Noetherian Polynomial FI-Algebra

• Noetherian polynomial FI-algebras are functors that assign polynomial rings to finite sets, establishing an FI-module theory with a uniform Noetherian property.
• They serve as a framework for asymptotic commutative algebra, ensuring finitely generated ideals and eventual polynomial behavior in dimensions.
• Their structure facilitates computational methods, including free FI-resolutions and Gröbner basis techniques via OI-categories for equivariant resolutions.

Searching arXiv for recent and foundational papers on FI-modules, polynomial FI-algebras, and Noetherianity. arxiv_search(query="Noetherian polynomial FI-algebra FI-modules OI-modules", max_results=10) Searching arXiv for relevant papers. A Noetherian polynomial FI-algebra is an FI-algebra whose values are functorially related polynomial rings or symmetric algebras and whose FI-module theory satisfies a Noetherian property. In one standard formulation, an FI-algebra is a covariant functor $$A:\mathrm{FI}\to\{\text{commutative, associative, unital }K\text{-algebras}\}$$, where FI is the category of finite sets and injections; a polynomial FI-algebra is then a functorial polynomial-ring construction such as $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$ with transition maps induced by injections on the column index. In the earlier FI literature, the expression also covers the prototype $A_n=R[x_1,\dots,x_n]$ and more generally symmetric algebras $\mathrm{Sym}(V)$ built from finitely generated FI-modules. The associated Noetherian condition appears in two closely related forms: module-theoretic Noetherianity, meaning that finitely generated FI-modules over the FI-algebra are Noetherian, and ideal-theoretic Noetherianity, meaning that FI-ideals are finitely generated as FI-modules (Church et al., 2012, Nagel et al., 2017, Morrow et al., 15 Jul 2025).

1. FI, FI-modules, and polynomial FI-algebras

The basic indexing category is FI: its objects are finite sets, usually identified with $[n]=\{1,\dots,n\}$ for $n\ge0$, and its morphisms are injections. The endomorphism group of $[n]$ is $S_n$. An FI-module over a commutative ring $R$ is a covariant functor $V:\mathrm{FI}\to R\text{-Mod}$; an FI-algebra over $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$0 is a covariant functor $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$1. If $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$2 is an FI-algebra, an FI-module over $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$3 is a covariant functor $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$4 such that each $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$5 is an $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$6-module and the transition maps are compatible with the algebra maps $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$7 for injections $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$8 (Church et al., 2012, Morrow et al., 15 Jul 2025).

Two families of free objects organize the theory. In the constant-coefficient setting, the free FI-module $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$9 is given by

$A_n=R[x_1,\dots,x_n]$0

Over an FI-algebra $A_n=R[x_1,\dots,x_n]$1, the free FI-module $A_n=R[x_1,\dots,x_n]$2 is defined by

$A_n=R[x_1,\dots,x_n]$3

with transition maps $A_n=R[x_1,\dots,x_n]$4. These free modules play the role of representable generators and control finite generation (Church et al., 2012, Morrow et al., 15 Jul 2025).

The literature uses “polynomial FI-algebra” in more than one closely related way. One usage treats functorial polynomial rings such as

$A_n=R[x_1,\dots,x_n]$5

and their multivariable analogues $A_n=R[x_1,\dots,x_n]$6 for finitely generated FI-modules $A_n=R[x_1,\dots,x_n]$7. Another, more specific, usage defines the polynomial FI-algebra $A_n=R[x_1,\dots,x_n]$8 by

$A_n=R[x_1,\dots,x_n]$9

for $\mathrm{Sym}(V)$0. A third, structurally equivalent, presentation uses the basic FI-algebras $\mathrm{Sym}(V)$1, whose variables are indexed by injections $\mathrm{Sym}(V)$2; polynomial FI-algebras are tensor products of the $\mathrm{Sym}(V)$3 (Nagel et al., 2017, Morrow et al., 15 Jul 2025).

