{U'})) \end{equation*}$$

of the same degree; here we use that $K$ is infinite. The domain and codomain of $P'_L$ are canonically $\operatorname {Hom}_L(L \otimes _K V_U,L \otimes _K V_{U'})$ and $\operatorname {Hom}_L(P_L(U),P_L(U'))$, respectively. Hence for $\varphi \in \operatorname {Hom}_L(U,U')$ we may set

$$\begin{equation*} P_L(\varphi )=P'_L(\psi _{U'} \circ \varphi \circ \psi _U^{-1}). \end{equation*}$$

A simple calculation shows that $P_L$ is indeed a polynomial functor $\mathbf{Vec}_L \to \mathbf{Vec}_L$.

Furthermore, if $X$ is a $\mathbf{Vec}$-closed subset of $P$, then we obtain a $\mathbf{Vec}$-closed subset $X_L$ of $P_L$ by letting $X_L(U)$ be the Zariski closure of $\{1 \otimes q \mid q \in X(V_U)\}$ in $P_L(U)=L \otimes _K P(V_U)$. The following lemma is straightforward.

Lemma 23.

The map $X \mapsto X_L$ from $\mathbf{Vec}$-closed subsets of $P$ to $\mathbf{Vec}_L$-closed subsets of $P_L$ is inclusion-preserving and injective. Consequently, Noetherianity of $P_L$ implies that of $P$.

2.9. Proof of Theorem 1

If $d=0$, then $P(V)$ is a finite-dimensional space independent of $V$, and for every linear map $\varphi :V \to W$ the map $P(\varphi )$ is the identity, so the theorem is the topological corollary to Hilbert’s basis theorem. We therefore may and will assume that $d>0$. Furthermore, by Lemma 23 we may assume that $K$ is algebraically closed, so that we can use Lemma 21.

We proceed by induction, assuming that the theorem holds for all polynomial functors $P' \prec P$ in the well-founded order from §2.2; this is our outer induction hypothesis. From §2.5 we recall the definition of $\delta _X$. We now order $\mathbf{Vec}$-varieties inside the fixed $P$ by $X>Y$ if either $X_Q \supsetneq Y_Q$ or else $X_Q=Y_Q$ and $\delta _X>\delta _Y$. Since $\delta _X$ takes values in a well-ordered set, for any strictly decreasing sequence $X_1>X_2>\ldots$, $(X_i)_Q$ must become strictly smaller infinitely often; but this is impossible since $Q$ is Noetherian by the outer induction hypothesis. Hence $>$ is a well-founded order on $\mathbf{Vec}$-subvarieties of $P$.

We set out to prove, by induction along this well-founded order, that each $\mathbf{Vec}$-variety $X \subseteq P$ is Noetherian as a $\mathbf{Vec}$-topological space. Our inner induction hypothesis states that this holds for each $\mathbf{Vec}$-variety $Y<X$ inside $P$.

First assume that $\delta _X=\infty$, which means that $X$ is the pre-image of its projection $X_Q$, and let $Y \subsetneq X$ be any proper closed $\mathbf{Vec}$-subset. Then either $Y_Q \subsetneq X_Q$ or else $Y_Q=X_Q$ and $\delta _Y < \delta _X$. Hence $Y < X$, so that $Y$ is Noetherian by the inner induction hypothesis. Since any inclusion-wise strictly decreasing chain of closed $\mathbf{Vec}$-subsets of $X$ must contain such a $Y$ as its first or second element, $X$ is Noetherian, as well.

So we may assume that $\delta _X \in \mathbb{Z}_{\geq 1}$. Take $U \in \mathbf{Vec}$ of minimal dimension for which $\mathcal{I}_X(U)$ contains a nonzero homogeneous element of degree $\delta _X$, and let $f \in K[P(U)]$ be a homogeneous polynomial representing this element. Regarding $f$ as a polynomial with coefficients from $K[Q(U)]$ in coordinates that restrict to a basis of $R(U)^*$, we may remove from $f$ all terms with coefficients that vanish identically on $X_Q(U)$, and then at least one nonconstant term survives.

