HHILBERT POLYNOMIAL OF LENGTH FUNCTIONS
VERSION 8
ANTONGIULIO FORNASIERO
Abstract.
Let L be a general length function for modules over a Noetherianring R. We use L to define Hilbert series and polynomials for R[X]-modules.The leading term of any such polynomial is an invariant of R[X]-modules,which refines the algebraic entropy.
Contents
1. Introduction 12. Length functions and their entropy 33. λ -Hilbert series and polynomials for Noetherian modules 44. Dimension and degree: the general case 95. Addition theorem for exact sequences 106. Modules over R -algebrae 137. Hilbert-Samuel polynomial for homogeneous modules 148. d -dimensional and receptive versions of entropy 159. Totally additive versions of µ and (receptive) entropy 1610. Fine grading 20Appendix A. Non-commuative rings 24Appendix B. Finite generation and Noetherianity according to λ Introduction
Let R be a commutative ring with unity.A generalized length function on the category R -mod of R -modules is a function λ : R -mod → R ≥ ∪ {∞} satisfying the following conditions:(1) λ (0) = 0 ;(2) λ ( M ) = λ ( M (cid:48) ) when M and M (cid:48) are isomorphic;(3) for every exact sequence → A → B → C → , λ ( B ) = λ ( A ) + λ ( C ); (4) for every M ∈ R -mod , λ ( M ) = sup { λ ( M (cid:48) ) : M (cid:48) ≤ M finitely generated R -sub-module } Date : 13/02/2020.2010
Mathematics Subject Classification.
Primary 13D40, 16S50; Secondary 16D10, 16P40.
Key words and phrases.
Length function, Hilbert polynomial, Hilbert series, algebraic entropy.This work was partially in part by GNSAGA of INdAM and by PRIN 2017, project2017NWTM8R, Mathematical Logic, models, sets, computability. a r X i v : . [ m a t h . A C ] F e b A. FORNASIERO
Generalized length functions were introduced in [NR65] and further studied (amongother places) in [Vám68, SVV13]: see also §2 for a brief introduction; we will simplysay “length” or “lenght function” in the rest of the paper.Fix k ∈ N and let S := R [ x , . . . , x k ] . Let M be an S -module.We show that, under some assumptions on M , we can define a Hilbert series and a
Hilbert polynomial for M , generalizing the known theory when R is a fieldand λ is equal to the linear dimension (see e.g. [Eis95, MS05]). More precisely, wefix an R -sub-module V ≤ M such that λ ( V ) < ∞ and SV = M ; if we define S n asthe set of polynomials in S of total degree less or equal to n , we see that the formalpower series ∞ (cid:88) n =0 λ ( S n V ) t n is a rational function (of t ), and the function n (cid:55)→ λ ( S n V ) is equal to a polynomial(for n ∈ N large enough). The leading coefficient µ ( M ) of this polynomial doesnot depend on the choice of V (Theorem 3.9): thus, this leading monomial is aninvariant of M , and describes the growth rate of λ on M .First we show the existence of a Hilbert series for the case when M is an N -gradedmodule (Theorem 3.2), then we move to the case when M is a upward filtered mod-ule (Theorem 3.4). Often Hilbert series are defined for downward filtered modules,but, in light of the applications in §3.4 and §5, it is more appropriate for our pur-poses to consider upward filtered modules (cf. [KLMP99, §1.3]).We then deduce the existence of a Hilbert polynomial associated to a suitablegenerating R -sub-module V of an S -module M .In §4 we generalize this construction to the case when M is not finitely gener-ated, and show that, under some suitable assumptions, the function µ is additive(Theorem 5.1).The coefficient of the k -term of the Hilbert polynomial is (up to a constantfactor) the algebraic entropy of the action of N k on M (see §2 for the definitionof algebraic entropy and its main properties). Therefore, the additivity of µ is arefinement of the known additivity of algebraic entropy (see Fact 2.5 and [SVV13,SV15, DFGB20]). However, the additivity of algebraic entropy has been provedunder weaker assumptions: one of the most general results considers the case whenthe acting monoid Γ is cancellative and amenable, and M is locally λ R -finite (as inDefinition 2.1): see [DFGB20].In §7 we show how the usual construction of Hilbert-Samuel polynomial can begeneralized to length functions.We also consider in §6 the case when S is a finitely generated R -algebra and wedefine a corresponding Hilbert polynomial: its degree will be an invariant of M .In §8 we introduce the d -dimensional entropy as a generalization of the receptiveentropy (see [BDGSa]) and relate it to the Hilbert polynomial.In §9 we employ a tecnique in [Vám68] and use µ to define an invariant ˆ µ whichis a length function on S -modules (assuming that R is Noetherian): a similarconstruction works for the ( d -dimensional) entropy.Besides, we consider in §10 a finer version of the Hilbert series where the gradingis no longer in N but in some suitable monoid Γ .Most of the proof are either adaption of “classical” proof about Hilbert series andpolynomials to this setting, or quite straightforward: hence we often skip them.In most of our results, we assume that R is Noetherian. We think it may bepossible to generalize them to some non Noetherian rings (see the Appendices);however, we don’t have (yet) a working theory of “Noetherianity-up-to- λ ” for rings,so we could not go far in this direction of inquiry. Another open line of inquiry is ILBERT POLYNOMIAL OF LENGTH FUNCTIONS V. 8 3 considering the case when either the ring R or the algebra S are not commutative(cf. [Nor68] for the case when R is not commutative, and [DFGB20] for the casewhen S is not commutative): some partial results are in Appendix A.2. Length functions and their entropy
Let R be a ring and λ be a length function on R -mod .2.1. Length functions.
It can happen that λ ( R ) is infinite: the following defini-tion deals with that situation. Definition 2.1.
Let N be an R -module. We say that N is locally λ R -finite ( ) ifeither of the following equivalent conditions hold:(1) For every v ∈ N , λ ( Rv ) < ∞ ;(2) For every N < N finitely generated R -sub-module, λ ( N ) < ∞ . Examples 2.2. (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.(c) The following are two “trivial” lengths:i) λ ( M ) = 0 for every M ;ii) λ ( M ) = ∞ for every M (cid:54) = 0 .(d) Given any ring R , the (classical) length of an R -module M is the lenght of acomposition series for M (see e.g. [Eis95]).A property we will use often in the rest of the paper is the following: Definition 2.3.
Given an R -algebra S and an S -module M , we say that M is λ S -small if A is finitely generated (as S -module) and locally λ R -finite.( ) Remark 2.4.
Assume that S is a Noetherian R -algebra and let M be an S -module.T.f.a.e.:(1) M is λ S -small(2) there exists V < M R -sub-module such that:i) SV = M ii) V is finitely generated (as R -module)iii) λ ( V ) < ∞ (3) every sub-module and every quotient of M is λ S -small.[Vám68, Theorem 5] charactterizes of length functions on Noetherian rings: forevery prime ideal P < R there is a canonical length function on R -mod l p , and anylenght function λ can be written as λ = (cid:88) P The content of this subsection can be skipped: it ismostly a motivation for the definitions and results in the paper. We recall thedefinition of algebraic entropy and its main properties.We fix a length λ on R -mod . Let M be an R -module and φ be an endomorphismof M . Given an R -sub-module V ≤ M , we define H ( φ ; V ) := lim n →∞ λ (cid:0) V + φ ( V ) + · · · + φ n − ( V ) (cid:1) n ( ) Also called “locally λ -finite” in [SVV13]: here we prefer to write explicitly the ring R too( ) A similar notion is called “Hilbert S -module” in [Nor68, C.7]. A. FORNASIERO (the limit always exists). The entropy of φ (according to the length λ ) is definedby h λ ( φ ) = sup { H ( φ ; V ) : V ≤ M R -sub-module of finite length } . Equivalently, we can see M as an R [ x ] -module, (with X acting on M as φ ) andconsider h λ as an invariant of M as R [ X ] -module. For the relationship betweenalgebraic entropy and multiplicity, see [SVV13, Nor68].More generally, given an S -module M (where S = R [ x , . . . , x k ] ), let S n be theset of polynomials in S of degree at most n . Define H ( M ; V ) := lim n →∞ λ ( S n V ) (cid:14)(cid:0) n + kk (cid:1) h λ ( M ) := sup { H ( φ ; V ) : V ≤ M R -sub-module of finite length } The limit in the definition of H ( M ; V ) exists, and h λ is the algebraic entropy(relative to the length function λ ): see [CSCK14, DFGB20].