Notion	Definition	Typical example
FI	finite sets and injections	objects $\mathrm{Sym}(V)$4, endomorphisms $\mathrm{Sym}(V)$5
FI-module	covariant functor $\mathrm{Sym}(V)$6-Mod	$\mathrm{Sym}(V)$7
FI-algebra	covariant functor $\mathrm{Sym}(V)$8-Alg	$\mathrm{Sym}(V)$9
Polynomial FI-algebra	tensor product of $[n]=\{1,\dots,n\}$0, or specifically $[n]=\{1,\dots,n\}$1	$[n]=\{1,\dots,n\}$2
Free FI-module over $[n]=\{1,\dots,n\}$3	$[n]=\{1,\dots,n\}$4	$[n]=\{1,\dots,n\}$5

2. Classical Noetherianity in FI theory

The foundational Noetherian theorem for FI-modules states that if $[n]=\{1,\dots,n\}$6 is a Noetherian ring, $[n]=\{1,\dots,n\}$7 is a finitely generated FI-module over $[n]=\{1,\dots,n\}$8, and $[n]=\{1,\dots,n\}$9 is a sub-FI-module, then $n\ge0$0 is finitely generated. Equivalently, the category of FI-modules over a Noetherian ring is locally Noetherian, and the full subcategory of finitely generated FI-modules is abelian (Church et al., 2012).

This theorem has two immediate consequences that are central for polynomial FI-algebras. First, over any field $n\ge0$1, if $n\ge0$2 is a finitely generated FI-module, then $n\ge0$3 is eventually given by a polynomial $n\ge0$4, and if $n\ge0$5 is generated in degree $n\ge0$6, then $n\ge0$7. Second, there exists $n\ge0$8 such that

$n\ge0$9

for all $[n]$0. Thus finitely generated FI-modules are governed by bounded-degree data and admit an inductive description from small sets (Church et al., 2012).

In the FI-algebra setting, an FI-ideal is a subfunctor $[n]$1 such that each $[n]$2 is an ideal and the FI-maps preserve these ideals. The earlier FI-algebra viewpoint calls $[n]$3 Noetherian as an FI-algebra when every FI-ideal is finitely generated as an FI-module. For the prototype

$[n]$4

the degree-1 part is the free FI-module $[n]$5, and the polynomial algebra is generated by that degree-1 piece. More generally, multivariable polynomial FI-algebras arise as symmetric algebras on free FI-modules. In these cases, Theorem A implies that FI-ideals are finitely generated as FI-modules, and quotients such as diagonal coinvariant constructions inherit eventual polynomiality of graded dimensions (Church et al., 2012).

A recurrent point in the literature is that the FI-Noetherian property is stronger than the ordinary statement that each individual ring $[n]$6 is Noetherian. The ordinary ring-theoretic statement is levelwise; FI-Noetherianity is uniform in $[n]$7 and respects the injection structure. This uniformity is what makes the theory useful for asymptotic commutative algebra, representation stability, and families of ideals compatible with variable relabeling (Church et al., 2012).

3. Polynomial FI-algebras with varying coefficients

The varying-coefficient theory of Nagel and Römer formalizes FI-algebras and FI-modules over them in a way that places polynomial FI-algebras at the center. For a commutative ring $[n]$8, they define $[n]$9 by taking, for each finite set $S_n$0,

$S_n$1

and then define a polynomial FI-algebra to be a tensor product $S_n$2. The most important concrete case is $S_n$3, whose value on $S_n$4 is the polynomial ring $S_n$5 (Nagel et al., 2017).

Their main Noetherian theorem says that if $S_n$6 is Noetherian and $S_n$7, then every finitely generated OI-module over $S_n$8 is Noetherian; the FI analogue states that every finitely generated FI-module over $S_n$9 is Noetherian. This recovers the constant-coefficient case $R$0 and extends it to polynomial coefficient rings varying with $R$1 (Nagel et al., 2017).

The technical engine is the passage from FI to OI, where OI is the category of totally ordered finite sets and order-preserving injections. Order allows the construction of monomial orders compatible with the functorial structure and of Gröbner bases for submodules of free OI-modules. For the polynomial OI-algebra

$R$2

every submodule of a finitely generated free OI-module has a finite Gröbner basis. From this, one deduces that free OI-modules are Noetherian, then that finitely generated OI-modules are Noetherian, and finally that the FI versions are Noetherian by restriction and comparison (Nagel et al., 2017).

