Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum

In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free R-modules to finitely generated R-modules, for any commutative ring R whose spectrum is Noetherian. As Erman–Sam–Snowden pointed out, when applying this with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R={{\,\mathrm{{\mathbb Z}}\,}}$$\end{document}R=Z to direct sums of symmetric powers, one of their proofs of a conjecture by Stillman becomes characteristic-independent. Our paper advertises and further develops the beautiful but not so well-known machinery of polynomial laws. In particular, to any finitely generated R-module M we associate a topological space, which we show is Noetherian when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{Spec}\,}}(R)$$\end{document}Spec(R) is; this is the degree-zero case of our result on polynomial functors.

spaces V , W the map P V ,W : Hom(V , W ) → Hom(P(V ), P(W )) is a polynomial map. In many respects, polynomial functors behave like univariate polynomials: they can be added (direct sums), multiplied (tensor products), and composed; they are direct sums of unique homogeneous polynomial functors of degrees 0, 1, 2, . . .; and-for the theory that we are about to develop quite importantly-they can be shifted by a constant: if P is a polynomial functor and U a constant vector space, then the functor Sh U (P) that assigns to V the vector space P(U ⊕ V ) and to ϕ ∈ Hom K (V , W ) the linear map P(id U ⊕ ϕ) is a polynomial functor. Furthermore, if P has finite degree, which we will always require, then-much like a univariate polynomial and its shift by a constant-Sh U (P) has the same degree, and the top-degree homogeneous components of P and Sh U (P) are canonically isomorphic.
From a different perspective, polynomial functors are the ambient spaces of "GL ∞equivariant algebraic geometry", a research area which has seen much activity over the last years. A closed subset of P is a rule X that assigns to a vector space V a Zariskiclosed subset X (V ) of P(V ) in such a manner that for each ϕ ∈ Hom(U , V ), the linear map P U ,V (ϕ) maps X (U ) into X (V ). In earlier work [8], the third author showed that if P has finite degree, then it is Noetherian in the sense that any descending chain of closed subsets P ⊇ X 1 ⊇ X 2 ⊇ · · · eventually stabilises. This was used in work by Erman-Sam-Snowden [11,13,14] and by Draisma-Lasoń-Leykin [9] in new proofs of the conjecture by Stillman that the projective dimension of a homogeneous ideal that is generated by a fixed number of forms of a fixed degree is uniformly bounded independently of the number of variables [20,Problem 3.14]. In this context, Erman-Sam-Snowden asked whether the Noetherianity of polynomial functors also holds over Z; this would show that their proof of Stillman's conjecture yields bounds that are independent of the characteristic, just like another proof by Erman-Sam-Snowden [11] and the original proof by Ananyan-Hochster [2].
In this paper, we settle Erman-Sam-Snowden's question in the affirmative. Indeed, rather than working over Z, we will work over a ring R whose spectrum is Noetherianthis turns out to be precisely the setting where topological Noetherianity also holds for polynomial functors.
So let R be a ring (commutative with 1). In Sect. 3 we will review the notion of polynomial laws from an R-module M to an R-module N . In the special case where N = R, these polynomial laws form a graded ring R[M] (see Sect. 3.2), where the notation is chosen to resemble that for the coordinate ring of an affine variety. This ring will be used in Sect. 4 to define a topological space A M , in such a manner that any polynomial law ϕ : M → N yields a continuous map, also denoted ϕ, from A M → A N . To be precise, A M is a topological space over the category Dom R of Rdomains with R-algebra monomorphisms. Here a topological space over a category C is not a single set, but a functor from C equipped with the notions of elements and (closed) subsets, and we let all definitions related to usual topological spaces stated in terms of their elements and (closed) subsets carry over to this setting; see Definition 28 for details.
If M is freely generated by n elements, then R[M] is the polynomial ring R[x 1 , . . . , x n ] and the poset of closed sets in A M is the same as that in the spectrum of R [M]. In general, however, we do not completely understand the relation between A M and the spectrum of R[M] (see Remark 41), and we work with the for-mer rather than the latter. The following result is a topological version of Hilbert's basis theorem in this setting.

Proposition 1 If R has a Noetherian spectrum and M is a finitely generated R-module, then the topological space A M over Dom R is Noetherian.
Interestingly, it is not true that if R is Noetherian and M is finitely generated, then R [M] is Noetherian (see Example 23), so "topologically Noetherian" is the most natural setting here. A special case of the theorem (taking M free of rank 1) is that if R has a Noetherian spectrum, then so does the polynomial ring R [x]. This special case, a topological version of Hilbert's basis theorem, is easy and well-known; e.g., it also follows from [12, Theorem 1.1] with a trivial group G.
Following [22], in Sect. 5 we will recall the notion of polynomial functors from the category fgfMod R of finitely generated free R-modules to the category Mod R of R-modules. These polynomial functors form an Abelian category. The subcategory of polynomial functors from fgfMod R to the category fgMod R of finitely generated, but not necessarily free, R-modules is not an Abelian subcategory when R is not Noetherian, but it is closed under taking quotients, and this will suffice for our purposes.
Given a polynomial functor P : fgfMod R → fgMod R , a closed subset of A P is a rule X that assigns to each finitely generated free R-module U a closed subset X (U ) of A P(U ) such that the continuous map corresponding to the polynomial law maps the pre-image of X (U ) under the projection on P(U ) in A Hom(U ,V )×P(U ) into X (V ) (see Sect. 5.8 for details). If Y is a second such rule, then we say that X is a subset of Y if X (U ) is a subset of Y (U ) for each U ∈ fgfMod R . Our main result, then, is the following. Theorem 2 Let R be a commutative ring whose spectrum is a Noetherian topological space and let P be a finite-degree polynomial functor fgfMod R → fgMod R . Then every descending chain X 1 ⊇ X 2 ⊇ . . . of closed subsets of A P stabilises: for all sufficiently large n we have X n = X n+1 .
Proposition 1 is the special case of Theorem 2 where the polynomial functor has degree 0, i.e., sends each U to a fixed module M and each morphism to the identity id M . Proposition 1 will be proved first, as a base case in an inductive proof of Theorem 2.

Structure of the paper
In Sect. 2, we establish and recall certain basic results. In Sect. 3 we define polynomial laws and the coordinate ring of a module over a ring. Section 4 is devoted to the topological space A M . Here we prove Proposition 1, the first fundamental fact needed for our inductive proof of Theorem 2.
Then, in Sect. 5 we recall the definition a polynomial functor P over a ring and several of its properties. Among these is the Friedlander-Suslin lemma that yields equivalences of Abelian categories between polynomial functors fgfMod R → fgMod R of degree ≤ d and finitely generated modules for the non-commutative Ralgebra R[End(U )] * ≤d (called the Schur algebra) for any U ∈ fgfMod R of rank ≥ d. We also prove the second fundamental fact needed for Theorem 2: if R is a domain and P a polynomial functor from fgfMod R to Mod R such that Frac(R) ⊗ P is irreducible, then Frac(R/p) ⊗ P is irreducible for all primes p in some open dense subset of Spec(R). This is an incarnation of the philosophy in representation theory that irreducibility is a generic condition.
Finally, in Sect. 6 we prove Theorem 2. The global proof strategy is as follows: we show that the induction steps in [8], where Theorem 2 is proved when R is an infinite field, can be made global in the sense that they hold for Frac(R/p) for all p in some open dense subset of Spec(R); and then we use Noetherian induction on Spec(R) to deal with the remaining primes p. The details of this approach are a quite subtle and beautiful.
The big picture is depicted in the following diagram: Building on the notion of finitely generated R-modules, on the left we pass to polynomial functors over R. Here many results carry over, such as the fact that the rank is a semicontinuous function on Spec(R); see Proposition 54. We regard this as "linear algebra in varying dimensions". In the other direction, we construct the topological space A M and enter the realm of algebraic geometry; the closed subsets generalise affine algebraic varieties. Finally, both constructs come together in the construction of the topological space associated to a polynomial functor P. Here we use both results from the "linear algebra" of polynomial functors, such as Friedlander-Suslin's lemma, and results about the topological spaces A M , to prove that A P is Noetherian. Furthermore, we establish the fundamental result that the dimension function of a closed subset of A P depends on primes in Spec(R) in a constructible manner; see Proposition 86.