( )One of the most important properties of algebraic entropy is its additivity: Fact 2.5. Let → A → B → C → be an exact sequence of S -modules. Assumethat B is locally λ R -finite. Then, h λ ( B ) = h λ ( A ) + h λ ( C ) . For a proof of the above fact, see [SV15] for the case when k = 1 , and [DFGB20]for the case when k ≥ (or even more generally of R -modules with an action ofa cancellative amenable monoid). Length functions on R -modules were introducedin [NR65]. For an introduction to lengths and algebraic entropy, see [SVV13].3. λ -Hilbert series and polynomials for Noetherian modules Let S := R [ x , . . . , x k ] . We will define the Hilbert series and polynomial associ-ated to the length function λ , loosely following the ideas in [KLMP99].3.1. Graded modules.Definition 3.1. Fix ¯ γ = (cid:104) γ , . . . , γ k (cid:105) ∈ N k . An N -graded S -module of degree ¯ γ is given by an S -module M and a decomposition M = (cid:77) n ∈ N M n , where each M n is an R -module, and, for every i ≤ k and n ∈ N , x i M n ≤ M n + γ i . We denote by ¯ M the module M with the given grading (including the tuple ¯ γ := (cid:104) γ , . . . , γ k (cid:105) ). Theorem 3.2. Let ¯ M be an N -graded S -module of degree ¯ γ ∈ N k . For every n ∈ N ,let a n := λ ( M n ) . Define F ¯ M ( t ) := (cid:88) n a n t n . Assume that: (1) γ i > for i = 1 , . . . , k ; (2) λ ( M n ) < ∞ for every n ∈ N ; (3) M is a Noetherian S -module. ( ) Let B kn be the set of monic monomials in S n . Then (cid:0) n + kk (cid:1) is the cardinality of B kn . Moreover,the family ( B kn ) n ∈ N , is a Følner sequence inside the monoid of all monic monomials in S , andtherefore we can apply the machinery of [CSCK14]. ILBERT POLYNOMIAL OF LENGTH FUNCTIONS V. 8 5 Then, there exists a polynomial p ( t ) ∈ R [ t ] such that F ¯ M ( t ) = p ( t ) (cid:81) ki =1 (1 − t γ i ) . 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 − . Let y : M → M bethe multiplication by x k and α := γ k . For every n ∈ N , let y n : M n → M n + α bethe 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 := (cid:76) 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 , . . . , x k − ] -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 −→ K n −→ M n y n −→ M n + α −→ C n −→ . Since λ is additive, we have a n + α − a n = λ ( K n ) − λ ( C n ) . Thus, (cid:88) n a n t n + α − (cid:88) n a n t n = (cid:88) n λ ( K n ) t n − (cid:88) n λ ( C n ) t n . Therefore, ( t α − F ¯ M ( t ) = F ¯ K ( t ) − F ¯ C ( t ) (where ¯ K is the R [ x , . . . , x k − -module with the given grading, and similarly for ¯ C ).Thus, by induction, there exist polynomials q, q (cid:48) ∈ R [ t ] such that F ¯ K ( t ) = q ( t ) (cid:81) k − i =1 (1 − t γ i ) F ¯ C ( t ) = q (cid:48) ( t ) (cid:81) k − i =1 (1 − t γ i ) . Therefore, F ¯ M ( t ) = q ( t ) − q (cid:48) ( t ) (cid:81) ki =1 (1 − t γ i ) (cid:3) Filtered modules. We move now from graded modules to upward filteredmodules. Definition 3.3. Let S = R [ x , . . . , x k ] , and let ¯ γ := (cid:104) γ , . . . , γ k (cid:105) ∈ N k .An increasing filtering on N with degrees ¯ γ is an increasing sequence of R -sub-modules of N N ≤ N ≤ N · · · ≤ N such that (cid:83) ∞ i =0 N i = N , and x i N j ≤ N j + α i for every j ∈ N , i ≤ k . We denote by ¯ N the S -module with the given tuple ¯ γ and the filtering ( N i ) i ∈ N .The blow-up of ¯ N is the following graded S [ y ] -module.( )As an R -module, B ( ¯ N ) := (cid:77) n ∈ N N n y n . The grading of B ( ¯ N ) is given by the decomposition B ( ¯ N ) = (cid:76) n ∈ N N n y n . ( ) A similar construction is widely used in algebraic geometry for downward filtrations: see e.g.[Eis95, §5.2]. A. FORNASIERO The multiplication by x i on B ( ¯ N ) is defined as: x i ( vy j ) := ( x i v ) y j + γ i , 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 on B ( ¯ N ) . The multiplication by y on B ( ¯ N ) isdefined 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 . Theorem 3.4. Let ¯ N be a filtering on N with degrees ¯ γ .For every n ∈ N , let a n := λ ( N n ) . Define F ¯ N ( t ) := ∞ (cid:88) n =0 a n t n . Then, F ¯ N = F B ( ¯ N ) . Therefore, if we assume that (1) γ i > for i = 1 , . . . , k ; (2) λ ( N n ) < ∞ for every n ∈ N ; (3) B ( ¯ N ) is Noetherian as S [ y ] -module.Then, there exists a polynomial p ( t ) ∈ R [ t ] such that F ¯ N ( t ) = p ( t )(1 − t ) (cid:81) ki =1 (1 − t γ i ) (the (1 − t ) -factor in the denominator is due to the action of y on B ( N ) of degree ). Polynomial coefficients for some rational functions. We will need thefollowing results: it is probably well known, but I could not find a reference. Proposition 3.5. Let ¯ t = (cid:104) t , . . . , t (cid:96) (cid:105) . Let p (¯ t ) ∈ K [¯ t ] for some ring K . Let γ , . . . , γ (cid:96) ∈ N . Define f (¯ t ) := p (¯ t ) (cid:81) (cid:96)i =1 (1 − t i ) γ i ∈ K (¯ t ) and expand f as f (¯ t ) = (cid:88) ¯ n ∈ N (cid:96) a ¯ n ¯ t ¯ n ∈ K [[¯ t ]] Then, there exists a polynomial q (¯ t ) ∈ K [¯ t ] such that: (1) For every ¯ n ∈ N (cid:96) large enough a ¯ n = q (¯ n ) (2) for every i = 1 , . . . , (cid:96) deg t i ( q ) ≤ γ i − (where we set deg 0 := −∞ )Proof. It’s clear that it suffices to treat the case when p = 1 . We proceed byinduction on (cid:96) . If (cid:96) = 0 , then f = 1 , and q = 0 . If (cid:96) = 1 , the result is easy: byfurther induction on γ , one can prove that a n = (cid:0) n + γ − γ − (cid:1) .Assume now that we have already proved the result for (cid:96) − : we want to proveit for (cid:96) . Denote ˜ t := (cid:104) t , . . . , t (cid:96) (cid:105) and g (˜ t ) := 1 (cid:81) (cid:96)i =2 (1 − t i ) γ i := (cid:88) ˜ n ∈ N (cid:96) − b ˜ n ˜ t ˜ n . ILBERT POLYNOMIAL OF LENGTH FUNCTIONS V. 8 7 By inductive hypothesis, there exists r (˜ t ) ∈ K [˜ t ] satisfying (1) and (2) for g . More-over, f = g · − t ) γ = (cid:88) ˜ n ∈ N (cid:96) − b ˜ n ˜ t ˜ n · (cid:88) m ∈ N c m t m , where − t ) γ = (cid:80) m ∈ N c m t m . Thus, a m, ˜ n = c m b ˜ n . By the case (cid:96) = 1 there exists s ( t ) ∈ K [ t ] of degree at most γ − , such that, forevery m ∈ N large enough, c m = s ( m ) . Thus, taking m ∈ N and ˜ n ∈ N (cid:96) − large enough, we have a m, ˜ n = s ( m ) r (˜ n ) , and the polynomial q ( t , ˜ t ) := s ( t ) r (˜ t ) ∈ K [ t ] satisfies the conclusion. (cid:3) Definition 3.6. Let p (¯ t ) , q (¯ t ) ∈ R [¯ t ] ; we write p = p + p + . . . p d , where each p i ∈ R [¯ t ] is homogeneous of degree i , and p d (cid:54) = 0 . We call p d the leading homogeneouscomponent of p . We write • p (cid:22) q if there exists ¯ c ∈ N (cid:96) such that, for every ¯ n ∈ N (cid:96) large enough, p (¯ n ) ≤ q (¯ n + ¯ c ) ; • p (cid:39) q if p (cid:22) q and q (cid:22) p ; • p (cid:23) if, for every ¯ n ∈ N (cid:96) large enough, p (¯ n ) ≥ Proposition 3.7. Let p, q ∈ R [¯ t ] such that: (1) p (cid:23) (2) q (cid:23) (3) p (cid:39) q Then, p and q have the same leading homogeneous component.Proof. Let p (cid:48) and q (cid:48) be the leading homogeneous component of p and q respectively,and h (cid:48) be the leading homogeneous component of h := p − q . If, by contradiction, p (cid:48) (cid:54) = q (cid:48) , then deg h = max(deg p, deg q ) . Let ¯ v ∈ N (cid:96) such that h (cid:48) (¯ v ) (cid:54) = 0 . Then, r := lim s →∞ ,s ∈ N h ( s ¯ v ) ∈ {±∞} . Since q (cid:22) p , we have r = + ∞ , but since since p (cid:22) q , we have r = −∞ , absurd. (cid:3) Growth function.Theorem 3.8. Let N be an S -module. Let V ≤ N be an R -sub-module. For every n ∈ N , let S n be the set of polynomial in S of total degree at most n , V n := S n V (notice that S = R , that V = S V , and that S n and V n are R -modules), and a n := λ ( V n ) . Define G V ( t ) := (cid:88) n ∈ N a n t n . Assume that: (1) λ ( V ) < ∞ ; (2) V is finitely generated as R -module; (3) R is a Noetherian ring.Then, each a n is finite, and there exists a polynomial p ( t ) ∈ R [ t ] such that G V ( t ) = p ( t )(1 − t ) k +1 . A. FORNASIERO Proof. First, we show that (1) implies that λ ( V n ) is finite for every n ∈ N . In fact, V n is a quotient of V (cid:96) , where (cid:96) := (cid:0) n + kn (cid:1) ∈ N is the number of monic monomials in S of degree less or equal to n , and λ ( V (cid:96) ) is finite.Let V := (cid:91) n V n . Notice that V is an S -module, and that ¯ V := (cid:0) V n : n ∈ N (cid:1) is an increasing filteringof V (as S -module), where each x i has degree . Moreover, F ¯ V = G V . Thus, itsuffices to show that B ( ¯ V ) is Noetherian as an S [ y ] -module to conclude. Since, by(3), R is a Noetherian ring, S [ y ] is also a Noetherian ring. Thus, it suffices to showthat B ( ¯ V ) is finitely generated (as an S [ y ] -module). It is easy to see that B [ ¯ V ] isgenerated by V t , and the latter is finitely generated (as R -module) by (2). (cid:3) Theorem 3.9 (Hilbert polynomial) . Let N , V , and V n be as in Theorem 3.8 andassume that (1), (2), and (3) as in there hold. Then, there exists a polynomial q V ( t ) ∈ R [ t ] of degree at most k , such that, for every n large enough, λ ( V n ) = q V ( n ) . Assume that N = (cid:83) n ∈ N V n = SV .Let V (cid:48) be another finitely generated R -sub-module of N such that λ ( V (cid:48) ) < ∞ and SV (cid:48) = N . Then, there exists a constant n such that q V (cid:48) ( t ) ≤ q V ( t + n ) , and therefore the leading coefficients of q V and of q V (cid:48) are the same. Therefore, ifwe define µ λ ( N ) to be the leading coefficient of q V , µ λ ( N ) does not depend on thechoice of V .Finally, the coefficient of q V of degree k is equal to h λ ( N ) k ! where h λ ( N ) is the algebraic entropy of N according to λ (see §2.2).Proof. By Theorem 3.8 and Proposition 3.5 (cid:3) Notice that many authors (e.g., [KLMP99]) use a slightly different construction:in the situation when R is a field, they consider the function H ( n ) := dim( V n +1 /V n ) . It is easy to see that there exists a polynomial ˜ G V ( t ) ∈ N [ t ] such that, for n largeenough, H ( n ) = ˜ G V ( t ) ; from the definition it follows that ˜ G V ( t ) = G V ( t + 1) − G V ( t ) . In the present situation, we found it easier to work with the function λ ( V n ) . Definition 3.10. Assume that R is Noetherian, and M is an S -module, such that M is λ S -small (see Definition 2.3). Thus, there exists V < M R -sub-module whichis finitely generated, of finite length, and generating M as an S -module.We define µ λ ( M ) as in Theorem 3.9: µ λ ( M ) does not depend on the choice of V . 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.( ) ( ) See [Eis95, §1.9] for the usual definitions of dimension and degree. There is an alternativeconstruction in [Eis95, C.12] using Hilbert-Samuel polynomial, where the “degree” becomes “mul-tiplicity”: see also §7. ILBERT POLYNOMIAL OF LENGTH FUNCTIONS V. 8 9 One reason of the normalizing coefficient d ! is the following Example 3.11. Assume that λ ( R ) < ∞ and fix d ≤ k . Let M := R [ x , . . . , x d ] asan S -module, by defining the action of x j on M as multiplication by for j > d .Let V = R as a sub-module of M . Then, q V ( n ) = (cid:18) n + dd (cid:19) λ ( R ) µ ( M ) = t d d ! λ ( R ) Therefore, the λ -dimension of M is d , and its λ -degree is λ ( R ) . Proposition 3.12. Assume that R is Noetherian and λ ( R ) = (cid:96) < ∞ . Let p (¯ x ) ∈ S of degree e > . Assume that p e , the leading homogeneous component of p (seeDef. 3.6), is not a zero-divisor (inside S ). Let M := S/ ( p ) . Then, the λ -dimensionof M is k − and it λ -degree is (cid:96)e .Proof. “Usual” proof. Let N := ( p ) < S . We choose M := R < M . We have SM = M . For each n ∈ N , we denote M n := S n M = S n / ( p ) < M . 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 p also is not a zero-divisor, and therefore the multiplication by p is an injectivefunction (on S ). Therefore, for every n ∈ N , the following sequence is exact: −→ S n · p −→ S n + e −→ M n + e −→ Therefore, if q ( S ) and q ( M ) are the Hilbert polynomials associated of 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. (cid:3) Corollary 3.13. Assume that R is Noetherian and λ ( R ) = 1 . Let p ∈ S be as inProposition 3.12. Then, the λ -degree of S/ ( p ) is equal to deg( p ) , and in particularit is independent from λ . The above corollary implies that, if R is an integral domain and λ ( R ) = 1 , thenthe λ -degree of S/ ( p ) does not depend on λ . However, this hardly surprising, sinceunder the above assumption λ is unique: Exercise . Let R be an integral domain. Then, there exists a unique lenght λ on R -modules satisfying λ ( R ) = 1 . Denoting by K the field of fractions of R , λ is defined by: λ ( M ) := dim K ( K ⊗ M ) . Dimension and degree: the general case We assume now that R is Noetherian (and S = R [ x , . . . , x k ] for some k ∈ N ).Let M be any R -module. Given M (cid:48) ≤ M S -sub-module which is λ S -small (seeDefinition 2.3), let µ ( M (cid:48) ) be as in Definition 3.10: it is a monomial if the form rt d ,where r ∈ R ≥ and d ∈ , . . . , k . Given a family I = { r i t d i } of monomials of theabove form, we can define sup( I ) as the monomial rt d , where d := max { d i : i ∈ I } ∈ { , . . . , k } r := sup { r i : i ∈ I & d i = d } ∈ R ≥ ∪ {∞} , or sup( I ) = 0 if I is empty.Thus, we can define µ ( M ) as the supremum of µ ( M (cid:48) ) , where M (cid:48) varies among allthe possible S -sub-modules M (cid:48) ≤ M which are λ S -small (notice that, by definition, µ ( M ) = 0 if is the only S -sub-module of M of finite length). We can then defineas before the λ -dimension and λ -degree of M : the latter can be infinite.5. Addition theorem for exact sequences Fix k ∈ N and let S := R [ x , . . . , x k ] . In this section we will prove the followingTheorem. Theorem 5.1. Let −→ A ι −→ B π −→ C −→ be an exact sequence of S -modules.Assume that (1) R is Noetherian; (2) B is locally λ R -finite.Then, µ ( B ) = µ ( A ) ⊕ µ ( C ) , where rt n ⊕ st m = rt n if n > mst m if n < m ( r + s ) t n if n = m. Notice that, under the assumptions of the above theorem, also A and C arelocally λ R -finite and finitely generated.The main ingredient is the following proposition. Proposition 5.2 (Additivity) . Let −→ A ι −→ B π −→ C −→ be an exactsequence of S -modules. Assume that (1) R is Noetherian; (2) B is λ S -small.Then, µ ( B ) = µ ( A ) ⊕ µ ( C ) . Notice that, under the assumptions of the above propositions, also A and C are λ S -small.Before proving the proposition, we need to extend the machinery of §3.Let M be an S -module and ¯ M = ( M n ) n ∈ N be a filtering of M with degrees ¯ γ .For every n ∈ N , we define M m := (cid:77) n ≤ m M n y n ≤ B ( M ) . We say that M m tightly generates ¯ M if: for every n ∈ N and v ∈ M n ( † ) There exist m , . . . , m r ∈ N with m j ≤ m , v , . . . v r ∈ M such that v j ∈ M m j ,and ¯ n , . . . , ¯ n r ∈ N k such that: v = ¯ x ¯ n v + · · · + ¯ x ¯ n r v r ,n ≥ ¯ n j · ¯ γ + m j , j = 1 , . . . , r. where we are using the notations ¯ n j · ¯ γ := n j, γ + · · · + n j,k γ k ¯ x ¯ n j := x n j, · · · · · x n j,k k . Notice that ( † ) is equivalent to:( † (cid:48) ) There exist m , . . . , m r ∈ N with m j ≤ m , v , . . . , v r ∈ M such that v j ∈ M m j ,and p , . . . , p r ∈ S such that: v = p v ) + · · · + p r v r ,n ≥ deg γ ( p j ) + m j , j = 1 , . . . , r. Lemma 5.3. Let m ∈ N . M m generates M (as an S [ y ] -module) iff M m tightlygenerates ¯ M . ILBERT POLYNOMIAL OF LENGTH FUNCTIONS V. 8 11 Proof. ⇒ ) Let n ∈ N and v ∈ M n . Since M m generates B ( ¯ M ) , there exist v y m , . . . , v r y m r ∈ M m and q (¯ x, y ) , . . . , q r (¯ x, y ) ∈ R [¯ x, y ] such that: vy n = q (¯ x, y ) v y m + · · · + q r (¯ x, y ) v r y m r . Thus, if we define