Hilbert polynomial of length functions

Let $\lambda$ be a general length function for modules over a Noetherian ring R. We use $\lambda$ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of~$\lambda$. We show that the leading term $\mu$ of the Hilbert polynomial is an invariant of the module, which refines both the algebraic entropy and the receptive algebraic entropy; its degree is a suitable notion of dimension for $R[X]$-modules. Similar to algebraic entropy, $\mu$ in general is not additive for exact sequence of $R[X]$-modules: we demonstrate how to adapt of certain entropy constructions to this new invariant. We also consider multi-variate versions of the Hilbert polynomial.


Introduction
Let R be a commutative ring with unity; for this introduction and most of the article R will be Noetherian.
Generalized length functions were introduced in [NR65] and further studied (among other sources) in [Vám68a;SVV13]: see also §3 for a brief introduction; we will simply say "length" or "lenght function" in the rest of the paper.Fix 1 ≤ k ∈ N and let S := R[x 1 , . . ., x k ].Let M be an S-module.
We generalize the theory of Hilbert series and Hilbert polynomial for S-modules when R is a field and the linear dimension is the length function (see e.g.[Eis95;MS05]), to the case of S-modules with an arbitrary length function λ.We begin by assuming that M is an λ S -small module, i.e., there exists a finitely generated R-submodule V of M with finite λ-length such that SV = M (we say that V witnesses that M is λ S -small).We denote by S n the set of polynomials in S of total degree less or equal to n, and consider the formal power series and prove that it is a rational function (of t).We also show that for large enough n, the function n → λ(S n V ) is a polynomial, whose leading term µ(M ) is independent of the choice of the witness V (Theorem 6.6): thus, µ λ (M ) is an invariant of M that measures the asymptotic growth of λ on M , and refines both the algebraic entropy and the receptive algebraic entropy.Moreover, the degree of µ(M ) gives a well-behaved notion of dimension (w.r.t.λ) for S-modules.
In §4 we review some basic notions and results about graded and filtered modules over a Noetherian ring.In §5, we construct the Hilbert series for two classes of modules: graded modules (Theorem 5.1) and upward filtered modules (Corollary 5.2).We choose to work with upward filtered modules instead of the more common downward filtered modules, because they are more suitable for the applications in §6.2 and §8 (see also [KLMP99,§1.3]).
In §6, we prove the existence of a Hilbert polynomial for λ S -small modules, and show that its leading term, denoted by µ(M ), is an invariant of M In §7, we extend the definition of µ(M ) to the case when M is not λ S -small, and show that µ is an additive function on the class of modules that are locally λ R -finite (see Def. 3.1 and Theorem 8.1).
The coefficient of the k-term of the Hilbert polynomial is (up to a constant factor) the algebraic entropy of the action of N k on M (see §3 for the definition of algebraic entropy and its main properties).Therefore, the additivity of µ is a refinement of the known additivity of algebraic entropy (see Fact 3.7 and [SVV13; SV15; DFG20]).However, the additivity of algebraic entropy has already been proved under weaker assumptions: one of the most general results considers the case when the acting monoid N k is replaced by a cancellative and amenable monoid (and M is locally λ R -finite): see [Vir19;DFG20].
In §10 we show how the usual construction of Hilbert-Samuel polynomial can be extended to length functions, thus obtaining another invariant of M .
In §9 we replace S with a finitely generated R-algebra T and define a corresponding Hilbert polynomial for each T -module M : its degree will be an invariant of M (while the leading coefficient will depend on the choice of a set of generators for T ).
In §11 we introduce the d-dimensional entropy as a generalization of the receptive entropy in [BDGS20], and relate it to the Hilbert polynomial.
It is well known that the algebraic entropy of modules that are not λ R -finite may fail to be additive, which is a desirable property.To overcome this limitation, some alternative notions of algebraic entropy have been introduced in the literature.We will explore how similar adaptations can be applied to µ, the leading term of the Hilbert polynomial.
In §12, we use a technique from [Vám68a] to define μ, an additive function on all S-modules that extends µ on locally λ R -finite modules.This construction also works for the (d-dimensional) entropy.
In §13, we define the "intrinsic" Hilbert polynomial, which is related to the intrinsic algebraic entropy introduced in [DGSV15]: see §3.3.We obtain another invariant μ from the intrinsic Hilbert polynomial.We conjecture that μ is additive on S-modules, and prove that it is sub-additive.Under this conjecture, we have two additive invariants, μ and μ, which may differ in general.
Moreover, in §14, we consider a finer version of the Hilbert series where the grading is given by a suitable monoid Γ instead of N.
We mostly adapt well-known results about Hilbert series and polynomials to our setting, or prove them by simple arguments.Therefore, we omit some proofs for brevity.
In most of the results, we assume that R is Noetherian ring (or at least that the relevant modules are Noetherian).
We conjecture that some of our results can be extended to non-Noetherian rings.However, we lack a satisfactory notion of "Noetherianity with respect to λ" for rings, which prevents us from pursuing this direction further.We remark that, without any Noetherian assumption, a Hilbert polynomial may not exist (see §6.3).On the other hand, the algebraic entropy and its intrinsic version have been studied for non-Noetherian rings (see e.g.[SV15;SV19]).
We also leave as an open problem the case when either R or S are non-commutative rings.See [Nor68] for the case when R is not commutative, and [DFG20; Vir19] for the case when S is not commutative.Some partial results are in Appendix B.
We believe that most results (except possibly Proposition 12.7 and its corollaries) can be generalized to the case when R is non-commutative, but we do not explore this possibility here due to our limited expertise in non-commutative rings.
Acknowledgments.We thank Simone Virili and Giorgio Ottaviani for the useful discussions, and the anonymous referee for the care and great amount of work that he put in refereeing this article.