A class of applications
Our original motivation for this paper is the following: let P, Q be (finite-degree) polynomial functors from the category of finitely generated free Z-modules to itself and let α : Q → P be a polynomial transformation; see Definition 46. Define the closed subset X of A P as the closure of the image of α. Specifically, for a natural number n, the pull-back along α Z n defines a ring homomorphism Z[P(Z n )] → Z[Q(Z n )], and X (Z n ) is the closed subset of Spec Z[P(Z n )] defined by the kernel of that ring homomorphism. Theorem 2 implies the following.

Corollary 3
There exists a uniform bound d such that for all n ∈ Z ≥0 and all fields K , This corollary has many applications; here is one. If V is a finite-dimensional vector space over a field K and T ∈ V ⊗ V ⊗ V is a tensor, then T is said to have slice rank ≤ r if T can be written as the sum of r terms of the form σ (v ⊗ A), where v ∈ V and A ∈ V ⊗ V , and σ is a cyclic permutation of 1, 2, 3 permuting the tensor factors. If K is algebraically closed, then being of slice rank ≤ r is a Zariski-closed condition [26].
Corollary 4 Fix a natural number r . There exists a uniform bound d such that for all algebraically closed fields K and for all n ∈ Z ≥0 , the variety of slice-rank-≤ r tensors in K n ⊗ K n ⊗ K n is defined by polynomials of degree ≤ d.
The same holds when the number of tensor factors is increased to any fixed number, possibly at the expense of increasing d, and similar results hold for the set of cubic forms of bounded q-rank [7] or for the closure of the set of degree-e forms of bounded strength in the sense of [2]. We stress, however, that "defined by" is intended in a purely set-theoretic sense. We do not know whether the vanishing ideals of these varieties are generated in bounded degree, even if the field K were fixed beforehand.

Proof of Corollary 4
Consider the polynomial functor P that sends a free Z-module Z n to Z n ⊗ Z n ⊗ Z n , and the polynomial functor Q that sends Z n to Z n ⊕(Z n ⊗ Z n ). For any r -tuple (σ 1 , . . . , σ r ) of cyclic permutations of 1, 2, 3 we have a polynomial transformation whose image closure is defined in uniformly bounded degree e by Corollary 3. The variety of slice-rank-≤ r tensors is the union of these image closures over all r -tuples of cyclic permutations, hence defined in degree at most e · 3 r , independently of the algebraically closed field and independently of n.
Remark 5 Over a field K of characteristic zero, the irreducible polynomial functors P are precisely the Schur functors, and any polynomial functor is isomorphic to a direct sum of Schur functors. These always admit a Z-form, i.e., a polynomial functor P Z over Z such that K ⊗ P Z ∼ = P, which moreover has the property that it maps free Z-modules to free Z-modules [1]. The Z-form need not be unique; e.g., the Schur functor over K that maps V to its d-th symmetric power S d V , comes both from the functor from free Z-modules to free Z-modules that sends U to S d U and from the functor that sends U to the sub-Z-module of U ⊗d consisting of symmetric tensors. These two functors are non-isomorphic Z-forms. In applications such as the above, where one looks for field-independent bounds, it is important to choose the Z-form that captures the problem of interest.

Example 6
Again over R = Z, consider the polynomial transformation α : (S 2 ) 4 → S 4 that maps a quadruple (q 1 , . . . , q 4 ) of quadratic forms to q 2 1 + · · · + q 2 4 . Let X be the image closure as above. If K is algebraically closed of characteristic zero, then X K (K 4 ) is a hypersurface in S 4 (K ) of degree 38475 [3], so the degree bound from Corollary 3 must be at least that large. On the other hand, if K is algebraically closed of characteristic 2, then the image of α is just the linear space spanned by all degree-four monomials that are squares, and hence only linear equations are needed to cut out this image.

Remark 7
Over algebraically closed fields of positive characteristic, irreducible polynomial functors are still parameterised by partitions, but polynomial functors are no longer semisimple, and the Z-forms from Remark 5 do not always remain irreducible; standard references are [6,16]. The typical example is that, in characteristic p, the functor S p contains a subfunctor that maps V to the linear space of p-th powers of elements of V .

Further relations to the literature
The polynomial functors that we study are often referred to as strict polynomial functors in the literature; we will drop the adjective "strict". We do not know whether the polynomial functors over finite fields studied in [21] admit a similar theory.
Much literature on polynomial functors is primarily concerned with representation theory, whereas our emphasis is on the geometry/commutative algebra of closed subsets in such polynomial functors.
We will use work of Roby on polynomial laws [22] and work of Touzé on polynomial functors [27]-but indeed only more elementary parts of their work, such as the generalisation of Friedlander-Suslin's [15,Theorem 3.2] to general base rings R; see [27,Théorème 7.2].
The paper [14] establishes finiteness results for (cone-stable and weakly upper semicontinuous) ideal invariants in polynomial rings over a fixed field. As Erman pointed out to us, at least part of their results carry over to arbitrary base rings with Noetherian spectrum. In particular, Erman-Sam-Snowden establish the Noetherianity of a space Y d that parameterises homogeneous ideals generated in degrees d = (d 1 , . . . , d r ). While they work with certain limit spaces, the "functor analogue" of their Y d in our setting would be a functor from fgfMod R to the category of functors from Dom R to sets that sends a finitely generated free R-module U = R n to the functor that maps an R-domain D to the set of GL n (D)-orbits of ideals in R[x 1 , . . . , x n ] generated by homogeneous polynomials of degrees d 1 , . . . , d r . Then Y d admits a surjective map from the space A S d 1 ⊕···⊕S dr -a functor from fgfMod R to functors from Dom R to topological spaces, and one can give Y d the quotient topology. Theorem 2 implies that Y d is then Noetherian, provided that Spec(R) is Noetherian.
Our work does not say much about Noetherianity of the coordinate rings R[A P ], let alone about Noetherianity of finitely generated modules over them. Currently, these much stronger results are known only when R is a field of characteristic zero and P is either a direct sum of copies of S 1 [23,24] or P = S 2 or P = 2 [19] or P = S 1 ⊕ S 2 or P = S 1 ⊕ 2 [25].
Like Ananyan-Hochster's work [2], recent work by Kazhdan and Ziegler [17,18] implies that polynomials of high strength, and high-strength sequences of polynomials, behave very much like generic polynomials or sequences. Like Corollary 3, their results are uniform in the characteristic of the field. But the route that Kazhdan and Ziegler take is entirely different: first a theorem is proved over finite fields by algebraiccombinatorial means, with uniform constants that do not depend on the finite field, and then model theory is used to transfer the result to arbitrary algebraically closed fields.
In [4] it is shown that in any closed subset of the polynomial functor S d defined over Z, the strength of polynomials over a ground field of characteristic 0 or characteristic > d is uniformly bounded from above. While of a similar flavour as Corollary 3, that result-in which the restriction on the characteristic cannot be removed-does not follow from our current work. Far-reaching generalisations of [4], but only over fields of characteristic zero, are the topic of the forthcoming preprint [5].

Rings and algebras
To see this, extend v 1 , . . . , v m with v m+1 , . . . , v l to a generating set of the R-module M. Then for each j = m + 1, . . . , l we have, in K ⊗ M, for certain coefficients c i j ∈ K . This identity means that there exists a non-zero element r ∈ R and suitable coefficients c i j 's in R such that

Polynomial laws
We follow [22,Chapter 1]. Let M, N be R-modules. Denote by Alg R the category of R-algebras.
such that for every R-algebra homomorphism α : A → B the following diagram commutes: Example 10 Suppose that M and N are the free modules R 2 and R, respectively, so that A ⊗ M and A ⊗ N are canonically identified with A 2 and A. Then the collection (ϕ A ) A defined by ϕ A (x, y) = x y + y 2 for x, y ∈ A is a polynomial law M → N , and indeed one that is homogeneous of degree 2 in the sense of Definition 13 below.
More generally, the name polynomial law derives from the following fact.

Lemma 11
Consider two R-modules M and N . Suppose that M is finitely generated and let {v 1 , . . . , v n } be a set of generators. Let ϕ : M → N be a polynomial law. Then ϕ is completely determined by the element: This gives an injective map ι from the collection of polynomial laws from M to N to the module R[x 1 , . . . , x n ] ⊗ N . In the case where M is free with basis v 1 , . . . , v n , this injective map is a bijection.
Proof Let A be an R-algebra, let a 1 , . . . , a n ∈ A be elements and let α : R[x 1 , . . . , x n ] → A be the R-algebra homomorphism sending x i → a i . Then the diagram associated to α shows that ϕ A (a 1 ⊗v 1 +· · ·+a n ⊗v n ) = (α⊗id N )ι(ϕ) and hence ι is injective. If M is free with basis v 1 , . . . , v n , then ϕ A (a 1 ⊗v 1 +· · ·+a n ⊗v n ) = j f j (a 1 , . . . , a n )⊗ w j defines a polynomial law ϕ : M → N for every j f j ⊗w j ∈ R[x 1 , . . . , x n ]⊗ N .