p j (¯ x ) := q j (¯ x, ∈ S , j = 1 , . . . , r , we have v = p v + · · · + p r v r . Moreover, deg γ ( p j ) + m j ≤ m , j = 1 , . . . , r , showing that M m tightly generates ¯ M . ⇐ ) Let n ∈ N and vy n ∈ M n y n . Let m , . . . , m r ∈ N v , . . . v r ∈ M and ¯ n , . . . , ¯ n r ∈ N k as in ( † ). For j = 1 , . . . , r , define d j := n − (¯ n j · ¯ γ + m j ) p j := ¯ x ¯ n j y d j . We have v j y m j ∈ M m and vy n = m (cid:88) i =1 p j · ( v j y d j ) ∈ S [ y ] M m (cid:3) Definition 5.4. Let M be an S -module. A good filtering of M is given by afiltering ¯ M := ( M n : n ∈ N ) if:(1) Each x i has degree 1;(2) For every n ∈ N , λ ( M n ) < ∞ ;(3) B ( ¯ M ) is finitely generated (as an S [ y ] -module). Theorem 5.5. Assume that R is a Noetherian ring.Let M be an S -module and ¯ M := ( M n ) n ∈ N be a good filtering of M . For every n ∈ N , let a n := λ ( M n ) . Then, for n ∈ N large enough, a n is equal to a polynomial q ¯ M ( n ) of degree at most k . Moreover, the leading monomial of q ¯ M does not dependon the choice of the 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 good filtering of M (if such good filtering exists). Proof. Notice that, under the above assumptions, B ( M ) is Noetherian.By Theorem 3.4, F ¯ M ( t ) = p ( t )(1 − t ) k +1 , for some polynomial p ( t ) ∈ R [ t ] . Thus, the coefficients a n of the power series F ¯ M are equal to some polynomial q ( n ) ∈ R [ n ] of degree at most k .Assume now that ¯ M (cid:48) := ( M (cid:48) n : n ∈ N ) is another good filtering of M and denote a (cid:48) n := λ ( M (cid:48) n ) . Since B ( ¯ M ) is Noetherian, there exists m ∈ N such that M m generates B ( M ) ; similarly, after enlarging m if necessary, we may assume that M (cid:48) m := (cid:76) n ≤ m M (cid:48) n y n generates B ( M (cid:48) ) . Thus, M m tightly generates ¯ M . Let v , . . . , v (cid:96) ∈ M (cid:48) m which generate M (cid:48) m (as R -module). Since M = (cid:83) n M i , thereexists d ≥ such that v , . . . , v (cid:96) ∈ M m + d . Thus, B (cid:48) m ≤ B m + d . Claim . For every n ≥ m , B (cid:48) n ≤ B n + d . Denote S m to be the set of polynomials in S (cid:48) of degree at most m . Since B (cid:48) m tightly generates ¯ M (cid:48) , for every i ∈ N we have B (cid:48) m + i ≤ S i B (cid:48) m ≤ S i B m + d ≤ B (cid:48) m + d + i proving the claim. Thus, a (cid:48) n ≤ a n + d : therefore, for every n large enough, q ¯ M (cid:48) ( n ) ≤ q ¯ M ( n + d ) . Similarly, q ¯ M ( n ) ≤ q ¯ M (cid:48) ( n + d (cid:48) ) for some d (cid:48) ∈ N and every n large enough, showing that q ¯ M and q ¯ M (cid:48) have the sameleading monomial. (cid:3) We can now prove the proposition and the theorem at the start of the section. Proof of Prop. 5.2. Let B be an R -sub-module of B such that B is finitely gen-erated, λ ( B ) < ∞ , and SB = B . For every i ∈ N , define B i := S i B A i := ι − ( B i ) C i := π ( B i ) , 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. Thus, for n ∈ N largeenough, λ ( A n ) = q ¯ A ( n ) , and similarly for B and C . Moreover, for every n ∈ N , λ ( A n ) + λ ( C n ) = λ ( B n ) : thus, q ¯ A + q ¯ C = q ¯ B , and therefore µ ( A ) ⊕ µ ( C ) = µ ( B ) . (cid:3) Proof of Thm. 5.1. Since B is locally λ R –finite, every sub-module of A , B , or C islocally λ R -finite. Claim . µ ( B ) ≤ µ ( A ) ⊕ µ ( C ) . Let B (cid:48) ≤ B be an S -sub-module which is finitely generated. Define A (cid:48) := ι − ( B ) , C (cid:48) := π ( A (cid:48) ) . Notice that the sequence −→ A (cid:48) −→ B (cid:48) −→ C (cid:48) −→ is exact, and therefore, by Proposition 5.2, µ ( B (cid:48) ) = µ ( A (cid:48) ) ⊕ µ ( C (cid:48) ) ≤ µ ( A ) ⊕ µ ( C ) . Taking the supremum among all the B (cid:48) , we get the Claim. Claim . µ ( B ) ≥ µ ( A ) ⊕ µ ( C ) . Let A (cid:48) ≤ A and C (cid:48) ≤ C be a S -sub-modules which are finitely generated. Since C is finitely generated and π is surjective, there exists B (cid:48) ≤ B finitely generated andsuch that π ( B (cid:48) ) = C . Define B (cid:48)(cid:48) := B (cid:48) + ι ( A (cid:48) ) , A (cid:48)(cid:48) := ι − ( B (cid:48)(cid:48) ) . We have that the sequence −→ A (cid:48)(cid:48) −→ B (cid:48)(cid:48) −→ C (cid:48) −→ is exact, and B (cid:48)(cid:48) is finitely generated and locally λ R -finite. Thus, by Proposition 5.2, µ ( A (cid:48) ) ⊕ µ ( C (cid:48) ) ≤ µ ( A (cid:48)(cid:48) ) ⊕ µ ( C (cid:48) ) = µ ( B (cid:48)(cid:48) ) ≤ µ ( B ) . Taking the supremum on the left side among all possible A (cid:48) and C (cid:48) , we get theClaim. (cid:3) ILBERT POLYNOMIAL OF LENGTH FUNCTIONS V. 8 13 Modules over R -algebrae Let T be a finitely generated commutative R -algebra. Let M be a T -module.We want to define the dimension of M as a T -module.Fix γ , . . . , γ k generators of T as R -algebra. Equivalently, we fix a surjectivehomomorphism of R -algebrae φ : R [ x , . . . , x k ] → T and denote γ i := φ ( x i ) , i = 1 , . . . , k . We can therefore see M as an R [ x , . . . , x k ] -module, and we denote it either by (cid:104) M ; φ (cid:105) or by (cid:104) M ; ¯ γ (cid:105) .We assume that R is Noetherian, and that there exists M ≤ M R -sub-modulesuch that T M = M , λ ( M ) < ∞ , and M is finitely generated as R -module. 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 doesnot. Thus, we can define the λ -dimension of M (as a T -module) as the degree of µ ( M ; φ ) .Repeating: Definition 6.1. 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 ≤ M finitely generated R -sub-module, such that λ ( M ) < ∞ and T M = M .Then, we can define as before the λ -dimension of M as a T -module, and thisdimension does not depend on the choice of M or of φ . Examples 6.2. Fix some length function λ on R such that λ ( R ) = 1 .a) Let M := T = R [ z ] , ¯ γ := (cid:104) (cid:105) , ¯ δ := (cid:104) , (cid:105) . Thus, (cid:104) M ; ¯ γ (cid:105) is R [ z ] seen as R [ z ] -module with the canonical action, while (cid:104) M ; ¯ δ (cid:105) is R [ z ] seen as R [ y , y ] -module, with y acting as multiplication by z and y as multiplication by z . Then, µ ( M ; ¯ γ ) = t ,while µ ( M ; ¯ δ ) = 3 t .b) Let M := T = R [ z, z − ] , ¯ γ := (cid:104) , − (cid:105) , ¯ δ := (cid:104) , − (cid:105) . Thus, (cid:104) M ; ¯ γ (cid:105) is T seen as R [ x , x ] -module with x acting as multiplication by z , and x as multiplication by z − , while (cid:104) M ; ¯ δ (cid:105) is T seen as R [ x , x ] -module, with x acting as multiplicationby z and x as multiplication by z − . Then, µ ( M ; ¯ γ ) = 2 t , while µ ( M ; ¯ δ ) = 4 t .In both examples, we see that the two modules have different degrees, but have thesame dimension.It remains to prove that the degree of µ ( M ; φ ) does not depend on the choiceof φ . It is clear that it suffices to prove the following: Lemma 6.3. Let ¯ δ ∈ T k (cid:48) be another tuple of generators of T . Then, (cid:104) M ; ¯ γ (cid:105) and (cid:104) M ; ¯ δ (cid:105) have the same dimension.Proof. After exchanging the rôles of ¯ γ and ¯ δ if necessary, we may assume that k ≥ k (cid:48) . After extending δ by setting δ i = 0 for i ≥ k (cid:48) , we may assume that k = k (cid:48) .As usual, we denote S := R [ x , . . . , x k ] and by S n the set of polynomials in S ofdegree at most n .We denote by ψ : R [ x , . . . , x k ] → M be the surjective homomorphism of R -algebrae corresponding to ¯ δ (and by φ the one corresponding to ¯ γ ).For every n ∈ N , define T n := φ ( S n ) T (cid:48) n := ψ ( S n ) M n := T n M M (cid:48) n := T (cid:48) n M Notice that both ( T n ) n ∈ N and ( T (cid:48) n ) n ∈ N are upward filterings of T as an R -module,that ( M n ) n ∈ N and ( M (cid:48) n ) n ∈ N are upward filterings of M as an R -module, and thateach T n , T (cid:48) n , M n and M (cid:48) n are finitely generated (as R -modules).Moreover, M generates both (cid:104) M ; ¯ γ (cid:105) and (cid:104) M ; ¯ γ (cid:48) (cid:105) as S -modules. Thus, we canapply Theorem 3.9; we denote by q (resp., q (cid:48) ) the Hilbert polynomial of (cid:104) M ; ¯ γ (cid:105) (resp., of (cid:104) M ; ¯ γ (cid:48) (cid:105) ).Let c ∈ N such that γ , . . . , γ k ∈ T (cid:48) c . Thus, T ≤ T (cid:48) c .The following is then clear Claim . For every n ∈ N , T n ≤ ( T (cid:48) c ) n Therefore, for every n ∈ N M n = T n M ≤ ( T (cid:48) c ) n M ≤ T (cid:48) cn M = M (cid:48) cn Therefore, for every n ∈ N , λ ( M n ) ≤ λ ( M (cid:48) cn ) . Therefore, for every n large enough, q ( n ) = λ ( M n ) ≤ λ ( M (cid:48) cn ) = q (cid:48) ( nc ) , proving that deg q ≤ deg q (cid:48) . Exchanging the rôles of φ and φ (cid:48) , we see that q and q (cid:48) have the same degree. (cid:3) We end this section with a comparison between λ -dimension and Krull dimensionfor affine rings. Lemma 6.4. Let R be a field and λ equal to the linear dimension (as R -vectorspaces). Let T be a finitely generated R -algebra. Then, the λ -dimension and theKrull dimension of T coincide.Proof. Let d be equal to the Krull dimension of T . By Noether Normalization (see[Eis95, Theorem 13.3], there exists an R -sub-algebra A < T such that:(1) A , as an R -algebra, is isomorphic to the polynomial ring R [ y , . . . , y d ] ;(2) T is finitely generated as A -module.Thus, T and A have the same Krull dimension d , and it is easy to see that (2)implies that T has the same λ -dimension as A . Finally, A has λ -dimension d . (cid:3) Hilbert-Samuel polynomial for homogeneous modules Let S := R [ x , . . . , x n ] and I := ( x , . . . , x n ) ideal of S . Let M be an S -module.For every n ∈ N , define c n := λ ( M/I n +1 M ) . Theorem 7.1. Assume that:(i) R is Noetherian;(ii) M is finitely generated (as S -module);(iii) λ ( M/IM ) is finite.Then, for every n ∈ N , c n is finite, and there exists a polynomial ¯ q ( t ) ∈ R [ t ] suchthat: (1) for every n ∈ N large enough, c n = ¯ q ( n ) ; (2) deg ¯ q ≤ k .Proof. Usual proof (see e.g. [Eis95, Proposition 12.2]) (cid:3) Assume morevoer, besides the hypothesis in the theorem, that there exists V Then, one can define a n := λ ( S n V ) as before. We notice that c n = λ ( M/I n +1 M ) = λ ( S n V /I n +1 M ) ≤ a n 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 ) .In general, it can happen that ¯ µ ( M ) < µ ( M ) . Example 7.2. Let K be a field, λ be the linear dimension over K , S := K [ x, y ] , M := K [ x, y ] / ( xy − . Then, µ ( M ) = 2 t , while ¯ µ ( M ) = 0 .It is easy to prove that for homogeneous ideals the situation is different. Exercise . Assume that R is Noetherian. Let J < S be a homogeneous ideal,and M := S/J . Then, µ ( M ) = ¯ µ ( M ) . More precisely, fix a finite set G generating J , and let n be the maximum degreeof the polynomials in G . Let V := R . Then, for every n > n , S n V and M/I n +1 are isomorphic (as R -modules), and therefore a n = λ ( S n V ) = λ ( M/I n +1 ) = c n . See also[Eis95, C.12] and [Nor68, C.7] for the “classical” version of the Hilbert-Samuel polynomial.8. d -dimensional and receptive versions of entropy Let R be Noetherian and M be an S -module. Let m be the coeffient of µ ( M ) .For every d ≤ k , define h ( d ) λ ( M ) := + ∞ if deg µ ( M ) > d if deg µ ( M ) < d md ! if deg µ ( M ) = d The value h (1) ( M ) is the receptive entropy of M w.r.t. the standard regularsystem generated by ( x , . . . , x k ) (see [BDGSa, BDGSb]); we call each h ( d ) ( M ) the d -dimensional entropy of M (and thus the algebraic entropy h is the k -dimensional entropy). Proposition 8.1. Let → A → B → C → be an exaxt sequence of S -modules.Assume that (1) R is Noetherian; (2) B is locally λ R -finite.Then, for every d ≤ k h ( d ) ( B ) = h ( d ) ( A ) + h ( d ) ( C ) . Proof. By Theorem 5.1. (cid:3) Let T be a finitely generated R -algebra. We can give similar definitions for MT -module. Fix ¯ γ = (cid:104) γ , . . . , γ k (cid:105) generators of T (as R -algebra). Let (cid:104) M, ¯ γ (cid:105) be the S -algebra defined in §6. Assume that R is Notherian. We define h ( d, ¯ γ ) λ ( M ) := sup { h ( d ) ( (cid:104) M (cid:48) , ¯ γ (cid:105) ) : M (cid:48) < M λ S -small S -sub-module } h (1 , ¯ γ ) λ ( M ) is the receptive entropy of M w.r.t. the the standard regular systemgenerated by ¯ γ (see [BDGSa, BDGSb]); we call each h ( d ¯ γ ) ( M ) the d -dimensionalentropy of M w.r.t. ¯ γ . Theorem 8.2. Let → A → B → C → be an exaxt sequence of T -modules.Assume that (1) R is Noetherian; (2) B is locally λ R -finite.Then, for every d ≤ k , h ( d, ¯ γ ) ( B ) = h ( d, ¯ γ ) ( A ) + h ( d, ¯ γ ) ( C ) . In particular, the receptive entropy h (1 , ¯ γ ) is additive (under the assumptions ofNoetherianity of R and locally λ R -finiteness).9. Totally additive versions of µ and (receptive) entropy The definition of µ ( M ) reflects the usual definition of algebraic entropy (see§2.2). Following a construction in [Vám68, Proposition 3], we propose an alternativedefinition, which is in some ways better behaved Definition 9.1. Let A be an S -module. A λ S -small chain in A is a sequence of S -sub-modules A = (cid:0) A ≤ A ≤ · · · ≤ A n − ≤ A n ≤ A (cid:1) , where, for every i ≤ n , ˆ A i := A i /A i − is λ S -small. We call n is the size of A . Definition 9.2. Let θ be a partial function from S -modules to the commutativemonoid of monomials of the form mt d , where d ∈ N and m ∈ R ≥ ∪{ + ∞} (examplesof such function are λ and µ ). We will be interested only in functions θ which satisfythe following conditions: Additivity: θ (0) = 0 and, for every exact sequence → A → B → C → of S -modules in the domain of θ , θ ( B ) = θ ( A ) ⊕ θ ( C ) ; Invariance: if A and B are isomorphic S -modules in the domain of θ , θ ( A ) = θ ( B ) .Assume that the domain of θ includes all the λ S -small S -modules. Let A be any S -module. Given a λ S -small chain A in A of size n , we define θ ( A ) := n (cid:88) i =1 θ ( ˆ A i )ˆ θ ( A ) := sup { θ ( A ) : A λ S -small chain in A } We will see later that ˆ θ can be defined in a simpler way (Proposition 9.7); seealso [Vám68, Proposition 3] for an equivalent approach.For “well-behaved” length functions λ , we have ˆ λ = λ (here we take S = R ).However, the following example shows that it is not always the case. Example 9.3. Let λ be the following function on Z -modules: λ ( A ) := (cid:40) if A is torsion ∞ otherwise . Then, ˆ λ = 0 . Proposition 9.4. Assume: (1) R is Noetherian; (2) the domain of θ includes all λ S -small S -modules; (3) θ is additive and invariant (on λ S -small S -modules).Then, ILBERT POLYNOMIAL OF LENGTH FUNCTIONS V. 8 17 (a) ˆ θ is also additive and invariant;(b) if A is a λ S -small S -module, then ˆ θ ( A ) = θ ( A ) ;(c) For every S -module A , ˆ θ ( A ) = sup { ˆ θ ( B ) : B < A finitely generated S -sub-module } (d) if A is a locally λ R -finite S -module, then ˆ θ ( A ) = sup { θ ( B ) : B < A finitely generated S -sub-module } (e) ˆˆ θ = ˆ θ Proof. The proof is quite straightforward; we will prove that ˆ θ is additive, andleave the remainder as an exercise (see also [Vám68]). Thus, let −→ A ι −→ B π −→ C −→ be an exact sequence of S -modules. Claim . ˆ θ ( B ) ≤ ˆ θ ( A ) ⊕ ˆ θ ( C ) . Let B = (cid:0) B ≤ B ≤ · · · ≤ B n − ≤ B n ≤ B (cid:1) be a λ S -small chain in B . Forevery i ≤ n , define A i := ι − ( B i ) C i := π ( A i ) . Then, A := (cid:0) A ≤ A ≤ · · · ≤ A n − ≤ A n ≤ A (cid:1) and C := (cid:0) C ≤ C ≤ · · · ≤ C m − ≤ C m ≤ C (cid:1) are λ S -small chains in A and C , respectively. Moreover, forevery i ≤ n , we have an exact sequence −→ ˆ A i −→ ˆ B i −→ ˆ C i −→ . Therefore, θ ( B ) = θ ( A ) ⊕ θ ( C ) , and the claim follows. Claim . ˆ θ ( B ) ≥ ˆ θ ( A ) ⊕ ˆ θ ( C ) Let A = (cid:0) A ≤ A ≤ · · · ≤ A n − ≤ A n ≤ A (cid:1) and C = (cid:0) C ≤ C ≤ · · · ≤ C m − ≤ C m ≤ C (cid:1) be λ S -small chains in A and C , respectively. For every i ≤ m + n ) B i := (cid:40) ι ( A i ) if i ≤ n ; π − ( C i − n ) if n < i ≤ n + m ) . Then, B := (cid:0) B ≤ B ≤ · · · ≤ B n +2 m − ≤ B n +2 m ≤ B (cid:1) is a λ S -small chain in B .Moreover, for every i ≤ n + m , ˆ B i = (cid:40) ˆ A i if i ≤ n ;ˆ C i if n < i ≤ n + m. Therefore, θ ( B ) = θ ( A ) ⊕ θ ( C ) , and the claim follows. (cid:3) For the