Which series are Hilbert series of graded modules over standard multigraded polynomial rings?

Consider a polynomial ring R with the Zn ‐grading where the degree of each variable is a standard basis vector. In other words, R is the homogeneous coordinate ring of a product of n projective spaces. In this setting, we characterize the formal Laurent series which arise as Hilbert series of finitely generated R‐modules. We also provide necessary conditions for a formal Laurent series to be the Hilbert series of a finitely generated module with a given depth. In the bigraded case (corresponding to the product of two projective spaces), we completely classify the Hilbert series of finitely generated modules of positive depth.


INTRODUCTION
Let be a field. We consider the polynomial ring = in indeterminates equipped with a standard multigrading, that is, a multigrading such that the degree of each variable is one of the standard basis vectors of ℤ . This setup includes the standard ℕ-grading, as well as the fine grading, where = and deg = for all ∈ [ ]. Let = ⨁ ∈ℤ be a finitely generated ℤ -graded -module. The Hilbert series of is the formal Laurent series where ∶= 1 1 ⋯ for all = ( 1 , … , ) ∈ ℤ . This is well-defined, because the graded components of are finitedimensional -vector spaces, and, since is positively graded, there exists a ∈ ℤ such that = 0 if ≱ (componentwise). The Hilbert series is known to capture important information about , including its dimension or its multiplicity. For general information about Hilbert series, we refer the reader to Bruns and Herzog [3,Chapter 4], Miller and Sturmfels [9,Chapter 8] or Kreuzer and Robbiano [7,Chapter 5]. In the present work, we consider the following question. Question 1.1. Which formal Laurent series arise as the Hilbert series for a given class of -modules?
The classical result of Macaulay [8] answers this question for cyclic modules in the standard-ℕ-graded situation. This work has recently been extended by Boij and Smith in [1], which classified (up to closure) Hilbert series of modules generated in degree 0 in the standard-ℕ-graded setup, as well as the Hilbert series of certain subclasses of this class of modules.
In the present work, we do not impose conditions on the generators, except that we assume the modules to be finitely generated. If we allow non-finitely generated modules then Question 1.1 is rather trivial: Any nonnegative series = ∑ is the Hilbert series of the -module . Therefore we restrict our attention to finitely generated modules. Under this assumption, every Hilbert series is a formal Laurent series with nonnegative coefficients which is a rational function with denominator ∏ ( 1 − deg ) . In the standard-ℕgraded situation, these conditions already characterize Hilbert series, as it was shown by the third author in [15,Corollary 2.3]. However, they are not sufficient in the multigraded situation, see Theorem 3.8.
One of the main results of the present paper is a complete answer to Question 1.1 in the ℤ -graded situation, namely: Theorem 3.6. The following statements are equivalent for a formal Laurent series ∈ ℤ(( 1 , … , )): 1. There exists a finitely generated graded -module whose Hilbert series equals .
2. satisfies the following two conditions: While this result is somewhat technical, we obtain very satisfying specializations in the fine-graded and the bigraded situations, respectively; see Theorems 3.15 and 5.1. Also, we clarify the relation between Hilbert series and merely nonnegative series in Theorem 3.14.
It seems natural to further generalize these ideas to arbitrary multigradings. However, in this generality, arithmetical issues arise. For instance, there exists a formal Laurent series with integral coefficients which is not a Hilbert series, but after multiplication by 2 it is, see Theorem 3.2; the example already shows that one cannot hope for a characterization using linear inequalities in this setting. In the present paper, we do not further pursue this direction.
One of the difficulties of Question 1.1 is that if is a Hilbert series of some module, then any nonnegative series which coincides with in all but finitely many coefficients is also a Hilbert series (to see this, if = for some module , then one might replace finitely many components of in the lowest degrees by copies of ). Thus it seems natural to rule this out, that is, to consider modules which do not contain a copy of the residue field as a submodule. Algebraically, this amounts to requiring that the depth of should be positive. Generalizing this idea leads to the notion of Hilbert depth, introduced by the third author in [15]. The Hilbert depth 1 of a formal Laurent series is defined by We consider the Hilbert depth only for those formal Laurent series which actually arise as Hilbert series of some module. Again, in the standard-ℕ-graded setting, Hilbert series of a given Hilbert depth have been classified in [15]. Our next main result is a quite general class of linear inequalities which are satisfied by every Hilbert series with a given Hilbert depth. We formulate our result in terms of the projective dimension, but at least over the polynomial ring, this is equivalent to the depth via the Auslander-Buchsbaum formula.
Theorem 4.1. Let = ⨁ be a commutative Noetherian ℤ -graded -algebra, such that dim 0 < ∞. Let ∈ ℕ and , be finitely generated -modules. If pdim ≤ and is a -th syzygy module, then the following inequality holds: In general, a classification of Hilbert series with a given Hilbert depth seems to be very difficult. Therefore, we content ourselves with an important special case. Our third main result is a complete characterization of the Hilbert series with positive Hilbert depth in the bigraded (i.e. ℤ 2 -graded) setup, see Theorem 5.3. The condition that we obtain is similar to the general inequalities of Theorem 4.1, but quite different in nature to the condition of Theorem 3.6.
In the previous paper [10], the second and the third author characterized Hilbert series of positive Hilbert depth over a bivariate polynomial ring [ , ] endowed with a non-standard ℤ-grading. In Theorem 6.1 we show that this result can be restated in a way analogous to Theorem 5.3.