Preliminaries, assumptions, and notation
2.1.Notation.N denotes the set of natural numbers, including 0. ∞ denotes some element that is greater than any real number.R is a ring (commutative with 1) and λ is a length function on R-mod.We write I ⊳ R if I is an ideal of R, and A ≤ M if A is a submodule of the module M .We fix 1 ≤ k ∈ N and denote x := x 1 , . . ., x k and S := R[x].Given n ∈ N, S (n) denotes the set of homogeneous polynomials in S of degree exactly n (plus the 0 polynomial), while S n is the set of polynomials in S of degree at most n (they are both finitely generated R-modules: notice that S 0 = R).From §6 to the appendices R will be a Noetherian ring (therefore, S and S[y] will also be Noetherian rings).

2.2.
Polynomial coefficients for some rational functions.In this subsection we gather some results: probably they are well known, but we could not find a reference.
The case ℓ = 1 of the next proposition is well-known, and the one we will use for most of the article (see e.g.[AM69,Ch.11]).
Proposition 2.1.Let K be a ring of characteristic 0, and and expand f as Then, there exists a polynomial q( t) ∈ K[ t] such that: (1) For every n ∈ N ℓ large enough (2) for every i = 1, . . ., ℓ Proof.It is clear that it suffices to treat the case when p = 1.We proceed by induction on ℓ.If ℓ = 0, then f = 1, and q = 0.If ℓ = 1, the result is easy: by further induction on γ 1 , one can prove that a n = n+γ1−1 γ1−1 .Assume now that we have already proved the result for ℓ − 1: we want to prove it for ℓ.Denote t := t 2 , . . ., t ℓ and By inductive hypothesis, there exists r( t) ∈ K[ t] satisfying (1) and (2) for g.Moreover, where and the polynomial q(t 1 , t) := s(t 1 )r( t) ∈ K[t] satisfies the conclusion.
is homogeneous of degree i, and p d = 0. We call p d the leading homogeneous component of p (if p = 0 then, by convention, the leading homogeneous component of p is 0).As usual, if ℓ = 1 we call the leading homogeneous component of p the leading term of p. Definition 2.3.We write • p q if there exists c ∈ N ℓ such that, for every n ∈ N ℓ large enough, p(n) ≤ q(n + c); • p ≃ q if p q and q p; • p 0 if, for every n ∈ N ℓ large enough, p(n) ≥ 0.
Then, p and q have the same leading homogeneous component.
Proof.If either p or q is zero, it is clear that the other is also zero (and therefore they have the same leading homogeneous component).Thus, without loss of generality, we may assume that they are both non-zero.Let p ′ and q ′ be the leading homogeneous component of p and q respectively, and h ′ be the leading homogeneous component of h := p − q.If, by contradiction, Since q p, we have r = +∞, but since since p q, we have r = −∞, absurd.

Length functions and their entropy
3.1.Length functions.It can happen that λ(R) is infinite: the following definition deals with that situation.
Examples 3.2.(a) If R is a field, then the linear dimension is the unique length λ on R-mod such that λ(R) = 1.(b) Let R = Z, and define λ(M ) to be the logarithm of the cardinality of M .Then, λ(R) = ∞, and an Abelian group is locally λ R -finite iff it is torsion.We call λ the standard length on Z-modules, and we will use it often in examples.(c) Given any ring R, the (classical) length of an R-module M is the length of a composition series for M (see e.g.[Eis95]).(d) The following are two "trivial" lengths: i) λ(M ) = 0 for every M ; ii) λ(M ) = ∞ for every M = 0, and λ(0) = 0. (e) The following function is a length on Z-modules: Exercise 3.4.Let R be an integral domain.Then, there exists a unique length λ 0 on R-modules satisfying λ 0 (R) = 1.Denoting by K the field of fractions of R, λ 0 is defined by: λ 0 (M ) := dim K (K ⊗ M ).
( 1 ) Also called "locally λ-finite" in [SVV13]: here we prefer to write explicitly the ring R too.
The two trivial lengths in 3.2 (d) and the one in 3.2(e) are particular cases of "singular" lengths, i.e. lengths taking values only 0 or ∞ (see [Spi20,§6] for a characterization): in the present treatment we will mostly ignore them, since the associated entropies, the invariant µ, and its modifications μ and μ are all 0.
A property we will use often in the rest of the paper is the following: Definition 3.5.Given an R-algebra T and a T -module M , we say that M is λ T -small if M is finitely generated (as T -module) and locally λ R -finite.( 2 ) Remark 3.6.Assume that T is a Noetherian R-algebra and let M be a T -module.The following are equivalent: ( (3) every submodule and every quotient of M is λ T -small.Any submodule V ≤ M satisfying (2) in the above remark is a witness of the λ T -smallness of M .
[Vám68a, Thm.5] characterizes length functions on Noetherian rings: for every prime ideal P ≤ R there is a canonical length function l P on R-mod, and any length function λ can be written as λ = P ⊳R prime ideal r λ P • l P for some r λ P ∈ R ≥0 ∪ {∞} (we use the convention that 0 • ∞ = 0).3.2.Algebraic entropy.The content of this and the following subsection can be skipped: it is mostly a motivation for the definitions and results in the paper.We recall the definition of algebraic entropy and its main properties.
Let M be an R-module and φ be an endomorphism of M .Given an R-submodule V ≤ M , we define n (the limit always exists by Fekete's Lemma, since the function n → λ V + φ(V ) + • • • + φ n−1 (V ) is subadditive: but see also later in this subsection).The entropy of φ (according to the length λ) is defined by Equivalently, we can see M as an R[X]-module (with X acting on M as φ), and consider h λ as an invariant of M as R[X]-module.For the relationship between algebraic entropy and multiplicity, see [SVV13;Nor68].
More generally, given an S-module M , and an R-submodule V ≤ M of finite length, define The limit in the definition of H λ (M ; V ) exists, and h λ is the algebraic entropy (relative to the length function λ).We prove the stronger result that λ(S n V ) is eventually equal to a polynomial as Theorem 6.5, and therefore the limit in (1) exists.However, the existence of the limit was already well-known: e.g., [CCK14; ( 2 ) A similar notion is called "Hilbert T -module" in [Nor68, Ch.7].DFG20] give a more general version.[DFG20] consider the action of a cancellative amenable monoid: in our case, we can identify S with the group ring R[N k ], and therefore an S-module is the same as an R-module M together with an action * of N k on M by endomorphisms, and N k is a cancellative amenable monoid.Let B n be the set of tuples m ∈ N k such that is the cardinality of B n .Moreover, the family (B n ) n∈N , is a Følner sequence for N k and therefore, as in [DFG20], we can apply the machinery in [CCK14] to obtain that the following limit exists (and is independent from the choice of the Følner sequence): .
See also [Vir19] for an proof in the case when the acting monoid is a finitely generated group.
One of the most important properties of algebraic entropy is its additivity: Fact 3.7.Let 0 → A → B → C → 0 be an exact sequence of S-modules.Assume that B is locally λ R -finite.Then, We prove a stronger version of the above fact as Theorem 8.1.However, the fact was well-known: see [SV15] for the case when k = 1; [DFG20] gives a general version for Z-modules with the action of an amenable cancellative monoid (but the proof generalizes to R-modules), while [Vir19] treats the case of the action of an amenable finitely generated group on R-modules.