This framework also yields homological finiteness. If $R$3 is a Noetherian FI- or OI-algebra and the relevant module category is Noetherian, then every finitely generated FI- or OI-module $R$4 over $R$5 admits a projective resolution

$R$6

with each $R$7 finitely generated. In the graded case, the corresponding Tor-modules form finitely generated graded FI- or OI-modules, and for each fixed homological degree $R$8 the internal degrees in which the Betti numbers $R$9 are nonzero are uniformly bounded and eventually stabilize as $V:\mathrm{FI}\to R\text{-Mod}$0 (Nagel et al., 2017).

The same paper notes that this Noetherianity extends beyond the basic polynomial algebras to certain monomial subalgebras, Veronese subalgebras, and Segre-type constructions. The general conjecture formulated there is broader: every finitely generated OI-module, respectively FI-module, over a Noetherian OI-algebra, respectively FI-algebra, should be Noetherian (Nagel et al., 2017).

4. EI-categorical foundations and the abstract source of FI Noetherianity

Gan and Li recast FI-module Noetherianity inside the theory of infinite EI categories. An EI category is a small category in which every endomorphism is an isomorphism; a locally finite EI category of type $V:\mathrm{FI}\to R\text{-Mod}$1 has objects $V:\mathrm{FI}\to R\text{-Mod}$2 and underlying quiver

$V:\mathrm{FI}\to R\text{-Mod}$3

For such a category $V:\mathrm{FI}\to R\text{-Mod}$4, a $V:\mathrm{FI}\to R\text{-Mod}$5-module is equivalently a covariant functor $V:\mathrm{FI}\to R\text{-Mod}$6. FI is the fundamental example: its objects are finite sets, its morphisms are injections, and $V:\mathrm{FI}\to R\text{-Mod}$7 (Gan et al., 2014).

The main theorem of Gan and Li states that if $V:\mathrm{FI}\to R\text{-Mod}$8 is a locally finite EI category of type $V:\mathrm{FI}\to R\text{-Mod}$9 satisfying the transitivity and bijectivity conditions, and $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$00 is a field of characteristic $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$01, then every finitely generated $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$02-module is Noetherian. The transitivity condition requires that $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$03 act transitively on $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$04; the bijectivity condition requires eventual bijectivity of the orbit maps

$P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$05

For FI, these conditions hold, and the orbit structure stabilizes for $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$06 (Gan et al., 2014).

This theorem generalizes the characteristic-zero FI result and does so without symmetric-group representation theory. Its significance for polynomial FI-algebras is indirect but decisive: it supplies the module-theoretic Noetherian input on which FI-algebra arguments rest. The paper itself does not develop FI-algebra theory, but it explains that if an FI-algebra is finitely generated and its underlying FI-module is finitely generated, then Theorem 3.7 implies that every FI-submodule of the underlying module is finitely generated. The same pattern extends to other representation-stability categories satisfying the EI hypotheses, including $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$07, $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$08, $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$09, VI, and VIC (Gan et al., 2014).

From the algebraic point of view, this EI formulation isolates the category-theoretic source of Noetherianity. The polynomial structure is not part of the hypotheses; the crucial data are the combinatorial stabilization properties of morphism spaces. For FI-algebras, this means that the fundamental constraint is often not the algebra structure itself but the behavior of injections and orbit types in the underlying category (Gan et al., 2014).

5. Free resolutions, algorithms, and infinite-variable symmetry

The 2025 work on equivariant free resolutions treats polynomial FI-algebras computationally, with the specific ambient FI-algebra $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$10. In that setting, $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$11 is the polynomial ring in $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$12 variables

$P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$13

graded by $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$14, and each $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$15 in an FI-module over $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$16 carries a natural $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$17-action compatible with the $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$18-module structure. The paper takes as input the Noetherian property that finitely generated FI-modules over $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$19 are Noetherian, and notes that this is precisely what ensures the existence of free FI-resolutions by finitely generated free FI-modules (Morrow et al., 15 Jul 2025).

Free FI-modules over $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$20 are defined by

$P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$21

A finitely generated FI-module over $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$22 therefore admits a free FI-resolution

$P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$23

and in the graded field case there is a graded free FI-resolution of minimal rank, unique in the sense recorded in the paper. Taking the width-$P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$24 component yields a $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$25-equivariant free resolution over the ordinary polynomial ring $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$26 (Morrow et al., 15 Jul 2025).