NOTATION AND PRELIMINARIES
Let us fix some notation. We will use boldface letters , , , …, to denote elements of ℤ or ℕ . For such a vector , we write for its -th component. For all ∈ [ ], we denote by the -th standard basis vector. Moreover, for = ( 1 , … , ) ∈ ℕ and = ( 1 , … , For a formal Laurent series ∈ ℤ(( 1 , … , )) we write [ ] for the coefficient of in . We call nonnegative if every coefficient of is nonnegative and we denote this by ≥ 0.
We consider the partial order on ℤ which is given by coordinatewise comparison. In other words, for , ∈ ℤ we write ≤ if and only if ≤ for all ∈ [ ]. Moreover, we denote the coordinate minimum of and by ∧ , and their coordinate maximum by ∨ . A multivariate generalization of this fact has been given by Sturmfels [14]. For our purposes we only need the following special case of [14, Theorem 1]:

Multivariate Hilbert polynomials
be a Laurent polynomial, ∈ ℕ and There exist a ∈ ℤ and a polynomial ∈ ℤ For completeness, we include the easy proof.
Proof. Write = ∑ ∈Ω for a suitable finite set Ω ⊆ ℤ . Recall the series expansion From this, it is clear that

WHICH SERIES ARE HILBERT SERIES?
Let be a field. We consider the polynomial ring = equipped with a ℤ -grading, such that each variable is homogeneous. In this section, we deal with the question of which formal Laurent series ∈ ℤ(( 1 , … , )) arise as Hilbert series of finitely generated -modules. There are two obvious necessary conditions: It is a consequence of [15, Theorem 2.1] that in the case of the standard ℕ-grading (that is, = 1), these conditions are already sufficient. In general, this is not true, as we will see below. Corollary 2.2 in [10] establishes the following: … , )) is the Hilbert series of a finitely generated -module if and only if it can be written in the form Sketch of proof. We briefly remind the reader the inductive argument (on ) which allows us to prove Theorem 3.1: Let be the finitely generated -module. The key point is the consideration of a descending filtration on of submodules Then the Hilbert series of can be written as a sum of 1 + 1 + ⋯ + for ∶= +1 ∕ , where the first summand is a polynomial and ( 1 − deg ) = for = ∕ . Since acts trivially on , we may regard as a module over a polynomial ring in − 1 indeterminates, so that the induction hypothesis applies.
A decomposition as in Equation (3.1) is called a Hilbert decomposition of . This result is stated and proven in [10] for ℤ-gradings only, but the proof given there can be easily extended for multigradings.
with the ℕ-grading given by deg 1 = 2, deg 2 = 3 and deg 3 = 5. Consider the series One sees immediately that 2 has a Hilbert decomposition and therefore it is indeed a Hilbert series of a finitely generated -module. In particular, 2 and thus satisfy the necessary conditions mentioned above. However, is not a Hilbert series, as it does not admit a Hilbert decomposition.
To see this, note that the rational function has a pole of order 2 at = 1. Considering the possible summands in Equation (3.1), it follows that {2,3,5} = 0 and that at least one of {2,3} , {2,5} and {3,5} is non-zero. One can compute that the -th coefficient of is of the order 30 + (1). On the other hand, the -th coefficient of is of the order 1 + (1) for coprime , ∈ ℕ. As 2 ⋅ 3, 2 ⋅ 5, 3 ⋅ 5 < 30, the series does not have a Hilbert decomposition and thus it does not arise as a Hilbert series.