Intrinsic algebraic entropy.
Let M be an R-module and φ be an endomor- )/V n (the limit always exists, again by Fekete's Lemma).The intrinsic entropy of φ (according to the length λ), introduced in [DGSV15], is defined by There is a corresponding addition theorem For a proof of the above fact, see [DGSV15; SV18]: we will consider a stronger version in a more general setting in §13 (however, we were not able to prove additivity but only sub-additivity).

Graded and filtered modules
In this section we gather a few definitions and facts about graded and filtered S-modules.The most important ones are: how to construct a graded S[y]-module B(A) starting from an upward filtered module A (Definitions 4.5 and 4.6), and a version of Artin-Rees Lemma for upward filtered modules (Proposition 4.9).

Graded modules.
Definition 4.1.Fix γ = γ 1 , . . ., γ k ∈ N k .An N-graded S-module of degree γ is given by an S-module M and a decomposition where each M n is an R-module, and, for every i ≤ k and n ∈ N, We denote by M the module M with the given grading (including the tuple γ := γ 1 , . . ., γ k ).
We will use implicitly the following lemma many times in the remainder of the article.
Proof.Let m 1 , . . ., m p ∈ M generate M (as S-module).Fix n ∈ N; we want to show that M n is finitely generated.Without loss of generality, we may assume that each m i is homogeneous of degree d i (i.e., m i ∈ M di ).
Let a ∈ M n .There exist s 1 , . . ., s p ∈ S such that notice that I  is finite (since each γ ℓ > 0) and, for every i ∈ I , Since a ∈ M n , we have that that in (3) only the c i, in M n contribute to the sum: that is, only the ones such that i ∈ I .Therefore, Thus, M n is generated (as R-module) by the finite set (1) A is finitely generated (as S-module); (2) each x i has degree 1. Proposition 4.4.Let A be an acceptable graded S-module.Then, there exists d ∈ N such that, for every n ∈ N, Proof.It is always true that S (n) A d ≤ A d+n .We want to show the opposite containment.
Let a 1 , . . ., a ℓ ∈ A be generators of A. Without loss of generality, we may assume that each a i is homogeneous of degree d i (i.e., a i ∈ A di ). Let for some s i ∈ S. For every i = 1, . . ., ℓ, write for some (unique) r i, ∈ R. Without loss of generality, as in the proof of Lemma 4.2, we may assume that r i, = 0 when Definition 4.5.Let γ := γ 1 , . . ., γ k ∈ N k and N be an S-module.
An increasing filtering on N with degrees γ is an increasing sequence of R-sub- We denote by N the S-module with the given tuple γ and the filtering From now on, unless explicitly specified, all filterings will be increasing.( 3 ) A similar construction is widely used in algebraic geometry for downward filtrations: see e.g.[Eis95, §5.2].
The multiplication by x i on B(N ) is defined as: for every i ≤ k, j ∈ N, v ∈ N j , and then extended by R-linearity on all B(N ): notice that the x i has degree γ i in B(N ).The multiplication by y on B(N ) is defined as: y(vy j ) := vy j+1 , for every j ∈ N, v ∈ N j , and then extended by R-linearity on all B(N ): notice that y has degree 1.
Let M be an S-module and M = (M n ) n∈N be a filtering of M with degrees γ.For every m ∈ N, we define We say that M m tightly generates M if: for every n ∈ N and v ∈ M n , ( †) There exist m 1 , . . ., m r ∈ N with m j ≤ m, and v 1 , . . ., v r ∈ M such that v j ∈ M mj , and n1 , . . ., nr ∈ N k such that: where we are using the notations and p 1 , . . ., p r ∈ S such that: Since M m generates B(M ), there exist v 1 y m1 , . . ., v r y mr ∈ M m and q 1 (x, y), . . ., q r (x, y) ∈ R[x, y] (remember that S = R[x]) such that: Thus, if we define p j (x) := q j (x, 0) ∈ S, j = 1, . . ., r, we have Moreover, deg γ (p j ) + m j ≤ m, j = 1, . . ., r, showing that M m tightly generates M .
Definition 4.8.Let M be an S-module.An acceptable filtering of M is given by a filtering M := (M n : n ∈ N) such that B(M ) is an acceptable graded module; that is: (1) each x i has degree 1; (2) B(M ) is finitely generated (as an S[y]-module).
The following is an upward version of Artin-Rees Lemma: however, as it can be easily seen, the proof does not require R to be a commutative ring.Proposition 4.9.Let M be an S-module.Let M be an exhaustive acceptable filtering of M .Then, there exists d ∈ N such that: Proof.Since B(M ) is finitely generated, Lemma 4.7 implies that there exists d ∈ N such that M d tightly generates M : thus, (i) is proven.
Claim 1. (ii) also holds (for the same d).
By assumption, B(M ) is an acceptable graded S[y]-module.Thus, by (the proof of) Proposition 4.4, for every n ∈ N, (as submodules of B(M )) which is equivalent to (ii).