Example 12
If R is an infinite field, then a polynomial law ϕ from M = R n to N = R m is in fact uniquely determined by ϕ R , which is required to be a polynomial map, i.e., a map all of whose m coordinate functions are polynomials in the n coordinates on M. So then the set of polynomial laws from M to N is precisely the set of polynomial maps from the vector space M to the vector space N .
For a general ring R, we denote by A n R the affine scheme Spec(R[x 1 , . . . , x n ]). The set of polynomial laws from R n to R m is the set of morphisms A n R → A m R defined over R. Of course, such a morphism need not be determined by its map ϕ R : R n → R m , but it is determined by the maps ϕ A : A n → A m for all R-algebras A. This motivates the definition of polynomial laws.
for all R-algebras A, a i 1 , . . . , a i d ∈ A and m i 1 ∈ M 1 , . . . , m i d ∈ M d .

Proof
The maps ϕ A are well-defined as the maps A d ×M 1 ×· · ·×M d → A⊗N sending (a 1 , . . . , a d , m 1 , . . . , m d ) → a 1 · · · a d ϕ(m 1 · · · m d ) are multilinear. The collection (ϕ A ) A is a homogeneous polynomial law of degree d and ϕ R = ϕ.

Remark 15
Composition of R-module homomorphisms is a bilinear map. By the proposition, we can thus view this operation as a polynomial law.
A homogeneous polynomial law ϕ : M → N of degree 0 is the same thing as an element of N (namely, the element ϕ R (0), which equals ϕ A (m) for any R-algebra A and any element m ∈ A⊗M); we call these polynomial laws constant. A homogeneous polynomial law M → N of degree 1 is the extension of an R-module homomorphism M → N as in the proposition above (namely, the map ϕ R : M → N , which in this case is R-linear and uniquely determines ϕ A for all A ∈ Alg R ); we call these polynomial laws linear.
The following proposition says that, in many ways, polynomial laws behave like ordinary polynomial maps between vector spaces. For proofs we refer to [22]. (1) The collection ϕ + ψ : Proof Suppose that such a decomposition exists and let A be an R-algebra. Then we have for all m ∈ A ⊗ M and m ∈ A ⊗ M . This shows that the ϕ (i, j) are unique. If ϕ is homogeneous of degree d, setting s = t, we see that ϕ = i+ j=d ϕ (i, j) and hence ϕ (i, j) = 0 for i + j = d. What remains to show the existence of the decomposition. In fact, defining ϕ (i, j),A (m, m ) to be the coefficient of s i t j in ϕ A [s,t] (sm, tm ), it is easy to show that the ϕ (i, j) are bihomogeneous polynomial laws of degree (i, j) adding up to ϕ.
The class of R-modules, in addition to its structure of Abelian category with Rmodule homomorphisms as morphisms, has the structure of a (non-Abelian) category with polynomial laws as morphisms. Both structures will be important to us, but we reserve the notation Mod R for the category in which the morphisms are R-module homomorphisms (i.e., homogeneous polynomial laws of degree 1).

Definition 18 (Base change).
If B is an R-algebra, then the tensor product functor Mod R → Mod B , which sends linear polynomial laws over R to linear polynomial laws over B, can be extended to a functor from the category of R-modules with polynomial laws over R to the category of B-modules with polynomial laws over B: on objects, the functor is just M → B ⊗ M, and a polynomial law

The coordinate ring of a module
Let M be a finitely generated R-module.

Remark 20
In [22, Chapitre III], various algebras associated to an R-module M are introduced, but they are different from our R-algebra R [M]. One important difference is that for us, the elements of M play the role of geometric objects, whereas there, the algebras consist of elements in divided or symmetric powers of M.
As usual with coordinate rings, the association M → R[M] is a contravariant functor from the category of R-modules with polynomial laws to the category of Ralgebras: a polynomial law ϕ : M → N has a pull-back map ϕ * : x n ] of Lemma 11 is a graded ring homomorphism. The following lemma says precisely which subalgebra its image is. The following example shows that, even when R is Noetherian and M is finitely generated, R[M] need not be Noetherian.

Lemma 21 Let ψ : N → M be a surjective R-module homomorphism. Then the map ψ * is a graded isomorphism from R[M] to the graded R-subalgebra of R[N ] whose degree-d part equals
where K is a field of characteristic zero, and let M : is not Noetherian, since the ideal span{t, t x, t x 2 , . . .} is not finitely generated. On the other hand, the quotient R[M] red of R[M] by its ideal of nilpotent elements is K .
However, we will see later that if Spec(R) is Noetherian and M is finitely generated, then a certain topological space A M defined using R[M] is also Noetherian. In Example 23, this is a consequence of the fact that Spec(R[M]) = Spec(K ) is Noetherian. See also Remark 41.

Example 24
Now consider a field K of characteristic 2 and set R : The same computation as above shows that cx i with odd i can only be in If B is an R-algebra, then the base change functor from Definition 18 sends The following example shows that this needs not be an isomorphism.
However, when B is a localisation of a domain R, then the map is an isomorphism:

Proposition 26
Suppose that R is a domain. Let M be a finitely generated R-module and let S be a multiplicative subset of R not containing 0. Set R := S −1 R. Then Proof The first and last isomorphisms are standard. For the middle isomorphism, we choose generators m 1 , . . . , m n of M and embed On the other hand, using the generators 1 ⊗ In particular, the multiplication by s gives g • t u = g over R for all u ∈ O ⊆ R n . Since R is a domain, the same holds over R and hence g ∈ A, again by Lemma 21 but now applied to the R-module M. Hence f = s −1 g ∈ S −1 A, as desired.
Like in ordinary algebraic geometry, the coordinate ring of a direct sum is the tensor product of the coordinate rings.

Proposition 27 Let M, N be finitely generated R-modules. Then
To see this, first suppose that M, N are free. In this case, we get , respectively. In general, let ϕ : M → M and ψ : N → N be surjective R-linear maps from finitely generated free R-modules. Then we see that Example 23 shows that the coordinate ring of a module is quite a subtle notion. However, we will see that in the proof of our Theorem 2, by a localisation we can always pass to a case where the module M is free. In that case, by Lemma 21,R[M] is just a polynomial ring over R.

The space A M
We now construct the topological space A M for M a finitely generated R-module. To be precise, A M is a topological space over the category Dom R of R-domains with R-algebra monomorphisms, in the sense of the following definition.

Definition 28
Let F : C → D be a functor and suppose that the objects of D are sets and the morphisms are maps (i.e, we have a forgetful functor Forget : D → Set). An element of F is an element of F(C) for some C ∈ C. A subset of F is a subfunctor of Forget •F, i.e., a rule X that assigns to each C ∈ C a subset X (C) ⊆ F(C) in such a manner that F C,D (ϕ) maps X (C) into X (D) for all morphisms ϕ : C → D. A topological space over C is a pair (F, T ) where F is a functor as above and T is a collection of subsets of F including the subsets ∅, F that is closed under taking arbitrary intersections and finite unions.

Remark 29
We note that all definitions that can be stated in terms of elements and (closed) subsets of a topological space carry over to topological spaces over C. We also note that a topological space (F, T ) gives rise to a functor from C to the category of topological spaces, which sends C to the set F(C) with the collection {X (C) | X ∈ T } of closed subsets. Clearly, not every functor from C to the category of topological spaces arises in this manner.
In what follows, we use the term "injections" to refer to R-algebra monomorphisms.
This collection of closed sets makes A M into a topological space over Dom R in the sense of Definition 28.

Remark 31
If D is an R-domain, then we can make D ⊗ M into an topological space by defining the closed subsets to be V(S)(D) for S ⊆ R[M]; we will call this the Zariski topology (over R) on D ⊗ M. To see that these sets are preserved under finite unions,

Remark 32
We think of A M as the "affine space" corresponding to M. Note that in the definition of closed subsets of A M we require S to be independent of D, i.e., not every rule assigning to D ∈ Dom R a subset of the form V(S)(D) is a closed subset of A M . To see that this is desirable, consider R = Z, M = R and let X n be the rule is a descending chain of rules and Let X be a subset of A M . Then we define the ideal of X to be As f D maps elements into a domain, we see that I X is a radical ideal of R [M]. We define the closure of X in A M to be the closed subset X : Let ϕ : M → N be a polynomial law between finitely generated R-modules. Then the maps (ϕ D ) D∈Dom R define a continuous map A M → A N , i.e., for every injection When M is free and finitely generated, we have the usual correspondence between closed subsets and radical ideals.