The standard multigrading
The criterion of Theorem 3.1 is very useful for showing that a given Laurent series is a Hilbert series: One only needs to construct a Hilbert decomposition. However, it is rather difficult to use this criterion to show that a given series is not a Hilbert series. Moreover it does not provide a good insight into the structure of the set of Hilbert series. We would like to have a characterization of the Hilbert series in terms of inequalities. In view of the preceding example, there is no hope for such a characterization in full generality.
As a consequence, we now specialize our considerations to the case that the degree of every variable of is a standard basis vector. More precisely, we consider the case that = [ In this setting, we give a characterization of the Hilbert series of finitely generated modules over in terms of certain inequalities. Roughly speaking, this can be seen as an implicitization of the set of Hilbert series. Before we can state our result we need to introduce some notation.
2), and the summands of | , are summands of . It is more convenient to consider | , instead of | , , because the former lives in a smaller ring of Laurent series.
be a polynomial. We call a monomial appearing in extremal if it does not divide any other monomial of . Moreover, we say that has positive extremal coefficients if the coefficient of every extremal monomial of is positive.
We caution the reader that the extremal monomials of a polynomial are not the same as the vertices of its Newton polytope: . The extremal monomials of are 3 , 3 and . Note that lies in the interior of the Newton polytope. On the other hand, 1 is a vertex of the Newton polytope, but not extremal in our sense.
The following characterization of Hilbert series of -modules is the main result of this section. Recall that is the number of variables of degree in .
2. satisfies the following two conditions: is a polynomial, and (b) for every ∈ ℤ and every ⊆ [ ], the Hilbert polynomial of the restriction | , of has positive extremal coefficients.
Remark 3.7. The condition that ≥ 0 is implicit in the last condition of Theorem 3.6 above, because  ( is a polynomial, and it is also clear that ≥ 0. So satisfies the obvious necessary conditions for being a Hilbert series. Moreover, ( ) = ( − ) 2 = 2 − 2 + 2 . Here, all three monomials are extremal, so in particular ( ) does not have positive extremal coefficients. Hence does not arise as Hilbert series of a finitely generated -module.
Although this can be obtained from Theorem 3.6, one way to see this directly is as follows: Assume to the contrary that there = for a finitely generated -module . We write deg 1 and deg 2 for the first and second component of the degree of an element ∈ , respectively. Let 1 , … , be a set of generators of and let deg = ( , ) . If < , then ( ) ( , ) = 0 and hence deg 1 < for any ∈ . Similarly, if > , then deg 2 < for any ∈ . Hence, in both cases we have that min ( for all ∈ . As is generated by 1 , … , , it follows that min ( This contradicts our assumption that = .
Example 3.9. Our next example shows that it is not sufficient to consider only the Hilbert polynomial of . Let It holds that ( ) = ( − ) 2 + , so all extremal coefficients are nonnegative. On the other hand, for = {1, 2} , = 0 it holds that  ( | , ) = ( − ) 2 and this polynomial does not have positive extremal coefficients. Thus is not a Hilbert series.
One common trait in the theory of Hilbert series is that many properties can be determined by examining only those exponents which are below the exponent which is the join of the exponents of the numerator. So one might hope to sharpen Theorem 3.6 by showing that one only needs to consider restrictions , for ≤ . However, the next example shows that this does not hold.
This series is nonnegative, because for , , ∈ ℕ it holds that The Hilbert polynomial clearly has nonnegative extremal coefficients. Moreover, for = {2, 3} and any ∈ ℤ , the Hilbert polynomial of the restriction is  ( | , ) = 2 + 2 + lower terms so it has nonnegative extremal coefficients. By symmetry, the same holds for = {1, 3}. Further, it follows from Theorem 3.14 below that the Hilbert polynomials of restrictions | , with | | = 1 have nonnegative extremal coefficients. So it remains to consider the case = {1, 2}. Let = (0, 0, ) ∈ ℤ 3 . Then and all three terms are extremal. So this restriction has nonnegative extremal coefficients if and only if ≠ + 1. In particular, is not a Hilbert series.