Hilbert series for graded and filtered modules
In this section we define the Hilbert series associated to the length function λ, following the ideas in [KLMP99] and [AM69,Ch.11]; in §6 we will define the corresponding Hilbert polynomial.
Theorem 5.1.Let M be an N-graded S-module of degree γ ∈ N k .For every n ∈ N, let a n := λ(M n ).Define .
Proof.By induction on k.
If k = 0, then, since M is Noetherian, only finitely many of the M n are nonzero.Thus, F M (t) is a sum of finitely many (finite) terms, and hence it is a polynomial.
Assume now that we have proven the conclusion for k − 1.Let y : M → M be the multiplication by x k and α := γ k .For every n ∈ N, let y n : M n → M n+α be the restriction of y to M n .Let K := Ker(y) and K n := K ∩ M n = Ker(y n ).Let C n := Coker(y n ) = M m+α /yM n , and C := n∈N C n .Notice that both λ(K n ) and λ(C n ) are finite.Therefore, both K and C are N-graded R[x 1 , . . ., x k−1 ]-modules, and satisfy the assumptions of the theorem (that is, they are Noetherian modules, and each K n and each C n has finite λ).
For every n ∈ N , consider the exact sequence Since λ is additive, we have Therefore, (where K is the R[x 1 , . . ., x k−1 ]-module with the given grading, and similarly for C).
Thus, by induction, there exist polynomials q, q ′ ∈ R[t] such that . Therefore, .
5.1.Filtered modules.We move now from graded modules to (upward) filtered modules.
Corollary 5.2.Let N be a filtering on N with degrees γ.Define Then, Therefore, if we assume that then, there exists a polynomial .
Proof.Apply Theorem 5.1 to the graded ring B(N ).The (1 − t)-factor in the denominator of (5) is due to the action of y on B(N ) of degree 1.

Hilbert polynomials for small modules
For the remainder of the article, excluding the appendices, we assume that R is a Noetherian ring (commutative with 1).
6.1.Hilbert polynomial for filtered modules.Definition 6.1.Let M be an S-module.A good filtering of M is given by an acceptable filtering M := (M n : n ∈ N) (see Definition 4.8) such that: Remark 6.2.Let M be an acceptable filtering of an S-module M .Then, since R is Noetherian, B(M ) is Noetherian.Theorem 6.3.Let M be an S-module and M := (M n ) n∈N be a good filtering of M .Then, for n ∈ N large enough, λ(M n ) is equal to a polynomial q M (n) of degree at most k.
If moreover M is an exhaustive (and good) filtering of M , then the leading monomial of q M does not depend on the choice of the exhaustive good filtering (but only on M and λ).
Therefore, we can denote by µ(M ) the leading monomial of the polynomial q M associated to some exhaustive good filtering of M (if such good filtering exists).
Proof.By Corollary 5.2, . Thus, by Proposition 2.1, for n large enough the coefficients λ(M n ) of the power series F M are equal to some polynomial q(n) ∈ R[n] of degree at most k.
Assume now M is exhaustive, and that M ′ := (M ′ n : n ∈ N) is another exhaustive good filtering of M .By Proposition 4.9, there exists d 0 ∈ N such that, for every n ∈ N, is finitely generated as R-module).Thus, for every n large enough, 1 ∈ N and every n large enough, showing that q M and q M ′ have the same leading monomial.6.2.Growth function.Definition 6.4.Let N be an S-module, and V 0 ≤ N be an R-submodule.For every n ∈ N, let V n := S n V 0 (notice that V 0 = S 0 V 0 , and that S n and V n are R-modules).We denote by the corresponding filtering of N (as S-module), where each x i has degree 1, and Notice that Gr(V 0 ; N ) depends not on N but only on SV 0 ≤ N .Theorem 6.5.Let N be an S-module.Let V 0 ≤ N be an R-submodule.Define

Assume that:
(1) λ(V 0 ) < ∞; (2) V 0 is finitely generated as R-module.Then, each λ(S n V 0 ) is finite, and there exists a polynomial p(t) ∈ R[t] such that Proof.First, we show that (1) implies that λ(S n V 0 ) is finite for every n ∈ N. In fact, S n V 0 is a quotient of V ℓ 0 , where ℓ := n+k n ∈ N is the number of monic monomials in S of degree less or equal to n, and λ(V ℓ 0 ) is finite.Notice that V := Filt(V 0 ; N ) is an filtering of N (as S-module).Moreover, F V = G V0 .Thus, by Theorem 6.3, it suffices to show that B(V ) = Gr(V 0 ; N ) is Noetherian as an S[y]-module to conclude (since then the filtering V is good).Since S[y] is a Noetherian ring, it suffices to show that B(V ) is finitely generated (as an S[y]-module).It is easy to see that B(V ) is generated by V 0 y 0 , and the latter is finitely generated (as R-module) by (2).Theorem 6.6 (Hilbert polynomial).Let N and V 0 be as in Theorem 6.5 and assume that (1), (2) as in there hold.Then, there exists a polynomial q V0 (t) ∈ R[t] of degree at most k, such that, for every n large enough, Assume moreover that N = SV 0 (that is, V 0 witnesses that N is λ S -small).Let V ′ 0 also witness that N is λ S -small.Then, q V0 and q V ′ 0 have the same leading term.Therefore, if we define µ λ (N ) to be the leading term of q V0 , then µ λ (N ) does not depend on the choice of the witness V 0 .
Finally, let c k be the coefficient of q V0 of degree k.Then, k! where h λ (N ) is the algebraic entropy of N according to λ (see §3.2).
Notice that V is a good filtering of N , and that V is exhaustive iff SV 0 = N .Therefore, Theorem 6.3 implies the existence of the polynomial q V0 , and that if V 0 witnesses that N is λ S -small, then the leading monomial of q V0 is independent of the choice of the witness.
If V 0 is a witness, then Notice that many authors (e.g., [KLMP99]) use a slightly different construction: in the situation when R is a field, they consider the function It is easy to see that there exists a polynomial GV0 (t) ∈ N[t] such that, for n large enough, H(n) = GV0 (t); from the definition it follows that GV0 (t) = G V0 (t + 1) − G V0 (t).In the present situation, we found it easier to work with the function λ(V n ) (but see §13).Definition 6.7.Let M be a λ S -small S-module.We define µ λ (M ) as in Theorem 6.6 (with µ λ (M ) = 0 iff q V0 = 0): then, µ λ (M ) does not depend on the choice of a witness.When λ is clear from the context, we will write µ instead of µ λ .
Let d be the degree of µ(M ) and m be the coefficient of µ(M ).We define the λ-dimension of M (as an S-module) to be equal to d, and its λ-degree as