Proposition 34
Let M be a finitely generated free R-module of rank n. Then the rule sending an element ) is surjective and maps closed subsets of A M to closed subsets of A n R . Moreover, that map from closed subsets of A M to closed subsets of A n R is a bijection. In particular, we have I V(S) = rad(S) for any subset S ⊆ R [M].
Clearly, every closed subset arises from a closed subset of A M . To see that this map is injective, we note that Hence V(S) is uniquely determined by its associated subset of A n R .
While we have defined closed subsets of A M by looking at all R-domains D, it actually suffices to look at algebraic closures K p where p ∈ Spec(R). For p ∈ Spec(R), we write K p := Frac(R/p) for the fraction field of R/p.

Proposition 35 Let X be a subset of A M . Then
Let D be an R-domain and let p be the kernel of the homomorphism R → D. Then there exists a field L containing Frac(D) and K p . By the Nullstellensatz, the fact that f K p ∈ I X (K p ) implies that f L ∈ I X (L) . It follows that f D vanishes on X (D).
Proof This follows from the previous proposition since X = V(I X ).
The proof of Theorem 2 in Sect. 6 follows a divide-and-conquer strategy in which the following two lemmas and their generalisations to closed subsets of polynomial functors (Lemmas 64 and 65), play a crucial role.

Lemma 37 Let R be a ring with Noetherian spectrum and r an element of R. Let
Let K be an R-field and let R → K be the corresponding homomorphism. If the image of r in K is zero, then R → K factors via R/p i for some i = 1, . . . , k and hence K is a (R/p i )-domain. In this case, we have X (K ) = X R/p i (K ) = Y R/p i (K ) = Y (K ). If the image of r in K is nonzero, then K naturally is an R[1/r ]-field. In this case, we have X (K ) = X R [1/r ]  Proof The extension R ⊆ R satisfies lying over, i.e., for every prime p ∈ Spec(R) there is a prime q ∈ Spec(R ) with p = q ∩ R. The lemma follows by Corollary 36.

Noetherianity of A M
We now prove Proposition 1. Thus let R be a ring.

Lemma 39 If Spec(R) is Noetherian, then so is Spec(R[x]).
Proof This is an application of [12, Theorem 1.1] with trivial group.
Lemma 40 Assume that Spec(R) is Noetherian and set N := R n . Then A N is Noetherian, i.e., any chain X 1 ⊇ X 2 ⊇ · · · of closed subsets of A N stabilises eventually.
Proof Consider the chain I X 1 ⊆ I X 2 ⊆ · · · of radical ideals in R[N ] ∼ = R[x 1 , . . . , x n ]. Since the latter ring has a topological spectrum, this chain stabilises. Since X i = V(I X i ), so does the chain X 1 ⊆ X 2 ⊆ · · · .
Proof of Proposition 1 Let R be a ring with Noetherian spectrum, let M be a finitely generated R-module, and let X 1 ⊇ X 2 ⊇ · · · be a chain of closed subsets of A M . Since M is finitely generated, there exists a surjective R-module homomorphism ϕ : N := R n → M for some n. This defines a (linear) polynomial law N → M and so a continuous map for every i and D, and therefore X n = X n+1 for all n 0.  Then by the Nullstellensatz, some power of g reduces to zero modulo f 1 , . . . , f k . But then that reduction holds modulo p for every p ∈ Spec(R[1/r ]), so g vanishes identically on X (K p ) for all such p.

Polynomial functors over a ring
For reasons that will become clear later, we will only be interested in polynomial functors from the category fgfMod R of finitely generated free R-modules into either Mod R or fgMod R .
Definition 45 A polynomial functor P : fgfMod R → Mod R consists of an object P(U ) ∈ Mod R for each object U ∈ fgfMod R and a polynomial law for each U , V ∈ fgfMod R such that the diagram Hom(P(V ), P(W )) ⊕ Hom(P(U ), P(V )) Hom(P(U ), P(W )) Here the bilinear horizontal polynomial laws are given as in Remark 15. Moreover, for every U ∈ fgfMod R , we require that P U ,U (id U ) = id P(U ) and we require that P has finite degree, i.e., there is a uniform bound d ∈ Z ≥0 such that for all U , V the polynomial law P U ,V has degree at most d.
Polynomial functors fgfMod R → Mod R form an Abelian category PF R in which a morphism α : Q → P is given by an R-linear map α U : Q(U ) → P(U ) for each U ∈ fgfMod R such that the diagram of polynomial laws for all R-algebras A and ϕ ∈ A ⊗ Hom(U , V ). Note that to check that the diagram commutes, it suffices to check that this equality holds for A = R[x 1 , . . . , x n ] and ϕ = x 1 ⊗ ϕ 1 + · · · + x n ⊗ ϕ n where ϕ 1 , . . . , ϕ n is a basis of Hom(U , V ).
Recall that for all R-modules U , V , there is a natural A-linear map For U , V ∈ fgfMod R , this map is an isomorphism. Thus an element ϕ of A ⊗ Hom(U , V ) can be thought of as an "element of Hom(U , V ) with coordinates in A". Viewing Q U ,V ,A (ϕ), P U ,V ,A (ϕ) as maps, (1) implies that the diagram is injective and so the reverse implication also holds. So the family (α U ) U is a morphism of polynomial functors if and only if the last diagram above commutes for all A, U , V ϕ. This is closer to the definition of polynomial functors over infinite fields, and generalises as follows.

Definition 46
Let P, Q be polynomial functors. We define a polynomial transformation α : Q → P be a rule assigning to every U ∈ fgfMod R a polynomial law α U : Q(U ) → P(U ) such that the last diagram above commutes for all R-algebras A and ϕ ∈ A ⊗ Hom(U , V ).
Just like polynomial laws generalise R-module homomorphisms, and the latter are precisely the linear polynomial laws, polynomial transformations generalise morphisms of polynomial laws, and the latter are precisely the linear polynomial transformations.

Remark 47
If R is an infinite field, then a polynomial functor P : fgfMod R → fgMod R = fgfMod R is a the same thing as a functor from the category of finite-dimensional R-vector spaces to itself such that for all U , V ∈ fgfMod R the map is a polynomial map. This is the set-up in [8]. If R is a field but not necessarily infinite, then a polynomial functor fgfMod R → fgfMod R is a strict polynomial functor in the sense of Friedlander-Suslin [15].
Many of our proofs will involve passing to the case of (infinite) fields and invoking arguments from [8]. This is facilitated by the following construction.
Definition 48 (Base change). Let B be an R-algebra and let P : fgfMod R → Mod R be a polynomial functor. Then P induces a polynomial functor P B from fgfMod B to Mod B as follows: first, for each finitely generated free B-module U fix a B-module isomorphism ψ U : U → B ⊗ U R , where U R is a free R-module of the same R-rank as the B-rank of U . Then, set P B (U ) := B ⊗ P(U R ). Next, for each B-algebra A, we need to assign to every ϕ ∈ A ⊗ B Hom B (U , V ) an image in A ⊗ Hom B (P B (U ), P B (V )). For this, note that where the isomorphism in the first step is 1 A ⊗ B (ψ V •−•ψ −1 U ) and the second isomorphism follows from the freeness of U R and V R . Via these isomorphisms, ϕ is mapped to an element of A⊗Hom(U R , V R ). Applying P U R ,V R ,A to this element yields an element of A ⊗ Hom(P(U R ), P(V R )) ∼ = A ⊗ B (B ⊗ Hom(P(U R ), P(V R ))), and applying the natural map B ⊗ Hom(P(U R ), P(V R )) → Hom B (B ⊗ P(U R ), B ⊗ P(V R )) in the second factor (which may not be an isomorphism since P(U R ), P(V R ) need not be free) yields an element of A ⊗ B Hom B (P B (U ), P B (V )). It is straightforward to check that P B thus defined is a polynomial functor from fgfMod B to Mod B . A different choice of isomorphisms ψ U yields a different but isomorphic polynomial functor P B .