On the other hand, writing as a rational function one sees that the degrees of all terms in the numerator are less or equal than ∶= (2, 2, 2) ∈ ℕ . Thus for ≥ 2 it is not sufficient to consider restrictions | , with ≤ .
We need some preparations before we give the proof of Theorem 3.6. First, note that Hilbert decompositions are compatible with restrictions in the following sense: … , )) be a formal Laurent series. If has a Hilbert decomposition, then so does every restriction Proof. If has a Hilbert decomposition, then there exists a finitely generated graded -module This is a module over ′ ∶= ⊆ in a natural way. We give it the structure of an -module by letting the other variables act as zero. Its Hilbert series equals | , , hence this series has a Hilbert decomposition. But then | , clearly has a Hilbert decomposition as well. □ Next, we show that polynomials with positive extremal coefficients admit a certain decomposition. This is the key step in our proof of Theorem 3.6.
, the following statements are equivalent: 1. has positive extremal coefficients.
2. can be written as follows: If, in addition, there exists a ∈ ℤ such that ( ) ∈ ℤ for all ≥ , then the coefficients ( , ) can be chosen to be natural numbers.
Proof. We start with the implication (1) ⇒ (2). For ∈ ℕ , let be an extremal monomial of , and let be its coefficient.
It is easy to see that is also an extremal monomial of , and in fact it is its only extremal monomial. Further, its coefficient Hence, the corresponding terms cancel in We show that 1 still satisfies the hypothesis (1), so the claim follows by induction. For this, note that the only possible new extremal monomials of 1 are the monomials ∕ for 1 ≤ ≤ , so we need to compute their coefficients in . We start with one factor of : This implies that .
For a sufficiently large choice of 1 , … , , the coefficients of ∕ become arbitrarily negative. Hence, for large , 1 still satisfies the hypothesis.
For the other implication, if can be written as in (2), then there can be no cancellation between the extremal monomials on the right-hand side. So the coefficients of the extremal monomials of are (sums of) multiples of the , and thus positive.
Finally, assume that ( ) ∈ ℤ for all ∈ ℤ which are greater than some fixed . This implies that ( ) ∈ ℤ for all ∈ ℤ , see Corollary I.1.2 and Corollary XI.1.5 in Cahen and Chabert [5]. So by a classical result of Ostrowski [11] (see also [5,Corollary XI.1.11]), can be written as an integral linear combination of polynomials of the form is an extremal monomial of , then only 1 ,…, contributes to this term, so its coefficient is a multiple of the corresponding coefficient in 1 ,…, , which is 1∕ 1 ! ⋯ !. It now follows from the construction above that the ( , ) are positive integers. □ Proof of Theorem 3.6. We start with the necessity. If has a Hilbert decomposition, then so does every restriction of . Hence we only need to consider the case = [ ] and thus we need to show that ( ) has positive extremal coefficients. Consider a Hilbert decomposition Expanding every summand into a series, it follows that Hence ( ) has positive extremal coefficients by Theorem 3.12. Now we turn to the sufficiency. We proceed by induction over the number of variables of , with the base case being trivial. First, assume that ( ) ≠ 0. By assumption, its extremal coefficients are nonnegative, so Theorem 3.12 yields a decomposition It is easy to see that  ( . It holds that ( ′ ) = 0 and we claim that ′ still satisfies the hypothesis on the extremal coefficients. To see this, consider ⊆ [ ] and ∈ ℤ . If + ℕ ∩ + ℕ = ∅, then ′ | , = | , ; otherwise, let ∈ + ℕ ∩ + ℕ . Then [ ] ′ = 0 for all ≥ and hence  ( ′ | , ) = 0. In both cases, the hypothesis is satisfied. Next, we consider the case that ( ) = 0. In this case, the exponent vectors of the nonzero terms of are contained in finitely many translates of coordinate hyperplanes. Hence we may decompose as a sum of series in − 1 variables as follows: Choose ∈ ℤ such that [ ] = 0 for all ≥ . For 1 ≤ ≤ and 0 ≤ ≤ − 1 let ( , ) ∶= ( 1 , … , −1 , , 0, … , 0) ∈ ℤ . We decompose as follows: Every restriction of is a series in at most − 1 variables, so the claim follows by induction. □