d! m
When λ is clear, we will simply say "dimension" and "degree", respectively.( 4) If µ(M ) = 0, by convention we say that M has dimension −1 and degree undefined.
One reason of the normalizing coefficient d! is the following: as an S-module, by defining the action of x j on M as multiplication by 0 for j > d.
Let V 0 = R as a submodule of M .Then, Therefore, the λ-dimension of M is d, and its λ-degree is λ(R).
Notice that, since p e is not a zero-divisor, for every n ∈ N we have p • S n = (p) ∩ S n+e and therefore S n+e /p • S n and M n+e are isomorphic (as R-modules).Moreover also p is not a zero-divisor, and therefore the multiplication by p is an injective function (on S).Therefore, for every n ∈ N, the following sequence is exact: Therefore, if q (S) and q (M) are the Hilbert polynomials associated to S and M respectively, we have that, for every n ∈ N large enough, q (M) (n + e) = q (S) (n + e) − q (S) (n).

The conclusion follows.
Corollary 6.11.Assume that λ(R) = 1.Let p ∈ S be as in Proposition 6.10.Then, the λ-degree of S/(p) is equal to deg(p), and in particular it is independent from λ.
The above corollary implies that, if R is an integral domain and λ(R) = 1, then the λ-degree of S/(p) does not depend on λ (since the leading homogeneous component of p is not a zero divisor).However, this is hardly surprising, since under the above assumption λ is unique (see Exercise 3.4).
( 4 ) We boorrow the nomenclature from algebraic geometry, where, in the case when R is a field and λ is the linear dimension, the λ-dimension of M is simply called the "dimension" of M , and the λ-degree is the "degree" of M : see [Eis95, §1.9].There is a different construction in [Eis95, Ch.12] using Hilbert-Samuel polynomial, where the "degree" becomes "multiplicity": see also §10.
6.3.Necessity of Noetherianity.We give an example of a λ S -small module over a non Noetherian ring T with no associated Hilbert polynomial.
Define the following ring where p varies among the set of primes.Thus, T is a direct sum of fields and it is not Noetherian (notice that it is also not unitary).Any T -module M can be decomposed uniquely into the direct sum of its p-components: where each M p is a Z/pZ-vector space with a certain dimension dim p (M p ). Fix a sequence α p : p prime of real numbers such that, for each prime p, 0 < α p < 1, and p α p = 1.Define the followng lenght function on T -mod as , where we see S as a ring and M as a Tmodule.To give to M a structure as S-module, we specify the action of x on M in the following way: where v p ∈ Z/pZ, and extend it by linearity to all M .Let V 0 := T ≤ M .Thus, V 0 is a finitely generated T -submodule of M of finite length, but λ(S n+1 V 0 /S n V 0 ) is a strictly decreasing sequence of real numbers in (0, 1), and therefore λ(S n V 0 ) is not eventually equal to any polynomial.Notice that the algebraic entropy, i.e. the limit lim n→∞ λ(S n V 0 )/n, still exists.

Dimension and degree: the general case
We defined µ(M ) when M is a λ S -small S-module.We will extend the definition to the case when M is not necessarily λ S -small.We need first to explain what is the range of µ.
7.1.The value monoid.(Remember that we fixed k ∈ N).We define the following ordered monoids V and V.An element of V is either 0 or a monomial rt d , where r ∈ R >0 ∪ {∞} and d ∈ {0, 1, . . ., k}.Given a monomial 0 = rt d ∈ V, its degree is d and its coefficient is r; for completeness we define the degree of 0 to be −1.V is the subset of V given by 0 and the monomials with coefficient which is not ∞.
Remember that we follow the convention that r + ∞ = ∞ for every r ∈ R ∪ {∞}.The sum of two monomials in V is defined as and 0 + µ = µ for every µ ∈ V. We also define an ordering ≤ on V with the rule that rt n ≤ st m iff either n < m or n = m and r ≤ s, and 0 ≤ µ for every µ ∈ V.
With the above definitions, V; ⊕, 0, ≤ is a commutative ordered monoid (with 0 the neutral element) and ≤ is a linear ordering.Moreover, V is a submonoid of V.
Notice that ≤ is a complete ordering on V: given I ⊆ V, its supremum sup(I) is 0 if I is empty or I = {0}; otherwise, sup(I) is the monomial rt d , where Moreover, 0 is the minimum of V and ∞t k is its maximum, and ∞t k is an absorbing element: as an ordered set, V is isomorphic to the real interval [0, 1].
We give now an equivalent description of the value monoid.Let We endow P with the (total) quasi-ordering defined in Def.2.3, and the binary operation + given by pointwise addition.It is easy to see that P ; +, 0, is an ordered commutative monoid, and that the equivalence relation ≃ on P in Def.2.3 is compatible with the structure of ordered monoid.Therefore, P/≃ is also an ordered monoid (and the induced quasi-ordering on P/≃ is a linear ordering).
Proposition 2.4 easily implies the following:( 5 ) Remark 7.1.P/≃ is isomorphic to V (as an ordered monoid), with the canonical isomorphism given by the function mapping the equivalence class of a polynomial p to the leading term of p.
Remark 7.2.V is the completion of V (as an ordered set).
7.2.Non-small modules.Let M be an S-module (which might not be λ S -small).Given M ′ ≤ M S-submodule which is λ S -small (see Definition 3.5), let µ(M ′ ) be as in Definition 6.7: notice that µ(M ′ ) ∈ V. Thus, we can define µ(M ) ∈ V as the supremum of µ(M ′ ), where M ′ varies among all the possible S-submodules M ′ ≤ M which are λ S -small.We can then define as before the λ-dimension and λ-degree of M as the coefficient (up to a multiplicative constant) and the degree of µ(M ), respectively: the latter can be infinite.
From Remark 6.9 the following follows immediately.
Remark 7.3.µ(M ) = 0 iff all submodules of M of finite length have length 0. In particular, if 0 is the only submodule of M of finite length, then µ(M ) = 0.
An analogy from geometry that might help the intuition is the following.A semi-algebraic set X ⊆ R k has a dimension d ∈ {0, . . ., k} and a corresponding d-dimensional (Hausdorff) measure r := H d (X) ∈ R >0 ∪ {∞}: we could define µ(X) := rt d ∈ V as the object encapsulating both the dimension and the measure of X (with µ(X) = 0 iff X is empty).The definition of ⊕ is such that if X and Y are disjoint manifolds, then µ(X ∪Y ) = µ(X)⊕µ(Y ); if X and Y are not necessarily disjoint, then Thus, the λ-dimension of M is the analogue of the dimension of X, and the λ-degree of M is the analogue of the measure of X.We will see in §8 that (6) has an analogue for λ S -small modules: the additivity of µ.
( 5 ) It is also quite easy to see it directly, since we are dealing with polynomials in 1 variable.