By Lemma 18 applied to $f$ with $W'=P(U)$ and $W=R(U)$, this implies that there exists an $r_0 \in R(U)$ such that the directional derivative $h:=\partial _{r_0} f \in K[P(U)]$ in the sense of §2.6 also has at least some coefficient in $K[Q(U)]$ that does not vanish on $X_Q(U)$. Let $e_0 \in \mathbb{Z}_{\geq 0}$ be the exponent $e$ in the Lemma 18; so the map $r \mapsto (\partial _r f)(q)$ lies in $\mathrm{Add}(R(U))_{e_0}$ for each $q \in P(U)$. Since coordinate functions on $R_d(U)$ were assigned degree $d$ (§2.3), we have $\deg (h)=\deg (f)-d p^{e_0}$ and in particular $\deg (h)<\deg (f)$. By minimality of the degree of $f$, the polynomial $h$ does not vanish identically on $X$.

Let $Y$ be the largest closed $\mathbf{Vec}$-subset of $X$ such that $h$ does vanish identically on $Y(U)$, i.e., $Y(V)=\{p \in X(V) \mid h(P(\varphi )p)=0 \text{ for all } \varphi \in \operatorname {Hom}(V,U)\}$. Then either $Y_Q \subsetneq X_Q$ or else $Y_Q=X_Q$ and $\delta _Y \leq \deg (h) < \delta _X$. Hence $Y<X$, so $Y$ is Noetherian by the inner induction hypothesis.

Define $Z$ by $Z(V):=X(V) \setminus Y(V)$. This is typically not a $\mathbf{Vec}$-subset of $X$, since for $\varphi \in \operatorname {Hom}_\mathbf{Vec}(V,V')$ the map $P(\varphi )$ might map points of $Z(V)$ into $Y(V')$, i.e., outside $Z(V')$. Indeed, since we chose $U$ of minimal dimension, when we pull back $f$ to a $P(V)$ for $\dim V<\dim U$, we obtain a polynomial that is zero modulo the ideal of $X_Q(V)$. This implies that the pull-back of $h$ is identically zero on $X(V)$, so that $Z(V)=\emptyset$. To remedy this, we now construct a $\mathbf{Vec}$-variety $Z'$ closely related to $Z$, by shifting our polynomial functor over $U$ as in §2.4.

Set $P':=P \circ \mathrm{Sh}_U$ and $X':=X \circ \mathrm{Sh}_U$ and consider the open $\mathbf{Vec}$-subset $Z'$ of $X'$ defined by $Z'(V):=\{q \in X(U \oplus V) \mid h(q) \neq 0\}$. Here we regard $h$ as a polynomial on $P(U \oplus V)$ via the map $P(U \oplus V) \to P(U)$ coming from the projection $\pi _U: U \oplus V \to U$ along $V$. As the maps $\pi _U \circ g$ with $g \in \mathrm{GL}(U \oplus V)$ are Zariski-dense in $\operatorname {Hom}(U \oplus V,U)$, $Z'$ and $Z \circ \mathrm{Sh}_U$ are related by

$$\begin{align} &Z(U \oplus V)=\{q \in X(U \oplus V) \mid \exists \psi \in \operatorname {Hom}(U \oplus V,U): h(P(\psi )q) \neq 0\} {\tag{*}}\\ &=\{q \in X'(V) \mid h(g(q)) \neq 0 \text{ for some } g \in \mathrm{GL}(U \oplus V) \} = \bigcup _{g \in GL(U \oplus V)} g Z'(V). \end{align}$$

Recall that $R$ is an irreducible subfunctor of $P_d$. Write $R':=R \circ \mathrm{Sh}_U=R'_0 \oplus \cdots \oplus R'_d \subseteq P'$ where $R'_e$ is homogeneous of degree $e$ and $R'_d = R$ by Lemma 14. Define $Q':= P' / R'_d$ and note that $Q' \prec P$ since $Q'$ has degree at most $d$ and the degree-$d$ part of $Q'$ is equal to $P_d/R$ by Lemma 14. In particular, $Q'$ is a Noetherian $\mathbf{Vec}$-topological space by the outer induction hypothesis.