Nonnegative series
Our next goal is to clarify the relation between series satisfying the hypothesis of Theorem 3.6 and series which are merely nonnegative. We will need the following convex geometric lemma.
Lemma 3.13. Let ⊆ ℝ be a polytope and let ∈ be a vertex, such that + ∉ for all ∈ ℝ ≥0 , ≠ 0. Then there exists a linear form ∈ (ℝ ) * which attains its maximum over exactly at and whose coefficients are nonnegative integers.
Proof. Let ′ ⊆ be the convex hull of all vertices of which are different from . Further, let ∶= + ℝ ≥0 . Both ′ and are convex sets, and our assumption implies that ′ ∩ = ∅. Then there exists a separating hyperplane, that is, a linear form ∈ (ℝ ) * such that We may assume that has rational coefficients, and after clearing denominators we may even assume that the coefficients of are integers. We show that has the claimed properties. It is clear that the maximum of over is attained only at . To see that the coefficients of are nonnegative assume that ( ) < 0 for some standard basis vector . Then ( + ) can be arbitrarily negative for large ≫ 0, contradicting (3.2). □ The following lemma shows that if ≥ 0 then some extremal coefficients are automatically nonnegative.

For all
which is also a vertex of its Newton polytope has a positive coefficient.
for all ∈ ℤ . So we only need to show the other implication.
Let be an extremal monomial of ∶=  ( | , ) which is also a vertex of its Newton polytope. By Theorem 3.13, there exists a linear form ∈ (ℝ ) * with nonnegative integral coefficients, which attains its maximum over the Newton polytope exactly at . Consider the linear map̂∶ ℤ given by  → ( ) . Then̂( ) is a univariate polynomial, which attains nonnegative values at sufficiently large integers. Hence its leading coefficient is nonnegative. On the other hand, by our choice of , this leading coefficient of̂( ) equals the coefficient of in . So the claim is proven. □

The fine graded case
If there are at most two variables with the same degree, then the obvious necessary conditions for Hilbert series are also sufficient. This includes in particular the case of the fine graded polynomial ring.

Corollary 3.15.
In the situation of Theorem 3.6, assume that ≤ 2 for all . Then the following two statements are equivalent for a formal Laurent series ∈ ℤ (( 1 , … , )): 1. There exists a finitely generated graded -module whose Hilbert series equals .

satisfies the following two conditions:
(a) ≥ 0, and . The hypothesis that ≤ 2 for all implies that every monomial of is squarefree. Hence its Newton polytope is a 0/1-polytope, so every lattice point in it is a vertex. In particular, all extremal monomials of are vertices of its Newton polytope, so the claim follows from Theorem 3.14. □

GENERAL INEQUALITIES FOR THE HILBERT DEPTH
In this section, we present a class of linear inequalities for the Hilbert series of modules with a given depth. We relax our assumptions on and allow it to be an arbitrary (commutative) -algebra with a positive ℤ -grading, such that dim 0 < ∞. The general idea is to compare the Hilbert series in question with all Hilbert series of modules from a certain class. An -module is called -th syzygy module if it can be realized as the -th syzygy of some -module ′ ; see [2] for alternative characterizations of syzygy modules.
The following is the main result of this section.
Theorem 4.1. Let = ⨁ be a commutative Noetherian ℤ -graded -algebra, such that dim 0 < ∞. Let ∈ ℕ and , be finitely generated -modules. If pdim ≤ and is a -th syzygy module, then the following inequality holds: Proof. Every free -module satisfies ⊗ = . This clearly holds for = ( ) with ∈ ℤ , and it is easily seen that the equality is preserved under direct sums.
Consider a free resolution of : We compute that On the other hand, it holds that Let ′ be an -module such that is the -th syzygy module of ′ . Then it holds for > 0 that Tor ( , ) = Tor + ( , ′ ) = 0 because pdim ≤ . We conclude that Let us consider some extremal cases of this theorem in the case that is the polynomial ring (with an arbitrary ℤ -grading).
1. If = dim , all -th syzygies modules are free. Hence (4.1) reduces to the statement that ≥ 0 for every -module .
2. For = 0, every module with pdim ≤ is free. On the other hand, let be a module satisfying the inequality (4.1) for all 0-th syzygy modules .
, then the free module has the same Hilbert series as . Hence, (4.1) exactly describes the Hilbert series of free modules.
In general, the inequalities (4.1) are not sufficient for a Hilbert series to have a given Hilbert depth, see for example part (2b) of Theorem 6.2. Nevertheless, in the next two sections, we consider two special situations where slightly stronger inequalities are indeed sufficient.
For later use, we also record a useful criterion for Hilbert series of modules of positive depth. … , )

) is the Hilbert series of a finitely generated -module of positive depth if and only if it can be written in the form
for Laurent polynomials The difference to Theorem 3.1 is that there is no term ∅ . We call a Hilbert decomposition as in Equation (4.2) a Hilbert decomposition without polynomial part. As in Theorem 3.1, this result is essentially contained in [10,Prop. 2.4], but there it is stated only for ℤ-graded rings; the proof in our context follows by the same method.