Addition theorem for exact sequences
In this section we will prove the following Theorem.Notice that, under the assumptions of the above theorem, also A and C are locally λ R -finite.Notice moreover that µ(B) might have coefficient ∞.
It is well-known that without the assumption that B is locally λ R -finite, the theorem may fail.
The main ingredient is the following proposition, which treats the case of λ Ssmallness (where in particular µ(B) has finite coefficient).
Notice that, under the assumptions of the above propositions, also A and C are λ S -small.
Proof.Let B 0 be an R-submodule of B such that B 0 is finitely generated, λ(B 0 ) < ∞, and SB 0 = B.For every i ∈ N , define and define B := (B i : i ∈ N), A := (A i : i ∈ N), and C := (C i : i ∈ N).
Notice that A, B, and C are good filterings of A, B, and C, respectively (see Definiton 6.1).Thus, by Theorem 6.3, for n ∈ N large enough, λ(A n ) = q A (n) and µ(A) is the leading term of q A , and similarly for B and C.Moreover, for every and therefore µ(A) ⊕ µ(C) = µ(B).
Let B ′ ≤ B be an S-submodule which is finitely generated.Define Notice that the sequence is exact, and therefore, by Proposition 8.3, Taking the supremum among all the B ′ , we get the Claim.
Claim 3. µ(B) ≥ µ(A) ⊕ µ(C).Let A ′ ≤ A and C ′ ≤ C be finitely generated S-submodules.Since C ′ is finitely generated and π is surjective, there exists B ′ ≤ B finitely generated and such that π(B ′ ) = C ′ .Define We have that the sequence is exact, and B ′′ is finitely generated and locally λ R -finite.Thus, by Proposition 8.3, Taking the supremum on the left-hand side among all possible A ′ and C ′ , we get the Claim.

Modules over R-algebrae
Let T be a finitely generated commutative R-algebra (therefore, T is Noetherian).Let M be a T -module.We want to define the of M as a T -module.
Fix γ 1 , . . ., γ k generators of T as R-algebra.Equivalently, we fix a surjective homomorphism of R-algebrae φ : S → T and denote γ i := φ(x i ), i = 1, . . ., k.We can therefore see M as an S-module, and we denote it either by M ; φ or by M ; γ .
We assume M is λ T -small.Thus, we can use the above data to compute µ(M ; φ) (which will depend on φ).
We prove now that, while the coefficient of µ may depend on φ, its degree does not.Thus, we can define the λ-dimension of M (as a T -module) as the degree of µ(M ; φ).Definition 9.1.Spelling out all the assumptions, assume that: (1) R is Noetherian; (2) T is a finitely generated commutative R-algebra; (3) M is a T -module; (4) there exists M 0 ≤ M finitely generated R-submodule, such that λ(M 0 ) < ∞ and T M 0 = M .Then, we can define as before the λ-dimension of M as a T -module, and this dimension does not depend on the choice of M 0 or of φ.
In both examples, we see that the two modules have different degrees, but have the same dimension.
It remains to prove that the dimension of M ; φ does not depend on the choice of φ.It is clear that it suffices to prove the following: Theorem 9.3.Let δ ∈ T k ′ be another tuple of generators of T .Then, M ; γ and M ; δ have the same dimension.
Proof.After exchanging the rôles of γ and δ if necessary, we may assume that k ≥ k ′ .After extending δ by setting δ i = 0 for i ≥ k ′ , we may assume that k = k ′ .
We denote by ψ : R[x 1 , . . ., x k ] → M the surjective homomorphism of R-algebrae corresponding to δ (and by φ the one corresponding to γ).
For every n ∈ N, define n∈N are filterings of M as an R-module, and that each T n , T ′ n , M n and M ′ n are finitely generated (as R-modules).Moreover, M 0 generates both M ; γ and M ; γ′ as S-modules.Thus, we can apply Theorem 6.6; we denote by q (resp., q ′ ) the Hilbert polynomial of M ; γ (resp., of M ; γ′ . Therefore, for every n large enough, nc), proving that deg q ≤ deg q ′ .Exchanging the rôles of φ and φ ′ , we see that q and q ′ have the same degree.
We end this section with a comparison between λ-dimension and Krull dimension for affine rings.
Lemma 9.4.Let R be a field and λ equal to the linear dimension (as R-vector spaces).Let T be a finitely generated R-algebra.Then, the λ-dimension and the Krull dimension of T coincide.
Proof.Let d be equal to the Krull dimension of T .By Noether Normalization (see [Eis95,Thm.13.3], there exists an R-subalgebra A ≤ T such that: (1) A, as an R-algebra, is isomorphic to the polynomial ring R[y 1 , . . ., y d ]; (2) T is finitely generated as A-module.Thus, T and A have the same Krull dimension d.
In fact, T is a quotient of A n (for some n ∈ N), and Theorem 8.1 implies that Lemma 9.4 answers positively and extends the conjecture in [BDGS20, Remark 5.9].
Here is another example of equality between Krull and λ dimensions.
Let A be a finitely generated R-algebra.Then, dim λ (A) is equal to the Krull dimension of A.
Proof.We denote by dim K the Krull dimension.Write the factorization of n into primes: , where dim is either dim K or dim λ , it suffices to treat the case when ℓ = 1, i.e. n = p e .
Let B i := p i A as R-submodule of A, for i = 1, . . ., e.We have 0 = B e ≤ B e−1 ≤ . . .≤ B 0 = A. Notice that, for every i < e, Bi := B i /B i+1 is a Z/(p)-algebra.Moreover, there exists a surjective homomorphism of Z-modules between B0 and Bi , mapping a + pA to p i a + p i+1 A. Thus, µ( Bi ) ≤ µ( B0 ) and therefore Moreover, for Z/(p)-modules, λ (up to a constant factor) is equal to the linear dimension, and Lemma 9.4 implies that Finally, pA is the unique minimal prime ideal of A, and therefore
Theorem 10.1.Assume that: (i) M is finitely generated (as S-module); (ii) λ(M/IM ) is finite.Then, for every n ∈ N, c n is finite, and there exists a polynomial q(t) ∈ R[t] such that: (1) for every n ∈ N large enough, c n = q(n); (2) deg q ≤ k.
Assume moreover, besides the hypothesis in the theorem, that V ≤ M witnesses that M is λ S -small.Notice that Therefore, denoting by q V the Hilbert polynomial associated to V , we have q(t) ≤ q V (t) for every t large enough.If μ(M ) is the leading term of q, we have therefore μ(M ) ≤ µ(M ).
Example 10.2.Let K be a field, λ be the linear dimension over K, S : It is easy to prove that for homogeneous ideals the situation is different.
Exercise 10.3.Let J ⊳ S be a homogeneous ideal, and M := S/J.Then, More precisely, fix a finite set G generating J, and let n 0 be the maximum degree of the polynomials in G. Let V := R.Then, for every n > n 0 , S n V and M/I n+1 are isomorphic (as R-modules), and therefore See also [Eis95,Ch.12]and [Nor68, Ch.7] for the "classical" version of the Hilbert-Samuel polynomial.