remainder of this section, we assume that R is Noetherian . Definition 9.5. Given an ideal I < R , we say that I is λ -cofinite if λ ( R/I ) < ∞ . Remark 9.6. Let I < R be a λ -cofinite ideal, and A be an R -module. Then, A/I is locally λ R -finite. Proposition 9.7. Let θ be as above and total. Assume that, for every S -module Aθ ( A ) = sup { θ ( B ) : B ≤ A λ S -small S -submodule } . Then, ˆ θ ( A ) ≥ θ ( A ) and, if A is finitely generated, ˆ θ ( A ) = sup { θ ( A/I ) : I < R λ -cofinite ideal } The proof of the above proposition is in the next subsection: for now we willrecord some consequences. Corollary 9.8. (1) ˆ µ satisfies the conclusions of Propositions 9.4 and 9.7; (2) ˆ µ ( A ) = µ ( A ) for every locally λ R -finite S -module A ; (3) if λ ( R ) < ∞ , then ˆ µ = µ . Corollary 9.9. For every d ≤ k , (1) the d -dimensional entropy (see §8) ˆ h ( d ) λ is a length functions (on all S -modules)and satisfies the conclusion of Proposition 9.7; (2) ˆ h ( d ) λ ( A ) = h ( d ) λ ( A ) for every locally λ R -finite S -module A ; (3) if λ ( R ) < ∞ , then ˆ h ( d ) λ = h ( d ) λ . Corollary 9.10. Let T be a finitely generated R -algebra, ¯ γ ∈ T k be a set of gen-erators of T . Given d ≤ k , let h ( d, ¯ γ ) be defined as in §8. Then: (1) ˆ h ( d, ¯ γ ) λ is a length functions (on all T -modules) and satisfies the conclusion ofProposition 9.7; (2) ˆ h ( d, ¯ γ ) λ ( A ) = h ( d, ¯ γ ) λ ( A ) for every locally λ R -finite S -module A ; (3) if λ ( R ) < ∞ , then ˆ h ( d, ¯ γ ) λ = h ( d, ¯ γ ) λ . The above Corollary answers [BDGSb, Question 5.10]. Moreover, Lemma 6.4answers the conjecture in [BDGSb, Remark 5.9]. Corollary 9.11. Let λ be the length on Z -modules introduced in Example 2.2(b).Then, for every finitely generated Z [¯ x ] -module A , ˆ µ ( A ) = sup { µ ( A/nA ) : 2 ≤ n ∈ N } = lim n →∞ µ ( A/n ! A ) . We cannot drop the assumption that A is finitely generated in Proposition 9.7. Example 9.12. Let R := Z and λ be the length introduced in Example 2.2b. Let A := Q [¯ x ] (see as Z [¯ x ] -module). Then, ˆ µ ( A ) = ∞ · t k sup { µ ( A/nA ) : 2 ≤ n ∈ N } = 0 . We now give another example of equality between Krull and λ dimensions (be-sides Lemma 6.4). Lemma 9.13. Fix ≤ n ∈ N . Let R := Z / ( n ) . Let λ be the length on R -mod given by λ ( M ) := log n ( | M | ) . Let A := S/J for some ideal J < S . Then, λ - dim( A ) is equal to the Krull dimensionof A .Proof. We proceed by induction on n .If n is prime, then R is a field, and λ is equal to the linear dimension: thus, theconclusion follows from Lemma 6.4.Otherwise, let p ∈ N be a prime divisor of n and let m := n/p . We have theexact sequence −→ pA −→ A −→ A/pA −→ Both pA and A/pA are quotients of Z / ( m )[¯ x ] : therefore, by inductive hypothesis,their λ -dimensions are equal to their Krull dimensions. By additivity of Krull and λ dimensions, the conclusion follows. (cid:3) ILBERT POLYNOMIAL OF LENGTH FUNCTIONS V. 8 19 Proof of Proposition 9.7.Lemma 9.14. Let I, J < R be λ -cofinite ideals. Then, I ∩ J is also λ -cofinite.Proof. R/I ∩ J embeds into R/I × R/J . (cid:3) Lemma 9.15. Let I, J < R be λ -cofinite ideals. Then, IJ is also λ -cofinite.Proof. Let ¯ a = ( a , . . . , a (cid:96) ) generate I . Then, I/IJ is a quotient of ( R/J ) (cid:96) via themap ( r + J, . . . , r (cid:96) + J ) (cid:55)→ r a + · · · + r (cid:96) a (cid:96) + IJ. Therefore, λ ( I/IJ ) ≤ (cid:96)λ ( R/J ) < ∞ , and λ ( R/IJ ) ≤ λ ( R/I ) + λ ( I/IJ ) < ∞ . (cid:3) Lemma 9.16. Let A be an S -module. Assume that A is λ S -small. Then, Ann R ( A ) := { r ∈ R : rA = 0 } is λ -cofiniteProof. Let a , . . . , a (cid:96) be geneators of A (as S -module). For every i ≤ (cid:96) , λ ( Ra i ) < ∞ .Moreover, Ra i is isomorphic (as R -module) to R/Ann R ( a i ) , and therefore Ann R ( a i ) is λ -cofinite. Finally, Ann R ( A ) = Ann R ( a ) ∩ · · · ∩ Ann R ( a (cid:96) ) and the conclusion follows from the previous lemma. (cid:3) Proof of Proposition 9.7. Let B ≤ A be a λ S -small S -sub-module. By definition, ˆ θ ( A ) ≥ θ ( B ) : therefore, by the assumption, ˆ θ ( A ) ≥ θ ( A ) .Define θ (cid:48) ( A ) := sup { θ ( A/I ) : I ≤ R λ -cofinite ideal } . We want to prove that, when A is finitely generated, ˆ θ ( A ) = θ (cid:48) ( A ) . It suffices toshow that θ (cid:48) is additive on finitely generated S -modules. Thus, let −→ A −→ B −→ C −→ be an exact sequence of finitely generated S -modules. Claim . ˆ θ ( B ) ≤ ˆ θ ( A ) ⊕ ˆ θ ( C ) .Let I ≤ R be a λ -cofinite ideal. We have the exact sequence of λ S -small S -modules −→ A/ ( A ∩ IB ) −→ B/IB −→ C/IC −→ . Since θ is additive on λ S -small S -modules, and IA ≤ A ∩ IB , we have ˆ θ ( A ) + ˆ θ ( C ) ≥ θ ( A/IA ) ⊕ θ ( C/IC ) ≥ θ ( A/ ( A ∩ IB )) ⊕ θ ( C/IC ) = θ ( B/IB ) , and the claim follows. Claim . ˆ θ ( B ) ≥ ˆ θ ( A ) ⊕ ˆ θ ( C ) .Let I, I (cid:48) ≤ R be λ -cofinite ideals. We want to prove that θ ( A/I ) + θ ( C/I (cid:48) ) ≤ ˆ θ ( B ) . Replacing I, I (cid:48) with I ∩ I (cid:48) , w.l.o.g. we may assume that I = I (cid:48) . By the Artin-ReesLemma (see e.g. [Nor68, §4.7]), there exists ≤ n ∈ N such that, for every m ∈ N ,(1) A ∩ I m + n B = I m ( A ∩ I n B ) . Let J := I n and A (cid:48) := A ∩ JB : notice that J is also λ -cofinite. Taking m := n in (1), we obtain: A ∩ J B = JA (cid:48) . Thus, we have the exact sequence −→ A/JA (cid:48) −→ B/J B −→ C/J C −→ . The modules appearing above are all λ S -small: therefore, θ ( A/IA ) ⊕ θ ( C/IC ) ≤ θ ( A/JA (cid:48) ) ⊕ θ ( C/J C ) = θ ( B/J B ) ≤ ˆ θ ( B ) , proving the Claim. (cid:3) Examples. Let R := Z , α be the length introduced in Example 2.2(b) and β be the length given by the rank (i.e., β ( M ) = dim Q ( M ⊗ Q ) ). Since β ( Z ) = 1 < ∞ , µ β = ˆ µ β .1) Let S := Z [ x ] , I be an ideal of S , and M := S/I . The following table showsthe values of µ α ( M ) , ˆ µ α ( M ) , and µ β ( M ) for some values of I : I µ α ( S/I ) ˆ µ α ( S/I ) µ β ( S/I )0 0 ∞ · t t S n ) ; ≤ n ∈ N log( n ) · t log( n ) · t p ( x )) ; deg p ≥ ∞ · t deg p · t ( p ( x ) , n ) ; ≤ n ∈ N , log( n ) deg p · t log( n ) deg p · t p monic, deg p ≥ 2) Let S := Z [ x, y ] and M := S/ ( xy ) . Then, µ α ( M ) = 0 , ˆ µ ( M ) = ∞ · t, µ β ( M ) = 2 t Fine grading Up to now, we have only considered the case when the degrees are naturalnumbers. As in the classical case when R is a field, one can consider gradings inany commutative monoid (see e.g. [MS05]).10.1. Graded modules. Let Γ be a commutative monoid.Remember that Γ has a canonical quasi-ordering, given by m ≤ n if there exists p ∈ Γ with m + p = n . The neutral element is a minimum of the (cid:104) Γ , ≤(cid:105) . Definition 10.1. 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 > for i = 1 , . . . , k . ¯ γ ∈ N k is verygood (in N ) iff γ i (cid:54) = 0 for i = 1 , . . . , k . Notice that, in general, if ¯ γ is good, theneach γ i is non-zero (and even non-torsion). Remark 10.2. Given ¯ γ = (cid:104) γ , . . . , γ k (cid:105) ∈ Γ k , if ¯ γ is good, then the followingexpression is well defined: (cid:81) ki =1 (1 − t γ i ) ∈ N [[ t Γ ]] Definition 10.3. Fix ¯ γ = (cid:104) γ , . . . , γ k (cid:105) ∈ Γ k . A Γ -graded S -module of degree ¯ γ isgiven by an S -module M and a decomposition M = (cid:77) n ∈ Γ M n , where each M n is an R -module, and, for every i ≤ k and n ∈ Γ , x i M n ≤ M n + γ i . ILBERT POLYNOMIAL OF LENGTH FUNCTIONS V. 8 21 We denote by ¯ M the module M with the given grading (including the tuple ¯ γ := (cid:104) γ , . . . , γ k (cid:105) ). Theorem 10.4. Let ¯ M be a Γ -graded S -module of degree ¯ γ ∈ Γ k . For every n ∈ N ,let a n := λ ( M n ) . Define F ¯ M ( t ) := (cid:88) n ∈ Γ a n t n . Assume that: (1) γ is good; (2) λ ( M n ) < ∞ for every n ∈ Γ ; (3) M is a Noetherian S -module.Then, there exists a polynomial p ( t ) ∈ R [ t Γ ] such that F ¯ M ( t ) = p ( t ) (cid:81) ki =1 (1 − t γ i ) . Proof. Mutatis mutandis , same proof as Thm. 3.2. (cid:3) Filtered modules. We move now from graded modules to upward filteredmodules. Definition 10.5. Let ¯ γ := (cid:104) γ , . . . , γ k (cid:105) ∈ Γ k and N be an S -module.An increasing Γ -filtering on N with degrees ¯ γ is a sequence of R -sub-modulesof N ( N i : i ∈ Γ) such that it is increasing (i.e., if i ≤ j , then N i ≤ N j ), (cid:83) i ∈ Γ N i = N , and x i N j ≤ N j + α i for every j ∈ Γ , i ≤ k . We denote by ¯ N the S -module with the given tuple γ and the filtering ( N i ) i ∈ Γ . Definition 10.6. Let ¯ δ = (cid:104) δ , δ , . . . (cid:105) be a tuple of generators of Γ (for simplicity,we assume Γ countable: later we will be interested only in the case when ¯ δ is afinite tuple). Let ¯ y be a tuple of variables indexed by ¯ δ (i.e., there is one variable y j for each chosen generator δ j ). The blow-up module associated to ¯ N and thetuple ¯ δ is the following graded S [¯ y ] -module. As an R -module, B ¯ δ ( ¯ N ) := (cid:77) n ∈ Γ N n t n . The Γ -grading of B ¯ δ ( ¯ N ) is given by the decomposition B ¯ δ ( ¯ N ) = (cid:76) n ∈ Γ N n t n .The multiplication by x i on B ¯ δ ( ¯ N ) is defined as: x i ( vt n ) := ( x i v )¯ t n + γ i , for every i ≤ k , n ∈ Γ , v ∈ N n , and then extended by R -linearity on all B ¯ δ ( ¯ N ) :notice that the x i has degree γ i on B ¯ δ ( ¯ N ) . For each δ j ∈ ¯ δ , the multiplication by y j on B ¯ δ ( ¯ N ) is defined as: y j ( vt n ) := vt n + δ j , for every n ∈ Γ , v ∈ N n , and then extended by R -linearity on all B ¯ δ ( ¯ N ) : noticethat y j has degree δ j . Theorem 10.7. Let ¯ N be a Γ -filtering on N with degrees ¯ γ . Let ¯ δ = (cid:104) δ , . . . , δ (cid:96) (cid:105) be a finite tuple of generators of Γ , with corresponding variables ¯ y = (cid:104) y , . . . , y (cid:96) (cid:105) .For every n ∈ N , let a n := λ ( N n ) . Define F ¯ N ( t ) := (cid:88) n ∈ Γ a n t n ∈ R [[ t Γ ]] Then, F ¯ N = F B ¯ δ ( ¯ N ) . Therefore, if we assume that: (1) ¯ γ ∪ ¯ δ is good; (2) for every n ∈ N , λ ( N n ) < ∞ ; (3) B ( ¯ N ) is Noetherian as S [¯ y ] -module.Then, there exists a polynomial p ( t ) ∈ R [ t Γ ] such that F ¯ N ( t ) = p ( t ) (cid:81) (cid:96)j =1 (1 − t δ j ) (cid:81) ki =1 (1 − t γ i ) (the (1 − t δ j ) -factor in the denominator is due to the action of y j on B ¯ δ ( ¯ N ) ofdegree δ k ). Growth function. In this subsection we fix a monoid Γ with a tuple ofgenerators ¯ δ = (cid:104) δ , . . . , δ (cid:96) (cid:105) .We also fix a tuple ¯ γ = (cid:104) γ , . . . , γ k (cid:105) ∈ Γ k . Given a monomial in S = R [ x , . . . , x k ] its ¯ γ -degree deg ¯ γ is defined in the “obvious” way: deg ¯ γ ( rx n · · · x n k k ) := n γ + · · · + n k γ k . Given a polynomial p (¯ x ) ∈ S = R [ x , . . . , x k ] , we say that its ¯ γ -degree is less orequal to n ∈ Γ , and write deg ¯ γ ( p ) ≤ n , if each monomial in p has ¯ γ -degree less orequal to ¯ γ (since Γ is not linearly ordered in general, it’s not clear how to definethe ¯ γ -degree of a polynomial). For every n ∈ Γ , we denote S n := (cid:104) p ∈ S : deg γ ( p ) ≤ n (cid:105) . Theorem 10.8. Let N be an S -module. Let V ≤ N be an R -sub-module. Forevery n ∈ Γ , let V n := S n V (notice that S = R , that V = S V , and that S n and V n are R -modules), and a n := λ ( V n ) . Define G V ( t ) := (cid:88) n ∈ Γ a n t n ∈ R [[ t Γ ]] Assume that: (1) ¯ γ ∪ ¯ δ is very good (inside Γ ); (2) λ ( V ) < ∞ ; (3) V is finitely generated as R -module; (4) R is a Noetherian ring.Then, each a n is finite, and there exists a polynomial p ( t ) ∈ R [ t Γ ] such that G V ( t ) = p ( t ) (cid:81) (cid:96)j =1 (1 − t δ j ) (cid:81) ki =1 (1 − t γ i ) Proof. The fact that γ is very good is equivalent to the fact that S n is finitelygenerated (as R -module) for every n ∈ Γ . The above plus the fact that λ ( V ) < ∞ easily implies that λ ( V n ) < ∞ for every n ∈ Γ . Let ¯ V := SV as filtered S [¯ y ] -module. Then, V t generates B ( ¯ V ) as S [¯ y ] -module, and therefore B ( ¯ V ) is aNoetherian S [¯ y ] -module.We can conclude as in the proof of Theorem 3.8, using Theorem 10.7. (cid:3) Example 10.9. Let ¯ x := (cid:104) x , . . . , x k (cid:105) , ¯ y := (cid:104) y , . . . , y (cid:96) (cid:105) , S := R [¯ x, ¯ y ] , Γ := N , γ i := δ := (cid:104) , (cid:105) for i = 1 , . . . , k , and γ i := δ := (cid:104) , (cid:105) for i := k + 1 , . . . , k + (cid:96) .Thus, each x i has degree δ and each y j has degree δ . A monomial in ¯ x ¯ y hastherefore a “double degree” (cid:104) m, n (cid:105) ∈ Γ , where m is its total degree in ¯ x and n is itstotal degree in ¯ y . A polynomial in R [ t Γ ] is the same object as a polynomial in the ILBERT POLYNOMIAL OF LENGTH FUNCTIONS V. 8 23 two variables t , t . Let N be an S -module and V < N be an R -sub-module whichsatisfies (2) and (3). Then, if R is Noetherian, we have a corresponding function G V ( t , t ) = p ( t , t )(1 − t ) k +1 (1 − t ) (cid:96) +1 where p ∈ R [ t , t ] .10.4. Multi-variate Hilbert polynomial. Let P := (cid:104) P , . . . , P (cid:96) (cid:105) be a partitionof { , . . . , k } into (cid:96) nonempty subsets; for every j ≤ (cid:96) , let p j be the cardinality of P j . In the following, we will assume that P = { , , . . . , p } , P = { p + 1 , p +2 , . . . , p + p } , . . . , P (cid:96) = { p + · · · + p (cid:96) − , . . . , k } .Let Γ := N (cid:96) ; for every j ≤ (cid:96) , let ˆ e j ∈ Γ be the vector with all coordinates except the j -th which is . Let ¯ x := (cid:104) x , . . . , x k (cid:105) be a k -tuple of variables; to eachvariable x i ∈ P j assign the weight ˆ e j , and define ¯ e := (cid:104) ˆ e , ˆ e , . . . , ˆ e (cid:96) (cid:105) ∈ N k , where each weight ˆ e j is repeated p j times.As usual, S := R [ x , . . . , x k ] ; for every ¯ m ∈ N (cid:96) , let S (¯ e )¯ m := { p ∈ S : deg ¯ e ( p ) ≤ ¯ m } . An equivalent way of describing S (¯ e )¯ m is the following. Let ¯ t := (cid:104) t , . . . , t (cid:96) (cid:105) . Let φ : S → R [¯ t ] be the homomorphism of R -algebrae mapping x i to t j when i ∈ P j .Then, q ∈ S (¯ e )¯ m iff, for every j ≤ (cid:96) , deg t j ( φ ( q )) ≤ m j .Let M be a module over S and V ≤ M be an R -sub-module. For every ¯ m ∈ N (cid:96) ,define V ¯ m := S ¯ e ¯ m V ≤ M and a ¯ m := λ ( V ¯ m ) . Theorem 10.10. In the above setting, assume: (1) λ ( V ) < ∞ ; (2) V is finitely generated as R -module; (3) R is a Noetherian ring; (4) SV = M Then, each a ¯ m is finite. Moreover, there exists a polynomial q (¯ t ) ∈ R [¯ t ] such that:(i) for every ¯ m ∈ N (cid:96) large enough, a ¯ m = q ( ¯ m ); (ii) for every j ≤ (cid:96) , deg t j ( q ) ≤ p j . Moreover, the leading homogeneous component of q (see Def. 3.6)is independentfrom V .Proof. Choose ¯ e as tuple of generators of Γ and apply Theorem 10.8. We obtainthat there exists a polynomial r (¯ t ) ∈ R [¯ t ] such that (cid:88) ¯ m ∈ N (cid:96) a ¯ m ¯ t ¯ m = p (¯ t ) (cid:81) (cid:96)j =1 (1 − t j ) p j +1 (the exponents p j + 1 in the denominator come from the combination of the p j variables in P j , each with degree ˆ e j , plus the generator ˆ e j ). By Proposition 3.5,there exists a polynomial q satisfying (i) and (ii). By Proposition 3.7, the leadinghomogeneous component of q is independent from V . (cid:3) Appendix A. Non-commuative rings In this section R is no longer assumed to be commuative. Let G be some (as-sociative) monoid, and T := R [ G ] . T -mod is the category of left T -modules (by“modules” we will mean left modules). θ is some function from T -mod to the familyof monomials of the form rt n , with n ∈ N and r ∈ R ≥ ∪ {∞} . We assume:(1) θ is additive on the category of λ T -small T -modules: θ (0) = 0 and, for everyexact sequences of λ T -small T -modules → A → B → C → , θ ( B ) = θ ( A ) ⊕ θ ( C ) ;(2) θ is invariant: if A and B are isomorphic T -modules, then θ ( A ) = θ ( B ) ;(3) θ ( A ) = sup { θ ( B ) : B ≤ A λ T -small T -sub-module } . We define ˆ θ ( A ) := sup { θ ( A ) : A λ T -small chain in A } Theorem A.1. Assume that T is (left) Noetherian. Then,(a) ˆ θ is invariant and additive on all T -mod ;(b) if A is locally λ R -finite, then ˆ θ ( A ) = θ ( A ) ;(c) ˆ θ ( A ) = sup { ˆ θ ( B ) : B ≤ A finitely generated