Remark 49
In this construction we have made use of the fact that P is a polynomial functor from finitely generated free R-modules to R-modules. The choice of ψ U 's could have been avoided as follows: instead of working with fgfMod R , we could have worked with the category whose objects are finite sets and whose morphisms J → I are given by I × J matrices with entries in R. Then P J ,I would have been a polynomial law from the module of I × J matrices to Hom(P(J ), P(I )). However, the set-up we chose stresses better that we are interested in phenomena that do not depend on a choice of basis in our free modules.
Definition 50 A polynomial functor P : fgfMod R → Mod R is called homogeneous of degree d if the polynomial law P U ,V is homogeneous of degree d for each U , V ∈ fgfMod R . Every polynomial functor P : fgfMod R → Mod R is a direct sum P 0 ⊕ · · · ⊕ P d , where P i : fgfMod R → Mod R is the homogeneous polynomial functor of degree i given on objects by and P i,U ,V is the restriction of the degree-i component of the polynomial law P U ,V to P i (U ). Here we identify R[t] ⊗ Hom(P(V ), P(V )) with Hom(P(V ), R[t] ⊗ P(V )).

Duality
Definition 51 Let P : fgfMod R → Mod R be a polynomial functor over R. Then we obtain another polynomial functor P * : fgfMod R → Mod R by setting, for each V ∈ fgfMod R , P * (V ) := P(V * ) * = Hom(P(V * ), R) and for each ϕ ∈ A ⊗Hom(U , V ), where ϕ * is the image of ϕ under the natural isomorphism (here we use that U , V are free) and the outermost * again represents a dual.
The dual functor P * of P has the same degree as P and will play a role in Sect. 6.10. To avoid having too many stars, we will there think of it as the functor that sends V * to P(V ) * . If P takes values in fgfMod R , then (P * ) * is canonically isomorphic to P.

Shifting
Let U be a finitely generated free R-module.

Definition 52
We define the shift functor Sh U : fgfMod R → fgfMod R that sends V → U ⊕ V and ϕ → id U ⊕ ϕ. For a polynomial functor P : fgfMod R → fgMod R we set Sh U (P) := P • Sh U , called the shift of P by U .

Lemma 53
The composition Sh U (P) is again a polynomial functor fgfMod R → fgMod R , the projection U ⊕ V → V yields a surjection of polynomial functors Sh U (P) → P and inclusion the V → U ⊕ V yields a section P → Sh U (P) to that surjection. In particular, Sh U (P) ∼ = P ⊕ (Sh U (P)/P). Furthermore, Sh U (P)/P has degree strictly smaller than the degree of P.
Proof The proof in [8,Lemma 14] (in the case where R is an infinite field) carries over to the current more general setting.

Dimension functions of polynomial functors
Let P : fgfMod R → fgMod R be a polynomial functor. For p ∈ Spec(R), set f p (n) := dim K p (K p ⊗ P(R n )). It turns out that these functions are polynomials in n, and depend semicontinuously on p. To formalise this semicontinuity, we order polynomials in Z[x] by f ≥ g if f (n) ≥ g(n) for all n 0; this is the lexicographic order on coefficients. Proof We proceed by induction on the degree of P. If P has degree 0, then P(R n ) is a fixed R-module U , and f p is the constant polynomial that maps n to dim K p (K p ⊗ U ).
In this case, if f ∈ Z[x] has positive degree, then f p > f and f p ≥ f are either both trivially true for all p or both trivially false for p (depending on the sign of the leading coefficient of f ), so we need only look at constant f . In this case, the result is classical; we recall the argument. Let R n → U be a surjective R-module homomorphism, and let N be its kernel. Since tensoring with K p is right-exact, 1 ⊗ N spans the kernel of the surjection K n p → K p ⊗ U for each p. The statement that dim K p (K p ⊗U ) is upper semicontinuous is therefore equivalent to the statement that dimension of the span of 1 ⊗ N in K n p is lower semicontinuous. And indeed, the locus where this dimension is less than k is defined by the vanishing of all k × k subdeterminants of all k × n matrices (with entries in R) whose rows are k elements of N .
For the induction step, assume that the proposition is true for all polynomial functors of degree < d and assume that P has degree d ≥ 1. Then consider the functor Sh R (P), which by Lemma 53 is isomorphic to P ⊕ Q for Q := Sh R (P)/P of degree < d.
By the induction hypothesis, the proposition holds for Q: the function g p (n) := dim K p (K p ⊗ Q(R n )) equals a polynomial with integral coefficients for all n ≥ 0, and p → g p is semicontinuous. Now we have This means that f p (n) is the unique polynomial with ( f p )(n) := f p (n+1)− f p (n) = g p (n) for n ≥ 0 and f p (0) = dim K p (K p ⊗ P(0)); this f p has integral coefficients and degree at most d.
For the semi-continuity statement, note that f p ≥ f is equivalent to either g p = f p > f , or else g p ≥ f and moreover f p (0) ≥ f (0). Both possibilities are closed conditions on p. Similarly, f p > f is equivalent to either g p > f or else g p ≥ f and f p (0) > f (0), which, again, are closed conditions.

Local freeness
We now generalise Lemma 8 to polynomial functors. Note that we do not claim that the complement is itself the evaluation of another subobject; i.e., S R[1/r ] needs not be a summand of P R [1/r ] in the category of polynomial functors over R [1/r ].
Proof Again, we proceed by induction on the degree of P. If P has degree 0, then so does S and then the statement is just Lemma 8. Suppose that the degree of P is d > 0 and that the proposition holds for all polynomial functors of degree less than d.
By Lemma 53, for each n we have where Q = Sh R (P)/P has degree < d. Similarly, we have Then r := r 0 · r 1 does the trick for the pair P, S.

The Friedlander-Suslin lemma
The Friedlander-Suslin lemma relates polynomial functors of bounded degree to representations of certain associative algebras called Schur Algebras. To introduce these, let U ∈ fgfMod R and let d ≥ 1 be an integer. The bilinear polynomial law given by composition yields an algebra homomorphism

R[End(U )] → R[End(U ) × End(U )] ∼ = R[End(U )] ⊗ R[End(U )]
which maps the part Taking the dual R-modules, we obtain a map We set S ≤d (U ) := R[End(U )] * ≤d . The first map is, in fact, an isomorphism due to the fact that S ≤d (U ) is finitely generated and free as an R-module. Indeed, if U is free with basis u 1 , . . . , u n , then End(U ) is free with basis (E i j ) n i, j=1 , where E i j u k = δ jk u i , and R[End(U )] ≤d is free with basis the monomials x α of degree ≤ d in the coordinates x i j dual to the E i j , and hence R[End(U )] * ≤d is free with the dual basis (s α ) α , where α runs over all multi-indices in Z n×n ≥0 such that |α| : be the bilinear map associated to the map above.

Definition 56
The R-module S ≤d (U ) with the bilinear map − * − is called the Schur algebra of degree ≤ d on U, and (given a basis of U ), the basis (s α ) α is called its distinguished basis.
The Schur algebra is associative (but not commutative unless n = 1); this follows from the associativity of composition in End(U ). Explicitly, the coefficient of s γ in the product s α * s β is computed as follows: First, expand the composition ( i j x i j E i j ) • ( kl y kl E kl ), where the x i j and y kl are variables, as i,l ( j x i j y jl )E il =: il z il E il . Then expand z γ as a polynomial in the x i j and the y kl , and take the coefficient of the monomial x α y β .
The map End(U ) → S ≤d (U ) that sends ϕ to the R-linear evaluation map is an injective homomorphism of associative R-algebras, so S ≤d (U )-modules M are, in particular, representations of the R-algebra End(U ). In fact, they are precisely the polynomial End(U )-representations of degree ≤ d, i.e., those for which the map End(U ) → End(M) is not just a homomorphism of (noncommutative) R-algebras but also a polynomial law making certain diagrams commute. Since we will not need this interpretation, we skip the details. Now suppose that P is a polynomial functor fgfMod R → Mod R of degree ≤ d. Then P(U ) naturally carries the structure of an S ≤d (U )-module as follows: the polynomial law P U ,U : End(U ) → End(P(U )) has degree ≤ d and therefore we have for certain endomorphisms ϕ α ∈ End(P(U )). Now the basis element s α of S ≤d (U ) acts on P(U ) via ϕ α ; it can be shown that this construction is independent of the choice of basis of U .
Theorem 57 (Friedlander- To conclude this section, we observe that Schur algebras behave well under base change: if A is an R-algebra, then we have a commuting diagram (up to natural isomorphisms): where the lower horizontal map is evaluation at A⊗U and the A-algebra A⊗S ≤d (U ) is canonically isomorphic to the Schur algebra S ≤d (A ⊗U ) on the free A-module A ⊗U .