d-dimensional and receptive versions of entropy
Let M be an S-module.Let m be the coefficient of µ(M ).For every d ≤ k, define The value h (1) (M ) is the receptive entropy of M w.r.t. the standard regular system generated by (x 1 , . . ., x k ) (see [BDGS20; BDGS21]); we call each h (d) (M ) the d-dimensional entropy of M (and thus the algebraic entropy h is the kdimensional entropy).
The case d = 1 of the following Proposition answers positively (and extends) [BDGS20, Question 5.10].
If d = dim(B) > dim(A), then h (d) (A) = 0 and Theorem 8.1 again implies that h (d) In all four cases, the conclusion follows.
Let T be a finitely generated R-algebra (thus, T is Noetherian).We can give similar definitions of entropies for T -modules.Fix γ = γ 1 , . . ., γ k generators of T (as R-algebra).Let M, γ be the S-algebra defined in §9.We define is the receptive entropy of M w.r.t. the standard regular system generated by γ (see [BDGS20; BDGS21]); we call each h (d,γ) (M ) the d-dimensional entropy of M w.r.t.γ.
Remark 12.6.Let I ⊳ R be a λ-cofinite ideal, and A be an R-module.Then, A/IA is locally λ R -finite.

Then, θ(A) ≥ θ(A)
and, if A is finitely generated, θ(A) = sup{θ(A/IA) : The proof of the above proposition is in the next subsection: for now we will record some consequences.
Remember that ĥλ = ĥ(k) and therefore from the above Corollary we obtain that ĥ is a length function S -mod, that ĥ(A) = h(A) when A is locally finite, and ĥ = h when λ(R) < ∞.
Corollary 12.10.Let T be a finitely generated R-algebra, γ ∈ T k be a set of generators of T .Given d ≤ k, let h (d,γ) be defined as in §11.Then: (1) ĥ (d,γ) λ is a length functions (on all T -modules) and satisfies the conclusion of Proposition 12.7; (2) ĥ(d,γ) Proof.Apply Proposition 12.4 to the function h (d,γ) .
Corollary 12.11.Let λ be the standard length on Z-modules introduced in Example 3.2(b).Then, for every finitely generated Z[x]-module A, We cannot drop the assumption that A is finitely generated in Proposition 12.7.
Proof.R/I ∩ J embeds into R/I × R/J.
Lemma 12.15.Let A be an S-module.Assume that A is λ S -small.Then, Proof.Let a 1 , . . ., a ℓ be generators of A (as S-module).For every and the conclusion follows from Lemma 12.13.
We want to prove that, when A is finitely generated, θ(A) = θ ′ (A).It suffices to show that θ ′ is additive on finitely generated S-modules.Thus, let be an exact sequence of finitely generated S-modules.
Let I ⊳R be a λ-cofinite ideal.We have the exact sequence of λ S -small S-modules Since θ is additive on λ S -small S-modules, and IA ≤ A ∩ IB, we have and the claim follows.
Let I, I ′ ⊳ R be λ-cofinite ideals.We want to prove that Replacing I, I ′ with I ∩ I ′ , without loss of generality we may assume that I = I ′ .By Artin-Rees Lemma (see e.g.[Nor68, §4.7]), there exists 1 ≤ n 0 ∈ N such that, for every m ∈ N , Let J := I n0 and A ′ := A ∩ JB: notice that J is also λ-cofinite.Taking m := n 0 in (7), we obtain: Thus, we have the exact sequence The modules appearing above are all λ S -small: therefore, proving the Claim.
12.2.Examples.Let R := Z, α be the standard length introduced in Example 3.2(b) and β be the length given by the rank (i.e., β(M , I be an ideal of S, and M := S/I.The following table shows the values of µ α (M ), μα (M ), and µ β (M ) for some values of I:

Intrinsic Hilbert polynomial
In [DGSV15] the authors introduced the "intrinsic" algebraic entropy, a variant of the more usual algebraic entropy: following a similar pattern, we introduce here the intrinsic Hilbert polynomial.
Let A be an S-module.Let A := (A i ) i∈N be a filtering on A. For each i ∈ N, define Ãi := A i+1 /A i (as R-modules).Define notice that Coker(•y) is an S[y]-module: however, y acts trivially on Coker(•y), hence we lose nothing in considering Coker(•y) as an S-module; moreover, the above isomorphism is of graded S-modules.In particular, if B(A) is Noetherian (as S[y]-module), then B(A) is also Noetherian (as S-module).
Proposition 13.2.Assume that A is a λ-inert filtering on A. Then, there exists a polynomial qA (t) ∈ Q[t] of degree at most k − 1 such that, for every n ∈ N large enough, λ( Ãn ) = qA (n).
Proof.Same proof as Theorem 6.6.
We call qA the intrinsic Hilbert polynomial of A, and denote by μ(A) its leading term.
Remark 13.3.Assume that A is a λ-inert filtering on A. Assume moreover that λ(A 0 ) < ∞.In this situation, we have defined the Hilbert polynomial q A .We have, for every n ∈ N and therefore qA = ∆q A , where ∆p is the difference of p: the polynomial defined by ∆p(t) = p(t + 1) − p(t).
The intrinsic Hilbert polynomial becomes interesting when λ(A 0 ) is infinite (and therefore we cannot compute the usual Hilbert polynomial).
Proof.Let B = (B n ) n∈N be a λ-inert filtering on B. For every n ∈ N, define It is clear that A, B, and C are filterings on A, B, C, respectively.
Claim 10.C and A are λ-inert.
In fact, for every n ∈ N, we have an exact sequence of R-modules Since B0 is finitely generated and R is Noetherian, both Ã0 and C0 are finitely generated.Since moreover (for n large enough) λ( Bn ) is finite, both λ( Ãn ) and λ( Cn ) are finite.Similarly, if A = (A n ) n∈N is a λ-inert filtering on A, then it is also a λ-inert filtering on B, and therefore μ[A] ≤ μ(B); thus; μ(A) ≤ μ(B).
Let A be an S-module.Let d ∈ N be the degree of μ(A) and s ∈ R be its coefficient.We define the intrinsic λ-dimension of A to be d + 1 if μ(A) = 0, 0 if λ(A) > 0 and μ(A) = 0, and −∞ if λ(A) = 0.For each i ≤ k, the intrinsic i-dimensional λ-entropy of A is The intrinsic λ-entropy is hλ (A) := h(k) λ (A) and has been studied already (at least, in the case R = Z and k = 1) in [DGSV15; GS15; SV18], and for some non-Noetherian rings in [SV19]): our definition of μ is clearly inspired by the intrinsic algebraic entropy (the notation ent is used elsewhere, but we prefer hλ for consistency).
In general, ĥλ and hλ are different.

Fine grading
Up to now, we have only considered the case when the degrees are natural numbers.As in the classical case when R is a field, one can consider gradings in any commutative monoid (see e.g.[MS05]).
Remember that Γ has a canonical quasi-ordering, given by m ≤ n if there exists p ∈ Γ with m + p = n.The neutral element 0 is a minimum of Γ, ≤ .Definition 14.1.We say that γ ∈ Γ k is good (inside Γ) if: for every λ ∈ Γ there exist at most finitely many n ∈ N k , such that n • γ = λ.
We say that γ ∈ Γ k is very good if: for every λ ∈ Γ there exist at most finitely many n ∈ N k , such that n • γ ≤ λ.
For example, γ ∈ Z k is good (in Z) if γ i > 0 for i = 1, . . ., k. γ ∈ N k is very good (in N) iff γ i = 0 for i = 1, . . ., k.Notice that, in general, if γ is good, then each γ i is non-zero (and even non-torsion).
Then, there exists a polynomial p(t) ∈ R[t Γ ] such that (1 − t γi ) (the (1 − t δj )-factor in the denominator is due to the action of y j on Bδ(N ) of degree δ k ).
We also fix a tuple γ = γ 1 , . . ., γ k ∈ Γ k .Given a monomial in S = R[x 1 , . . ., x k ] its γ-degree deg γ is defined in the "obvious" way: Given a polynomial p(x) ∈ S = R[x 1 , . . ., x k ], we say that its γ-degree is less or equal to n ∈ Γ, and write deg γ (p) ≤ n, if each monomial in p has γ-degree less or equal to n (since Γ is not linearly ordered in general, it's not clear how to define the γ-degree of a polynomial).For every n ∈ Γ, we denote S n := p ∈ S : deg γ (p) ≤ n .
Theorem 14.8.Let N be an S-module.Let V 0 ≤ N be an R-submodule.For every n ∈ Γ, let V n := S n V 0 (notice that S 0 = R, that V 0 = S 0 V 0 , and that S n and V n are R-modules), and a n := λ(V n ).Define Assume that: (1) γ ∪ δ is very good (inside Γ); (2) λ(V 0 ) < ∞; (3) V 0 is finitely generated as R-module.Then, each a n is finite, and there exists a polynomial p(t) ∈ R[t Γ ] such that The fact that γ is very good is equivalent to the fact that S n is finitely generated (as R-module) for every n ∈ Γ.The above plus the fact that λ(V 0 ) < ∞ easily implies that λ(V n ) < ∞ for every n ∈ Γ.Let V := SV 0 as filtered S[ȳ]-module.Then, V 0 t 0 generates B(V ) as S[ȳ]-module, and therefore B(V ) is a Noetherian S[ȳ]-module.
We can conclude as in the proof of Theorem 6.5, using Theorem 14.7.

Definition 4. 6 .
The blow-up of the filtered S-module N is the following graded S[y]-module B(N ).( 3 ) As a graded R-module, B(N ) := n∈N N n y n .

Example 8. 2 .
Let R = Z with the standard length λ (see Example 3.2b).Let A := B := Z[x] and C := Z/2Z[X].Let ι : A → B, a → 2a and let π : B → C be the canonical projection.Then, 0 − S-module (where all the x i have degree 1).An equivalent description of B(A) is the following.Remember that B(A) is an S[y]-module.Let •y : B(A) → B(A) be the multiplication by y.Then, B(A) = Coker(•y);