Remark 24.

Note that for $e<d$, $Q'_e$ typically will have dimension larger than $P_e$! Also, $Q'$ is not equal to $Q \circ \mathrm{Sh}_U$: there is a surjection $Q' \to Q \circ \mathrm{Sh}_U$ with kernel $R'/R'_d$.

The map $P(\pi _U): P(U \oplus V) \to P(U)$ has $R'_d(V)$ in its kernel, so we can regard $h$ as a polynomial on $Q'(V)$, as well. Consider the basic open $\mathbf{Vec}$-subset of $Q'$ defined by $B(V):=\{q \in Q'(V) \mid h(q) \neq 0\}.$ As $Q'$ is Noetherian, so is $B$ with its induced topology.

Lemma 25.

For every $K$-vector space $V$ the projection $Z'(V) \to B(V)$ is a Zariski-homeomorphism with a closed subset of $B(V)$.

Before proving this lemma, we use it to complete the proof of Theorem 1. First, since $B$ is Noetherian, Lemma 25 implies that so is $Z'$.

Then suppose that $X=X_0 \supseteq X_1 \supseteq \cdots$ is a sequence of closed $\mathbf{Vec}$-subsets of $X$. By Noetherianity of $Y$ there exists an $n_0$ such that for all $V \in \mathbf{Vec}$ the sequence $(X_n(V) \cap Y(V))_n$ is constant for $n \geq n_0$. By Noetherianity of $Z'$ there exists an $n_1$ such that for all $V \in \mathbf{Vec}$ the sequence $(X_n(U \oplus V) \cap Z'(V))_n$ is constant for $n \geq n_1$. Using * we find

$$\begin{align*} X_n(U \oplus V) \cap Z(U \oplus V) &= \bigcup _{g \in \mathrm{GL}(U \oplus V)} X_n(U \oplus V) \cap gZ'(V)\\ &= \bigcup _{g \in \mathrm{GL}(U \oplus V)} g(X_n(U \oplus V) \cap Z'(V)), \end{align*}$$

where in the last step we used that $X_n(U \oplus V)$ is $\mathrm{GL}(U \oplus V)$-stable. We find that, for each $V$ of dimension at least $\dim U$, the sequence $(X_n(V) \cap Z(V))_n$ is constant for $n \geq n_1$. Since $Z(V)=\emptyset$ for $V$ of dimension less than $\dim U$, we find that $X_n$ is constant for $n \geq \max \{n_0,n_1\}$. This proves Noetherianity of $X$ and concludes the proof of the inner induction step. $\square$

Example 26.

We pause and see what Lemma 25 says in the running Examples 11,13,16. Here $U=K^2$ and we take $V=K^n$. Then $Z'(V)$ is the variety of $(n+2) \times (n+2)$-matrices

$$\begin{equation*} M=\begin{bmatrix} C & D \\ E & F \end{bmatrix} \end{equation*}$$

of rank $1$ such that the upper-left $2 \times 2$-submatrix $C$ has a nonzero coordinate $z_{12}$ (recall that $h=2z_{12}$).

The lemma says that forgetting the skew-symmetric part of $F$ is a closed embedding of $Z'(V)$ into the open subset of $U \otimes U + V \otimes U + U \otimes V + Q(V)$ where $h$ is nonzero. We prove this by showing that, on $Z'$, each entry of the skew-symmetric part of $F$ can be expressed as a rational function in the entries of $C,D,E$ and the entries of the symmetric part of $F$, with a denominator equal to $h$.