Irreducibility in an open subset of Spec(R)
Let R be a domain and let P : fgfMod R → fgMod R be a polynomial functor. As before, for each prime p ∈ Spec(R) we set K p := Frac(R/p); in particular, K := K (0) is the fraction field of R. Recall that the base change functor yields a polynomial functor P K p over the field K p for each p ∈ Spec(R), and also a polynomial functor P K p over the algebraic closure K p of K p . The goal of this section is to transfer certain properties of P K to P K p for p in an open dense subset of Spec(R). Remark 59 Note that we don't require that Q is a functor into fgMod R ; we may not be able to guarantee this if R is not a Noetherian ring.
In order to prove this proposition, we use the following lemma.
Lemma 60 Let A be a (not necessarily commutative) associative R-algebra and N an A-module that is, as an R-module, finitely generated and free. Suppose that K ⊗ N is an irreducible (K ⊗ A) For each k = 1, . . . , n − 1, we will construct a constructible subset Z k of the Grassmannian Gr R (k, n) over R whose set of K p -points, for p ∈ Spec(R), is the set of k-dimensional (K p ⊗ A)-submodules of K p ⊗ N . The construction is as follows: for each J ⊆ [n] of size k consider the k × n matrix X J whose entries on the columns labelled by J are a k × k identity matrix over R and whose other entries are variables n) has an open cover of affine spaces A k×(n−k) R,J over R on which the coordinates are precisely these x i j with j / ∈ J . For j ∈ J we write x i j ∈ {0, 1} for the corresponding entry of X J . Note that, for each m = 1, . . . , k and each a ∈ A, we have  ∈ A and m = 1, . . . , k. For each prime p ∈ Spec(R), the subset [n] that map surjectively to K p J . In particular, by the assumption that K ⊗ N is still irreducible, the image of C J in Spec(R) does not contain the prime 0, for any k and any k-set J ⊆ [n]. In other words, the morphism C J → Spec(R) is not dominant. Set Z k := J ⊆[n],|J |=k C J , a finite union of locally closed subsets of the Grassmannian. Then Z k → Spec(R) is still not dominant, and neither is n−1 k=1 Z k → Spec(R). Hence there exists a nonzero r ∈ R that lies in the vanishing ideal of the image; the open dense subset Spec(R[1/r ]) ⊆ Spec R then has the desired property.

Proof of Proposition 58
By the Friedlander-Suslin Lemma (Theorem 57) and the fact that the Schur algebra behaves well under base change, it suffices to prove the corresponding statement for all d ∈ Z ≥0 , U := R d , and all S ≤d (U )-modules that are finitely generated over R (which, of course, is equivalent to being finitely generated as an S ≤d (U )-module).
So let M be a finitely generated S ≤d (U )-module and let N be an irreducible (K ⊗ S ≤d (U ))-submodule of K ⊗M that remains irreducible when tensoring with K . Define A straightforward computation shows that N is a (not necessarily finitely generated) S ≤d (U )-submodule of M.
By Lemma 8 there exist a nonzero r ∈ R and elements v 1 , . . . , v n ∈ N such that R[1/r ] ⊗ N is a free R[1/r ]-module with basis 1 ⊗ v 1 , . . . , 1 ⊗ v n . Then Lemma 60 applied with R equal to R[1/r ] and A equal to R[1/r ] ⊗ S ≤d (U ) shows that K p ⊗ N is an irreducible (K p ⊗ S ≤d (U ))-submodule of K p ⊗ M for all p in some nonempty open subset Spec R[1/(rs)] ⊆ Spec(R[1/r ]) ⊆ Spec(R).

Closed subsets of polynomial functors
Closed subsets of a polynomial functors play the role of affine varieties in finitedimensional algebraic geometry. In this subsection, P is a fixed polynomial functor fgfMod R → fgMod R of finite degree.
For any U , V ∈ fgfMod R we have a sequence of polynomial laws whose composition we denote by U ,V . We also let U ,V : Hom(U , V ) × P(U ) → P(U ) be the linear polynomial law given by projection. Recall that U ,V and U ,V both yield continuous maps from A Hom(U ,V )×P(U ) → A P(V ) .

Definition 61
We define A P to be P. A subset of A P is a rule X that assigns to each U ∈ fgfMod R a subset X (U ) of A P(U ) (see Definition 30) in such a manner that It is worth spelling out what this means. Let U , V be finitely generated free Rmodules, let D be an R-domain and let ϕ ∈ D ⊗ Hom(U , V ). Then the condition is that In the particular case where V = U , this condition can be informally thought of as the condition that X (U ) is preserved under the polynomial action of End(U ). Let α : Q → P be a polynomial transformation and let X be a subset of Q. Then α(X ) = (U → α U (X (U ))) is a subset of P.
Definition 62 For X ⊆ A P , we define the ideal I X of X to be the rule assigning Definition 63 (Base change). If X ⊆ A P is a closed subset and B is an R-algebra, then we obtain a closed subset X B of A P B by letting, for a U ∈ fgfMod B , X B (U ) be the closed subset X (U R ) B of A P B (U ) = A B⊗P(U R ) , where U R is the free R-module such that U ∼ = B ⊗ U R from the definition of P B .
We will use the following lemmas very frequently in our proof of Theorem 2.
Lemma 64 Let R be a ring with Noetherian spectrum and r an element of R. Let p 1 , . . . , p k be the minimal primes of R/(r ). Then two closed subsets X , Y ⊆ A P are equal if and only if X R[1/r ] = Y R[1/r ] and X R/p i = Y R/p i for all i = 1, . . . , k.
Proof It is easy to check that Y (V ) is a subset of A P(V ) for all V ∈ fgfMod R and that Y is a subset of A P . We need to check that Y is a closed subset of A P , i.e., that Y (V ) is a closed subset of A P(V ) for every V ∈ fgfMod R .
Let ϕ 1 , . . . , ϕ n be a basis of Hom(V , U ). For every R-algebra A, consider the map We have Let D be an R-domain and take p ∈ Y (V )(D). Then, viewing p as an element of Y (V ) (D[x 1 , . . . , x n ]), we see that Specializing the x i to elements of D, we find that g D (P V ,U ,D (a 1 ⊗ ϕ 1 + · · · + a n ⊗ ϕ n )( p)) = 0 for all a 1 , . . . , a n ∈ D.

Remark 67 It is not true in general that
For an example, take . Then the right hand side above consists of all p ∈ D ⊗ V ∼ = D n such that x p = x for every coordinate of p while the left hand side also has the requirement that (αx) p = αx for all α ∈ E for every D-domain E. So Y (V )(D) = 0.

Gradings
Let P : fgfMod R → fgMod R be a polynomial functor. For each U ∈ fgfMod R , the R-algebra R[P(U )] has two natural gradings: first, the ordinary grading that each coordinate ring R[M] of a module M has (see Definition 19); and second, a grading that takes into account the degrees of the homogeneous components P, as follows. Write P = P 0 ⊕ P 1 ⊕ · · · ⊕ P d , so that R[P(U )] is the tensor product of the R[P i (U )] by Proposition 27. Then multiply the ordinary grading on R[P i (U )] by i and use these to define a grading on R[P(U )], called the standard grading. The standard grading has an alternative characterisation, as follows: Lemma 68 For any closed subset X ⊆ A P and any U ∈ fgfMod R , the ideal I X (U ) is homogeneous with respect to the standard grading.
Proof Take f ∈ I X (U ) and let D be an R-domain. Then Hence the homogeneous parts of f are also contained in I X (U ).

Proof of the main theorem
In this section we prove Theorem 2. Let R be a ring whose spectrum is Noetherian and let P : fgfMod R → fgMod R a polynomial functor of finite degree. We will prove that any chain A P ⊇ X 1 ⊇ X 2 ⊇ · · · of closed subsets eventually stabilises.

Reduction to the case of a domain
Since Spec(R) is Noetherian, the ring R has finitely many minimal primes p 1 , . . . , p k . By Lemma 64 with r = 1, the sequence A P ⊇ X 1 ⊇ X 2 ⊇ · · · stabilises if and only if the sequence A P R/p i ⊇ X 1,R/p i ⊇ X 2,R/p i ⊇ · · · stabilises for each i ∈ [k]. So from now on we assume that R is a domain, we write K p := Frac(R/p) for p ∈ Spec(R), K := K (0) = Frac(R), and we let K , K p be algebraic closures of K , K p , respectively.