The algorithmic novelty is that kernels are computed after restriction from FI to OI. Restriction preserves enough structure that a free FI-module becomes a controlled direct sum of free OI-modules: $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$27 In the OI world one has Gröbner bases and an OI-Schreyer theorem. For a map of finitely generated free OI-modules over $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$28, a finite generating set of the kernel can be computed in finite time by the OI-Buchberger algorithm together with the OI-Schreyer theorem. A lifting lemma then transfers the generating set back to the FI kernel. Iterating this procedure yields truncated free FI-resolutions (Morrow et al., 15 Jul 2025).

The same paper passes to colimits. The colimit of $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$29 is the infinite-variable polynomial ring

$P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$30

equipped with the action of the infinite symmetric group on the column index. Exactness of colimits sends FI-modules over $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$31 to $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$32-modules over $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$33, and truncated free FI-resolutions become $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$34-equivariant free resolutions. The free modules occurring in such resolutions are finitely generated up to symmetry. This places Noetherian polynomial FI-algebras directly inside equivariant computational commutative algebra on infinitely many variables (Morrow et al., 15 Jul 2025).

6. Topological Noetherianity, limitations, and neighboring frameworks

A standard source of confusion is the distinction between algebraic and topological Noetherianity. Draisma proved that over an infinite field, every finite-degree polynomial functor $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$35 is topologically Noetherian: every descending chain of closed $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$36-subspaces stabilizes, or equivalently the inverse-limit space $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$37 is $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$38-Noetherian. For $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$39, this yields topological Noetherianity for spaces of tuples of forms and underlies the geometric input in Stillman-type arguments (Draisma, 2017).

Bik, Danelon, and Draisma extended this from infinite fields to commutative rings $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$40 with Noetherian spectrum. For a finite-degree polynomial functor $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$41, the associated topological space $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$42 is Noetherian: every descending chain of closed subfunctors stabilizes. In particular, for functors built from symmetric powers, parameter spaces of homogeneous generators behave topologically Noetherian over $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$43 and more general base rings. This is a geometric analogue of FI- and tca-type Noetherian phenomena, but it does not assert that the coordinate ring is Noetherian as an algebra (Bik et al., 2020).

That distinction matters. Draisma explicitly asks whether, for a finite-degree polynomial functor $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$44 over an infinite field, every ascending chain of ideals in the tca $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$45 stabilizes. The paper answers this for degree $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$46 and a few special cases, but records the general problem as open. A closely related recent result in positive characteristic goes in the opposite direction: in the category of strict polynomial functors, there exists a finitely generated commutative algebra object whose associated $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$47-algebra is not $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$48-noetherian. The counterexample uses the algebra $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$49 of polarizations of elementary symmetric polynomials inside the multisymmetric invariant ring in $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$50 variables, together with Frobenius splitting. This is not itself a theorem about FI-algebras, but it shows that polynomial-functor noetherianity can fail at the ideal-theoretic level in characteristic $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$51 (Draisma, 2017, Ganapathy, 18 Jun 2025).

A neighboring additive-category result sharpens the contrast. For functors from finitely generated projective $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$52-modules to $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$53-modules, with $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$54 a finitely generated commutative ring and $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$55 Noetherian, every finitely generated polynomial functor is Noetherian and admits a finitely generated projective resolution. This is not phrased in FI language, but it is structurally close to the FI story: polynomial degree, local Noetherianity, and finite projective resolutions coexist in a functor category governed by representation-theoretic and homological methods (Djament et al., 2022).

Taken together, these developments define the modern scope of the subject. In the FI setting proper, polynomial FI-algebras such as $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$56, $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$57, and $P^{FI,c}_n=K[x_{i,j}\mid i\in[c],\,j\in[n]]$58 are Noetherian in the strong module-theoretic sense needed for ideals, syzygies, and equivariant resolutions. In adjacent polynomial-functor settings, topological Noetherianity is considerably broader than algebraic Noetherianity, and positive characteristic introduces genuine ideal-theoretic pathologies. The theory of Noetherian polynomial FI-algebras therefore sits at the intersection of representation stability, Gröbner methods on combinatorial categories, infinite-symmetry commutative algebra, and the still unresolved boundary between topological and algebraic Noetherianity (Nagel et al., 2017, Morrow et al., 15 Jul 2025, Ganapathy, 18 Jun 2025).