We assume $n \geq 2$ and consider a surjective linear map $\varphi :V \to U$. Thinking of $\varphi$ as a $2 \times n$-matrix, we have

$$\begin{equation*} P(1_U \oplus t\varphi )M= C + t(\varphi E + D \varphi ^T) + t^2 \varphi F \varphi ^T. \end{equation*}$$

Since $Z'$ is a $\mathbf{Vec}$-closed subset, for all $M \in Z'(V)$, the $2 \times 2$-determinant $f$ vanishes on the latter $2 \times 2$-matrix for all choices of $t$ and of $\varphi$. Hence expanding $f(P(1_U \oplus t \varphi )M)$ as a polynomial in $t$, the coefficient $k$ of $t^2$ also vanishes for all $M,\varphi$. Take $i<j$ in $\{1,\ldots ,n\}$ and specialise $\varphi$ to the linear map sending the $i$th standard basis vector to $e_1$, the $j$th standard basis vector to $e_2$, and all other standard basis vectors to zero. Then the matrix above reads

$$\begin{equation*} \begin{bmatrix} C_{11} + t E_{i1} + t D_{1i} + t^2 F_{ii} & C_{12} + t E_{i2} + t D_{1j} + t^2 F_{ij} \\ C_{21} + t E_{j1} + t D_{2i} + t^2 F_{ji} & C_{22} + t E_{j2} + t D_{2j} + t^2 F_{jj} \end{bmatrix}. \end{equation*}$$

The coefficient of $t^2$ in the determinant of this matrix is

$$\begin{equation*} k(C,D,E,F)=F_{ii} C_{22} + F_{jj} C_{11} - F_{ij} C_{21} - F_{ji} C_{12} + \cdots \end{equation*}$$

where the remaining terms do not involve $F$. Using, as in Example 11, the variables $y,z$ for the symmetric and skew-symmetric parts of $C$, and using the variables $y',z'$ for the symmetric and skew-symmetric parts of $F$, this reads

$$\begin{align*} &y'_{ii} y_{22} + y'_{jj} y_{11} -(y'_{ij}+z'_{ij})(y_{12}-z_{12}) -(y'_{ij}-z'_{ij})(y_{12}+z_{12}) + \cdots \\ &= 2 z'_{ij} z_{12} + \cdots = z'_{ij} h(C) + \cdots \end{align*}$$

where the dots in the last two expressions contain only variables that we do not discard in the projection $P'(V) \to Q'(V)$. This shows that, on $Z'$, the coordinate $z'_{ij}$ can be expressed in the entries of $C,D,E$ and the coordinates $y'_{ij}$ on the symmetric part of $F$, as desired. $\clubsuit$

Proof of Lemma 25.

First assume that $\dim V \geq \max \{\dim U,d\}$. We want to apply Lemma 21 with $W'$ equal to $P'(V)$, $W$ equal to $R'_d(V)=R(V)$, $B$ equal to $B(V)$, and $Z$ equal to $Z'(V)$. Let $J$ be the ideal of $Z'(V)$ in the coordinate ring of the pre-image of $B(V)$ inside $P'(V)$, and let $J_{e_0}$ be as in the text preceding Lemma 21.

Fix any surjective linear map $\varphi :V \to U$. For $t \in K$ and $e \in \{0,\ldots ,d\}$ consider the linear map $\Phi _e(t):=P_e(1_U \oplus t\varphi ): P_e(U \oplus V) \to P_e(U)$. This map depends on $t$ as a polynomial of degree at most $e$, hence decomposes as $\Phi _e(t)=t^0 \Phi _{e0} + \cdots + t^e \Phi _{ee}$ for unique linear maps $\Phi _{ei}:P_e(U \oplus V) \to P_e(U)$. Note that $\Phi _{ee}=P_e(0 \oplus \varphi )$ and $\Phi _{e0}=P_e(1_U \oplus 0)=P_e(\pi _U)$. On the other hand, decompose $P'_e:=P_e \circ \mathrm{Sh}_u=P'_{e0} \oplus \cdots \oplus P'_{ee}$ where $P'_{ei}$ is a homogeneous polynomial functor of degree $i$. Then $\Phi _{ei}$ is zero on $P'_{ej}(V)$ except when $i=j$. To see this, take a $q \in P'_{ej}(V)$, compute

$$\begin{align*} \Phi _e(t)q&=P_e(1_U \oplus t\varphi )q= P_e(1_U \oplus \varphi ) P_e(1_U \oplus t1_V) q\\ &= P_e(1_U \oplus \varphi ) P'_e(t 1_V)q= t^j P_e(1_U \oplus \varphi ) q, \end{align*}$$

and observe that the right-hand side is homogeneous of degree $j$ in $t$.

The $\Phi _e(t)$ together form the map $\Phi (t):=P(1_U \oplus t\varphi ): P(U \oplus V) \to P(U)$. Since $X$ is a $\mathbf{Vec}$-subset of $P$ and $f$ vanishes on $X(U)$, $f(\Phi (t)q)=0$ for all $q \in X(U \oplus V)$ and $t \in K$. This implies that the coefficient $k(q)$ of $t^{dp^{e_0}}$ in $f(\Phi (t)q)$ also vanishes identically on $X(U \oplus V)$.

To determine how $k(q)$ depends on $R_d'(V)$, consider $r \in R'_d(V)$ and $q=q_0 + \cdots + q_d \in P(U \oplus V)$ with $q_e \in P_e(U \oplus V)$, and for variables $t,s$ compute

$$\begin{align*} &f\left(\Phi (t) (q_0 + \cdots + q_{d-1} + (q_d+s r))\right)\\ &= f(\Phi _0(t) q_0 + \cdots + \Phi _{d-1}(t) q_{d-1} + \Phi _d(t)(q_d+s r))\\ &= f(\Phi _0(t) q_0 + \cdots + \Phi _{d-1}(t) q_{d-1} +\Phi _d(t)q_d+t^d s \Phi _{dd} r)\\ &= f(\Phi _0(t) q_0 + \cdots + \Phi _{d-1}(t) q_{d-1} +\Phi _d(t)q_d+t^d s P_d(0 \oplus \varphi ) r)\\ &\equiv f(\Phi _0(t) q_0 + \cdots + \Phi _{d-1}(t) q_{d-1} +\Phi _d(t)q_d)\\ &+(t^d s)^{p^{e_0}} (\partial _{P_d(0 \oplus \varphi )r} f)(\Phi _0(t) q_0 + \cdots + \Phi _{d-1}(t) q_{d-1} +\Phi _d(t)q_d) \mod ((t^d s)^{p^{e_0}+1}) \end{align*}$$

where in the second equality we have used that $r \in P_{dd}'(V)$ and in the last step we have used Lemma 18. We see that $t^{dp^{e_0}}$ is the lowest power of $t$ to whose coefficient $r$ contributes, and that this contribution is additive in $r$. More specifically, for each $s \in K$ we have $k(q+s r)=k(q)+s^{p^{e_0}} (\partial _r k)(q)$, and $(\partial _r k)(q)$ equals the value of the directional derivative $\partial _{R (\varphi )r} f$ at the point $\sum _{e=0}^d \Phi _{e0} q_e=P(\pi _U)(q)$. In particular, $k \in J_{e_0}$.

From now on, assume that the image of $q$ in $Q'(V)$ lies in $B(V)$. Since $\varphi$ is surjective, so is $R(\varphi ):R(V) \to R(U)$. In particular, there exists an $r \in R(V)$ such that $R(\varphi )r=r_0$. For such an $r$ we have $\ell (r):=(\partial _r k)(q)=h(q) \neq 0$, so $\ell \in \mathrm{Add}(R(V))_{e_0}$ is not zero.

Keeping $q$ fixed but replacing $\varphi$ by $\varphi \circ g$ for $g \in \mathrm{GL}(V)$, $\ell$ transforms into the additive function $r \mapsto \ell (R(g)r)$. Hence by varying $g$ we find that the image of $J_e$ in $\mathrm{Add}(R(V))_e$ under the map $\tilde{k} \mapsto (r \mapsto (\partial _r \tilde{k})(q))$ from Lemma 21 contains a nonzero $\mathrm{GL}(V)$-submodule $L$ of $\mathrm{Add}(R(V))_e$. Since $R$ is irreducible and $\dim V \geq d$, $R(V)$ is an irreducible $\mathrm{GL}(V)$-module 18, Lemma 3.4, and this implies the irreducibility of $(R(V))^*$ and of $\mathrm{Add}(R(V))_{e_0}$—indeed, raising to the power $p^{e_0}$ gives a bijection from $\mathrm{GL}(V)$-submodules to $\mathrm{GL}(V)$-submodules of the latter.