A stronger statement
We will prove the following stronger statement which clearly implies Theorem 2.
Theorem 69 Let (R, P, X ) be a triple consisting of a domain R with Noetherian spectrum, a polynomial functor P : fgfMod R → fgMod R of finite degree and a closed subset X ⊆ A P . Then (R, P, X ) satisfies the following conditions: (1) Every descending chain X = X 1 ⊇ X 2 ⊇ · · · of closed subsets of X eventually stabilises. (2) There exists a nonzero r ∈ R such that the following holds for all U ∈ fgfMod R : if f ∈ R[P(U )] vanishes identically on X (U )(K ), then f vanishes identically on X (U )(K p ) for all primes p ∈ Spec(R[1/r ]). (2) of the theorem means that I X R [1/r ] is determined by I X K . More precisely, setting R = R[1/r ], for every U ∈ fgfMod R , the ideal

Remark 70 Condition
is the pull-back of the ideal in K [P R (K ⊗ U )] of the affine variety X R (K ⊗ U ).
The proof of Theorem 69 is a somewhat intricate induction, combining induction on P, Noetherian induction on Spec(R) and induction on minimal degrees of functions in the ideal of X -for details, see below.

The induction base
If P has degree zero, then X is just a closed subset of A P(0) . Here, the Noetherianity statement is Proposition 1 and the statement about vanishing functions is Proposition 44.

The outer induction
To prove the theorem for P of positive degree, we will show that (R, P, X ) is implied by (R , P , X ) where X is a closed subset of A P and (R , P ) ranges over pairs that have one of the following forms: (i) (R , P ) = (R/p, P R/p ) for some nonzero prime p of R; or (ii) (R , P ) where R is a domain that is a finite extension of a localisation R [1/r ] of R, deg P ≤ deg P =: d, for K := Frac(R ) we have P K P K and for the largest e such that the homogeneous parts P e,K and P e,K are not isomorphic, the former is a quotient of the latter.
In both cases, we write (R, P) → (R , P ). We consider the class of all the pairs (R, P). The reflexive and transitive closure of the relation → is a partial order on .

Lemma 72 The partial order on is well-founded.
Proof Suppose that we had an infinite sequence of such steps. By the Friedlander-Suslin lemma, any sequence of steps of type (ii) only must terminate (see also [8,Lemma 12]). So our sequence contains infinitely many steps of type (i).
Each step (R, P) → (R , P ) induces a morphism α : Spec(R ) → Spec(R). This morphism α has the property that for irreducible closed subsets C D ⊆ Spec(R ), we have α(C) α(D). This holds trivially for steps of type (i), where the morphism α : Spec(R/p) → Spec(R) is a closed embedding, and also for steps of type (ii) by elementary properties of localisation and of integral extensions of rings (see, e.g., [10, Corollary 4.18 (Incomparability)]). Let . Then the maps β i have the same incomparability property as the α i . Hence, whenever the step there is the inclusion of irreducible closed sets im α i Spec(R i−1 ) and therefore im β i im β i−1 is a strict inclusion. This contradicts the Noetherianity of Spec(R 0 ).
By Lemma 72 we can proceed by induction on , namely, in proving that (R, P, X ) holds, we may assume (R , P , X ) whenever (R , P ) ← (R, P).
Lemma 73 Let r ∈ R be a nonzero element and let p 1 , . . . , p k be the minimal primes hold. Then (R, P, X ) holds as well.
Proof By Lemma 64, we see that condition (1) for (R, P, X ) follows from condition (1)  Combining this lemma with our induction hypothesis, we see that in order to prove (R, P, X ) it suffices to prove (R[1/r ], P R[1/r ] , X R[1/r ] ) for some r ∈ R. So we may replace (R, P, X ) by (R[1/r ], P R[1/r ] , X R [1/r ] ) whenever this is convenient.

Finding an irreducible factor
Now let P : fgfMod R → fgMod R be a fixed polynomial functor of degree d > 0 over a domain R with Noetherian spectrum. Recall that K is the fraction field of R.
Suppose first that the base change P K has degree < d. Then K ⊗ P d (U ) = 0 for all U ∈ fgfMod R . In particular, this holds for U = R d . So since P d (U ) is a finitely generated R-module, there exists a nonzero r ∈ R such that R[1/r ] ⊗ P d (U ) = 0. By the Friedlander-Suslin lemma (Theorem 57), we then find (P d ) R[1/r ] = 0. In this case, we replace (R, P, X ) by (R[1/r ], P R[1/r ] , X R [1/r ] ). By repeating this at most d times, we may assume that the base change P K has the same degree as P.
We want a polynomial subfunctor M of the top-degree part P d of P whose base change with K is an irreducible polynomial subfunctor of (P d ) K . In the next lemma, we show that such an M exists after passing from R to a suitable finite extension of one of its localisations. Let r ∈ R and R be as in the previous proposition. We would like to reduce to the case where R = R. As before, we can replace (R, P, X ) by

Proposition 74 There exist a finite extension R of a localisation R[1/r ] of R and a polynomial subfunctor M of the top-degree part of the polynomial functor P R such that the base change M K is an irreducible polynomial subfunctor of P d,K .
, so that R is a finite extension of R. We now prove a version of Lemma 73 for such extensions.
Lemma 75 Assume that (R , P R , X R ) holds. Then (R, P, X ) holds as well.
Proof By Lemma 65, condition (1) for (R , P R , X R ) implies condition (1) for (R, P, X ). Let r ∈ R be a nonzero element as in condition (2) for (R , P R , X R ), i.e., for every U ∈ fgfMod R , every f ∈ R [P R (U )] vanishing identically on X R (U )(K ) also vanishes identically on X R (U )(K p ) for every prime ideal p ∈ Spec(R [1/r ]). Now (r ) ∩ R is not the zero ideal, since r is nonzero and integral over R. Pick any nonzero r ∈ (r ) ∩ R. We claim that condition (2) holds for (R, P, X ) with this particular r .
Indeed, let U R ∈ fgfMod R and take U := R ⊗ U R . Let f be an element of R[P(U R )] vanishing identically on X (U R )(K ). Then f is naturally induces an element of R [P R (U )] vanishing identically on X R (U )(K ) = X (U R )(K ). So we see that f vanishes on X R (U )(K q ) for each q ∈ Spec(R [1/r ]). Since R is integral over R, for any p ∈ Spec(R) there exists an q ∈ Spec(R ) with q ∩ R = p; and if, moverover, the prime ideal p does not contain r , then the prime ideal q does not contain r . Hence f vanishes identically on K p , as desired.
We replace (R, P, X ) by (R , P R , X R ), so that there exists a polynomial subfunctor M of the top-degree part P d of P such that the base change M K is an irreducible polynomial subfunctor of P d,K .

Splitting off M
Proposition 55 guarantees that after passing to a further localisation (and using Noetherian induction for the complement), we may assume that for each U ∈ fgfMod R , the R-module P(U ) is the direct sum of a finitely generated free R-module and the (also finitely generated free) R-module M(U ). In particular, both P and P := P/M are polynomial functors fgfMod R → fgfMod R .
Let π : P → P be the projection morphism. For a closed subset X ⊆ A P , we define the closed subset X ⊆ A P as the closure of π(X ). Note that (R, P) → (R, P ) and hence (R, P , X ) holds. In particular, we may and will replace R by a further localisation R[1/r ] which ensures that, if f ∈ R[P (U )] vanishes identically on X (U )(K ), then it vanishes identically on X (U )(K p ) for all p ∈ Spec(R).

The inner induction
We perform the same inner induction as in [ Note that, by the outer induction hypothesis for (R, P , X ) and since {0, 1, . . . , ∞} is well-ordered, this order is well-founded. Hence when proving (P, R, X ), we may assume that (P, R, Y ) holds for all Y < X .
First suppose that δ X = ∞. Then, for all proper closed subsets Y of X , we have Y < X and so (R, P, Y ) holds by the inner induction hypothesis. It follows that condition (1) holds for (R, P, X ). Condition (2) for (R, P, X ) follows from condition (2) for (R, P , X ), with the same r ∈ R to be inverted.