T -sub-module } ;(d) ˆ θ ≥ θ .Proof. Same as Propositions 9.4 and 9.7. The assumption that T = S [ G ] is usedin the following way: Claim . Let A be a T -module, generated (as a T -module) by a , . . . , a (cid:96) . Assumethat λ ( Ra i ) is finite, for i = 1 , . . . , (cid:96) . Then, A is locally λ R -finite. (cid:3) It is unclear if, under the same assumptions as in Thm. A.1, we can concludethat, when A is a finitely generated T -module, ˆ θ ( A ) = sup { ˆ θ ( A/I ) : I ≤ R λ -cofinite ideal } . Example A.2. Let G be an amenable cancellative monoid and T := R [ G ] . Let λ be a length on R -mod . Let h λ ( A ) be the algebraic entropy of the action of G on A for the length λ (see e.g. [DFGB20]). Assume that T is Noetherian. Then, θ := h λ satisfies the assumptions of this section. Thus, the function ˆ h λ satisfies theconclusion of Theorem A.1, and in particular is a length function on all T -mod. Appendix B. Finite generation and Noetherianity according to λ Let S be an R -algebra, and N be an S -module.Given A ⊆ N , we denote by SA the S -sub-module of N generated by A . Wedefine λ S ( b | A ) := λ (( SA + Sb ) /SA ) = λ ( Sb/ ( SA ∩ Sb )) . We also define c(cid:96) S,λN ( A ) := { b ∈ N : λ S ( b | A ) = 0 } . Lemma B.1. Let N (cid:48) < N be an S -sub-module. We have λ ( N/N (cid:48) ) > iff thereexists a ∈ N \ c(cid:96) S,λN ( N (cid:48) ) .Proof. ⇐ ) Assume that a ∈ N \ c(cid:96) S,λN ( N (cid:48) ) , that is, λ (( Sa + N (cid:48) ) /N (cid:48) ) > . We have λ ( N/N (cid:48) ) ≥ λ (( Sa + N (cid:48) ) /N (cid:48) ) > . Conversely, assume that λ ( N/N (cid:48) ) > . Since λ is continuous, there exists a , . . . , a n ∈ N such that λ (( Sa + · · · + S a n ) /N (cid:48) ) > ILBERT POLYNOMIAL OF LENGTH FUNCTIONS V. 8 25 We have λ (( Sa + · · · + S a n ) /N (cid:48) ) ≤ n (cid:88) i =1 λ ( a i | N (cid:48) ) . Thus, for some i ≤ n , λ ( a i | N (cid:48) ) > . (cid:3) Lemma B.2. c(cid:96) S,λN ( A ) is an S -sub-module of N . c(cid:96) S,λN is a closure operator (on subsets of N ). c(cid:96) S,λN ( A ) is the largest S -sub-module N (cid:48) of N such that λ ( N (cid:48) /SA ) = 0 .Proof. Let a, b ∈ c(cid:96) S,λN ( A ) . Thus, λ (( Sa + Sb + N ) /N ) ≤ λ (( Sa + N ) /N ) + λ (( Sb + N ) /N ) = 0 . Therefore, every c ∈ Sa + Sb satisfies λ ( c | N ) = 0 , showing that Sa + Sb ⊆ c(cid:96) S,λN ( A ) ,that is c(cid:96) S,λN ( A ) is an S -sub-module of N . Claim . c(cid:96) S,λN is increasing.In fact, let A ⊆ B ⊆ N and a ∈ c(cid:96) S,λN ( A ) . We have λ ( a | B ) = λ ( Sa/ ( Sa ∩ B )) ≤ λ ( Sa/ ( Sa ∩ A )) = 0 . Claim . A ⊆ c(cid:96) S,λN ( A ) .Clear, since λ ( A/A ) = 0 . Claim . λ ( c(cid:96) S,λN ( A ) /SA ) = 0 .By LemmaB.1. Claim . c(cid:96) S,λN ( c(cid:96) S,λN ( A )) = c(cid:96) S,λN ( A ) .We have only to show that c(cid:96) S,λN ( c(cid:96) S,λN ( A )) = c(cid:96) S,λN ( A ) . Thus, let C := c(cid:96) S,λN ( A ) and b ∈ c(cid:96) S,λN ( C ) . We want to prove that λ ( b | A ) = 0 .We have the surjective natural map φ : Sb/ ( Sb ∩ SA ) → Sb/ ( Sb ∩ C ) (remember that SC = C ). Its kernel is Ker φ = ( Sb ∩ C ) / ( Sb ∩ SA ) ≤ C/SA. Thus, λ (Ker φ ) ≤ λ ( C/SA ) = 0 . By additivity, λ ( b | A ) = λ ( Sb/ ( Sb ∩ SA )) = λ ( b | C ) + λ (Ker φ ) = 0 + 0 = 0 . Thus, c(cid:96) S,λN is a closure operator, and λ ( c(cid:96) S,λN ( A ) /SA ) = 0 . Now let N (cid:48) ≤ N be another S -sub-module such that λ ( N (cid:48) /N ) = 0 . Thus, for every a ∈ N (cid:48) , λ ( a | SA ) ≤ λ ( N (cid:48) /SA ) = 0 , showing that N (cid:48) ⊆ c(cid:96) S,λN ( A ) . (cid:3) Definition B.3. • N (cid:48) ⊆ N is λ S -closed (in N ) if N (cid:48) = c(cid:96) S,λN ( N ) . • A ⊆ N λ S -generates N if c(cid:96) S,λN ( A ) = N . • N is λ S -finitely generated (or λ S -f.g. for short) if there exists a finite subset A of N which λ S -generates N . • N is λ S -Noetherian if every S -sub-module of N is λ S -f.g.. Lemma B.4. T.f.a.e.: (1) N is λ S -Noetherian; (2) there doesn’t exists an infinite sequence N < N < N < · · · < N of S -sub-modules of N such that, for every n ∈ N , λ ( N n +1 /N n ) > ; (3) there doesn’t exists an infinite sequence a , a , . . . of elements of N such that,for every n ∈ N , λ ( a n +1 | a , . . . , a n ) > . It’s not clear if in the above Lemma it suffices to assume:(4) every λ S -closed sub-module of N is λ S -f.g..B.1. λ -Noetherian algebrae. The definitions and results in this subsection willnot be used later: it’s mostly conjectures which may be used to generalize resultson Hilbert polynomials to some non-Noetherian rings. Definition B.5. We say that S is λ -Noetherian (as an R -algebra) if it is λ S -Noetherian (as an S -module). Remark B.6. If N is finitely generated as S -module, then it is λ S -f.g..If N is Noetherian (as S -module), then it is λ S -Noetherian.If S is a Noetherian ring, then it is λ -Noetherian. Remark B.7. It’s not true in general that if λ ( N ) < ∞ , then N is λ S -Noetherian.We say that λ has positive gap if inf { λ ( M ) : M is an R -module and λ ( M ) > } > . If λ has positive gap and λ ( N ) < ∞ , then N is λ S -Noetherian. Conjecture B.8. If S is λ -Noetherian and M is λ S -f.g., then M is λ S -Noetherian. Conjecture B.9. If S is λ -Noetherian, then S [ x ] is also λ -Noetherian. Appendix C. Less assumptions In this section we reprove some of the results in §3 weakening some of the as-sumptions. Theorem C.1. In Theorem 3.2, we can replace Assumption (3) with(3)’ M is an λ S -Noetherian S -module.Proof. The proof proceeds in parallel, by induction on k .If k = 0 , then, since M is λ S -Noetherian, only finitely many of the M i have λ ( M i ) > . Thus, F ¯ M ( t ) is a polynomial.Assume now that we have proven the conclusion for k − . Let y, α, y n , ¯ K, ¯ C beas in the proof of Thm. 3.2. Claim . Let S (cid:48) := R [ x , . . . , x k − ] . Both K and C are λ S (cid:48) -Noetherian.We can then apply the inductive hypothesis to ¯ K and ¯ C and conclude the proofas in Thm. 3.2.In fact, K is an S -sub-module of M and C is a quotient of M , and thereforethey are both λ S -Noetherian. However, the action of x k on K and C is trivial (i.e., x k a = 0 for every a ). Thus, if e.g. N (cid:48) < K is an S (cid:48) -sub-module, then N (cid:48) is an S -sub-module, and hence (since K is λ S -Noetherian) N (cid:48) is λ S -finitely generated, and(since again the action of x k is trivial on K ), N (cid:48) is λ S (cid:48) -finitely generated, showingthat K is λ S (cid:48) -Noetherian. Identical argument works for C . (cid:3) Corollary C.2. In Theorem 3.4 we can weaken Assumption (3) to(3)’ B ( N ) is λ S [ y ] -Noetherian.Moreover, in Theorem 3.8, we can weaken the Assumptions (2) and (3) to(2)” B ( ¯ V ) is λ S [ y ] -Noetherian. ILBERT POLYNOMIAL OF LENGTH FUNCTIONS V. 8 27 References [BDGSa] Andrzej Biś, Dikran Dikranjan, Anna Giordano Bruno, and Luchezar Stoyanov, Alge-braic entropies of commuting endomorphisms of torsion abelian groups . Preprint. ↑ Metric vs topological receptive entropy of semigroup actions . To appear inQualitative Theory of Dynamical Systems. ↑ 15, 18[CSCK14] Tullio Ceccherini-Silberstein, Michel Coornaert, and Fabrice Krieger, An analogue ofFekete’s lemma for subadditive functions on cancellative amenable semigroups , J.Anal. Math. (2014), 59–81. 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Reufel, A generalization of the concept of length , TheQuarterly Journal of Mathematics (1965), no. 4, 297–321, available at https://academic.oup.com/qjmath/article-pdf/16/4/297/4369078/16-4-297.pdf . ↑ 2, 4[SV15] Luigi Salce and Simone Virili, The addition theorem for algebraic entropies inducedby non-discrete length functions , Forum Mathematicum (2015), 1143–1157. ↑ 2, 4[SVV13] Luigi Salce, Peter Vámos, and Simone Virili, Length functions, multiplicities and al-gebraic entropy , Forum Mathematicum (2013), no. 2, 255–282. ↑ 2, 3, 4[Vám68] Peter Vámos, Additive functions and duality over noetherian rings , Q. J. Math. (1968), no. 1, 43–55. ↑ 2, 3, 16, 17 Università di Firenze Email address : [email protected] URL ::