We conclude that $L=\mathrm{Add}(R(V))_{e_0}$, and since $q$ was arbitrary in the pre-image of $B(V)$, the conditions of Lemma 21 are fulfilled. Hence the projection $Z'(V) \to B(V)$ is a Zariski-homeomorphism with a closed subset of $B(V)$, as desired.

Finally, if $\dim V < \max \{\dim U,d\}$, then take any embedding $\iota : V \to V'$ where $V'$ does have sufficiently high dimension. Then we have a commuting diagram

$$\begin{equation*} \vcenter{\img[][81pt][49pt][{$\xymatrix@R-1pc{ Z'(V) \ar[d] \ar[r]^{P'(\iota)} & Z'(V') \ar[d] \\ B(V) \ar[r]_{P'(\iota)} & B(V') }$}]{Images/img75ae4ecf5f2cb51169abcdc8941ab4bf.svg}} \end{equation*}$$

where all arrows except, a priori, the left-most one are homeomorphisms with closed subsets of the target space. But then so is the left-most one.

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2.10. Comments on the proof

The idea to do induction on $P$ is not completely new: it is also used, in special cases, in 491115. In these papers, more information than just Noetherianity is extracted from the proof, e.g., that the tuple rank of a tuple of matrices is bounded in a proper closed subvariety of a polynomial functor capturing matrices 1115, or that the $q$-rank of a cubic is bounded 9, or that the strength of a homogeneous form is bounded 4. Our proof above does not directly yield such qualitative information. However, in 6 we repair this defect for symmetric, alternating, and ordinary tensors and characteristic $0$ or sufficiently large.

If $K$ has characteristic $0$, then symmetric powers $S^d V$ are irreducible $\mathrm{GL}(V)$-modules, and one can prove Noetherianity for direct sums of these without the need for more general polynomial functors—though also without the proof becoming any easier. But in general characteristic, symmetric powers need not be irreducible, and polynomial functors need not be completely reducible into irreducible summands, so reducing modulo an irreducible subfunctor is the only natural thing to do.

The idea further to do induction on $\delta _X$ and to use directional derivatives is new, but inspired by techniques used earlier in 101112, where a determinant is regarded as an affine-linear polynomial in one matrix entry, whose coefficient is a determinant of lower order, and induction is done over that order.

2.11. An open problem

The most tantalising open problem in this area is the following.

Question 27.

Let $P$ be a finite-degree polynomial functor over an infinite field $K$. Does any sequence $I_1 \subseteq I_2 \subseteq \cdots$ of ideals in $K[P]$ eventually become constant?

For $P$ of degree at most $1$, the answer is yes, and it follows from the stronger statement that the ring $R[y_{ij} | i=1,\ldots ,k, j \in \mathbb{N}]$, acted upon by $\operatorname {Sym}(\mathbb{N})$ via $\pi y_{ij}=y_{i \pi (j)}$, is $\operatorname {Sym}(\mathbb{N})$-Noetherian for any Noetherian ground ring $R$ 5819. For $P=S^2$ and $P=\bigwedge ^2$ in characteristic $0$, the answer is also yes, since we know all $\mathrm{GL}(V)$-stable ideals from 2 and 1, respectively. In 22 a much stronger result than this is established for $S^2$ and $\bigwedge ^2$, namely, that finitely generated modules over $K[P]$ are also Noetherian. These questions were first raised, in the setting of twisted commutative algebras, in 30. They remain widely open even for more general degree-two functors, and also for, say, $\bigwedge ^2$ in positive characteristic.