A directional derivative
be a homogeneous polynomial of degree δ X in the standard grading, which lies in the ideal of X (U ) but not on the preimage in A P(U ) of X (U ).  . . . , x n , y 1 , . . . , y m , t] explicitly reads as t y 1 , x 2 + t y 2 , . . . , x m + t y m , x m+1 , . . . , x n ).
Take p = 1 if char R = 0 and p = char R otherwise. A Taylor expansion in t turns this expression into for some integer e ≥ 0, polynomial g ∈ R[x 1 , . . . , x n , y 1 , . . . , y m , t] and homogeneous polynomials h i ∈ R[P(U )] of (standard) degree δ X − p e d not all vanishing identically on X (U )(K ). Specialising the variables y i to values a i ∈ {0, 1}, we get that does not vanish identically on X (U )(K ).
Let p ∈ K ⊗ P(U ) be a point in X (U )(K ) such that h K ( p) = 0. Relative to the chosen basis of P(U ), we may write p = (α 1 , . . . , α n ). Reasoning as before, let r ∈ R be the product of all the denominators appearing in the minimal polynomials of the α i over K so that R = R[1/r ][α 1 , · · · , α k ] is a finite extension of R[1/r ] containing all α i . Replacing R by R and using Lemma 75, we can therefore assume that p ∈ X (U )(R) satisfies h R ( p) = 0. Further replacing R by R[1/h R ( p)], we find that h D ( p) = 0 for all R-domains D. Define Y to be the biggest closed subset of X where h does vanish.

Lemma 76 We have
for all V ∈ fgfMod R and R-domains D.
Proof The closed subset Y is the intersection of X with the biggest closed subset of A P where h vanishes. So the lemma follows from Lemma 66.
Let X = X 1 ⊇ X 2 ⊇ · · · be a sequence of closed subsets of X . Since Y < X , the statement (R, P, Y ) holds by the inner induction. In particular, the intersections of the X i with Y stabilise. This settles part of condition (1) of (R, P, X ). We now develop the theory to deal with the complement of Y . This will afterwards be used to settle both condition (2) for (R, P, X ) in Sect. 6.10 and complete the proof of condition (1) in Sect. 6.11.

Dealing with the localised shift
In [8,Lemma 25], it is proved that for all p ∈ Spec(R) and V ∈ fgfMod R , the projection Sh U (P) → Sh U (P)/M induces a homeomorphism of Sh U (X ) [

1/h](V )(K p ) with a closed subset of the basic open (Sh U (P)/M)[1/h](V )(K p )
. This proof uses that M K p is irreducible, which is why we have localised so as to make this true. The proof shows that, indeed, for each linear function x ∈ (K p ⊗ M(V )) * , the p e -th power x p e lies in the sum of the ideal of Sh U (X ) [1/h] . We globalise this result as follows: for all V ∈ fgfMod R , define There is a slight abuse of notation here: M(V ) is a submodule of P(U ⊕ V ), so M(V ) * is naturally a quotient of P(U ⊕ V ) * rather than a submodule. But the projection Conversely, by specializing the x i to a i ∈ A for any R-algebra A, this in fact suffices. As M is a subfunctor of P, we may here replace M by P.
Since P(V ) is free, the R-linear map

/h] (W ) + R[P(U ⊕ W )/M(W )][1/h]).
So we now need to show that (y) p e = (y p e ) is contained in this latter set. Since y ∈ N (V ), we have y p e = g 1 + g 2 for some g 1 ∈ I Sh U (X ) [1/h]  We now replace R by the localisation R[1/r ] and may henceforth assume that N (V ) = M(V ) * .

Proof of condition (2)
To establish condition (2) for (P, R, X ), we will first prove an analogous statement for the localised shift.
Lemma 80 There exists a nonzero r ∈ R such that the following holds for all V ∈ fgfMod R : if g ∈ R[P(U ⊕ V )] vanishes identically on Sh U (X ) [1/h] [1/h] and an element in the ideal of Sh U (X ) [1/h] (V ). We then find that also g p e = g 1 +g 2 with g 1 ∈ R[P(U ⊕V )/M(V )][1/h] and g 2 ∈ I Sh U (X ) [1/h] (V ) . Let Z be the closure of the projection of Sh U (X ) [1/h] to (Sh U (P)/M) [1/h]. Since both g and g 2 vanish identically on Sh U (X ) [1/h](V )(K ), g 1 vanishes identically on Z (V )(K ). By the outer induction hypothesis, after a localisation that doesn't depend on g 1 or on V , one concludes that g 1 vanishes identically on Z (V )(K p ) for all p ∈ Spec(R). But then g p e , and hence g itself, vanish identically on Sh U (X ) [1/h] Now we can establish condition (2) of (R, P, X ): Proposition 81 There exists a nonzero r ∈ R such that the following holds for all V ∈ fgfMod R : if g ∈ R[P(V )] vanishes identically on X (V )(K ), then g vanishes identically on X (V )(K p ) for all primes p ∈ Spec(R[1/r ]).

Remark 82
For each fixed V , such an r exists by Proposition 44. Taking the product of such r 's, the same applies to a finite number of V 's, so we may restrict our attention to all V of sufficiently large rank; we will do this in the proof.

Proof of Proposition 81
By the inner induction hypothesis, after replacing R by a localisation R[1/r ], we know that if g ∈ R[P(V )] vanishes identically on Y (V )(K ), then it vanishes identically on Y (V )(K p ) for all p ∈ Spec(R).
For any V ∈ fgfMod R and p ∈ Spec(R), define Z (V )(K p ) := X (V )(K p ) \ Y (V )(K p ). It suffices to show that with a further localisation we achieve that for any V ∈ fgfMod R , if g ∈ R[P(V )] vanishes identically on all points of Z (V )(K ), then it vanishes identically on all points of Z (V )(K p ) for all p ∈ Spec(R). In proving this, by Remark 82 above, we may assume that V has rank at least that of U . Hence we may replace V by U ⊕ V .
Such a g that vanishes identically on Z (U ⊕ V )(K ) vanishes, in particular, identically on Sh U (X )[1/h](V )(K ). Lemma 80 says that (after replacing R by a localisation that does not depend on g or V ), g also vanishes identically on Sh U (X ) [1/h](V )(K p ) for all p ∈ Spec R. This basic open is actually dense in Z (U ⊕ V )(K p ), as one sees as follows: Z (U ⊕ V )(K p ) is the image of the action If the basic open were contained in the union of a proper subset of the irreducible components of Z (U ⊕ V )(K p ), then, by irreducibility of GL(K p ⊗ (U ⊕ V )), so would the image of that action, a contradiction. Hence g then vanishes identically on Z (V )(K p ) for all p ∈ Spec(R).
Remark 83 Note that, unlike Y , the Z defined in the proof is not a subset of X in the sense of Definition 61.

Proof of the Noetherianity of X
Finally, we prove condition (1) of (R, P, X ). Let X = X 1 ⊇ X 2 ⊇ · · · be a sequence of closed subsets of X . Recall from Sect. 6.8 that the intersections of the X i with Y Since both the sequence of X i ∩ Y 's and Z i 's stabilize, using Corollary 36, the sequence of X i 's also stabilizes. So the closed subset X is Noetherian. This concludes the proof of condition (1) for (R, P, X ) and hence the proof of Theorem 2.

Dimension functions of closed subsets of polynomial functors
To illustrate that the proof method for Theorem 2 can be used to obtain further results on closed subsets of polynomial functors, we establish a natural common variant of Propositions 43 and 54. For each p ∈ Spec(R) define the function f p : Z ≥0 → Z ≥0 as f p (n) := dim(X (R n )(K p )).
Proposition 86 For each p ∈ Spec(R), f p (n) is a polynomial in n with integral coefficients for all n 0. Furthermore, the map that sends p to this polynomial is constructible.
Proof (Proof sketch) Both statements follow by inductions identical to the one for Theorem 2, using that, in the most interesting induction step, for n ≥ m := rk(U ) the dimension of X K p (K p n ) is the maximum of the dimensions of Y K p (K p n ) and Furthermore, for the case where X K p is the pre-image of X K p , we use Proposition 54, and for the base case in the induction proof for the constructibility statement we use Proposition 43.
Example 87 Take R = Z, take P = S 3 , and let X be the closed subset defined as the image closure of the polynomial transformation (S 1 ) 2 → S 3 , (v, w) → v 3 + w 3 ; see Sect. 1.3 for similar polynomial transformations. Then X K p (K p n ) has dimension 2n for p = (3) and dimension n for p = (3), since in the latter case the set of cubes of linear forms is a linear subspace of the space of cubics. This is an instance of Proposition 86.
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