THE BIGRADED CASE
In this section we consider the ℤ 2 -graded situation. More precisely, let be the polynomial ring with a ℤ 2 -grading given by deg = (1, 0) for all ∈ [ ] and deg = (0, 1) for all ∈ [̃]. Specializing Theorem 3.6 to this situation we obtain the following characterization of Hilbert series over : Proposition 5.1. For a formal Laurent series ∈ ℤ (( 1 , 2 )), there exists a finitely generated graded -module with = if and only if satisfies the following conditions:  ) is a univariate polynomial, so its only extremal monomial is the leading one, which is also a vertex of its Newton polytope. Thus the claim follows from Theorem 3.14. □ Our next main result will be the characterization of Hilbert series of positive Hilbert depth over . For this we will consider certain pairs of sequences of indices in ℤ 2 .
The set of all declining sequences will be denoted by .
3. Although condition (ST) resembles its counterpart in the non-standard-ℤ-graded case, namely the condition (⋆) of Theorem 6.1 below, there are important differences: The inequalities required by condition (⋆) only relate coefficients lying within one common period of the module's Hilbert function, they have the same number of terms on both sides, and this number is bounded above by min{deg , deg }.
By contrast, a declining sequence may have any number of entries, which may be arbitrarily separated, and the right-hand side of (ST) has always one term more than the left-hand side. One might think that it could be possible to weaken condition (ST) by restricting it to a subset of  consisting of somehow bounded sequences, but this turns out to be a vain hope. For instance, the examples show that we cannot afford to restrict condition (ST) to those sequences = Before we present the proof of Theorem 5.3 we give several lemmata. First of all, it turns out to be convenient to consider a slightly larger class of inequalities. For this, we call a sequence then ( ) = ( ) ∧ ( +1) , so we may delete ( ) from . After finitely many such deletions, we obtain a declining sequence ′′ ∈  with ( ) = ′′ ( ). As ′′ ( ) ≥ 0 by assumption, the claim follows. □ The following lemma essentially reduces the question to the fine-graded situation. Recall that [ ] denotes the coefficient of in the series . Lemma 5.6. Assume that ∈ ℕ (( 1 , 2 )) admits a Hilbert decomposition without polynomial part. Then there exists an 0 ∈ ℤ 2 with the following property: For any ≥ 0 , there exists a decomposition = 1 + 2 such that Proof. Choose a Hilbert decomposition of without polynomial part: Choose 0 ∈ ℤ 2 which is strictly larger than the degrees of all monomials in the numerator polynomials (1,0) and (0,1) . By repeatedly using the relation we may modify the Hilbert decomposition such that it satisfies the following: • For every ∈ ℕ 2 , ≠ (0, 1), (1, 0) such that ≠ 0, it holds that contains no monomials of degree strictly less than .
• The polynomial 0) contains only monomials which are strictly less than . The third lemma is the key step in our proof of Theorem 5.3. Here we show how to decompose a nonnegative Laurent series of the form 0 + 1 ∕ ( so we may assume that only intersects with , . In this case it is not clear a priori whether the corresponding inequality still holds for̃, because only the right-hand side of the original inequality is diminished. (A similar problem occurs in [10], where the analogous inequalities are called critical.) Let ( ) = ( , ′ ) denote the intersection of and , . Since we assume that ( ) and , do not intersect, the sequence either ends in ( ) or the point ( ) ∧ ( +1) = ( , ′′ ) , and hence all further points of and ( ) lie in the half-plane {( , ) | < }. Since all coefficients of and̃in this half-plane vanish, we may assume that the staircase ends in ( ) . We amend bỹ( +1) ∶= ( , ) to build̃. Note that̃is weakly declining, hencẽ( ) ≥ 0 by Theorem 5.5. As ( ) = ( , ′ ) , it follows that ( ) ∧̃( +1) = ( , ) and thus Now we turn to the proof of the second claim. By the choice of the serieŝ=∶ ∑ ,ĥ 1 2 is nonnegative. Similar to the proof of the first claim the verification of condition (ST) for̂reduces to an inspection of sequences ∈  intersecting +1, ∶= {( , ) | ≥ + 1} in , but not in . Let ( ) = ( ′ , ) be this intersection; again we may assume that ends in ( ) . Let̃be the new weakly declining sequence obtained from by replacing the last element ( ) = ( ′ , ) bỹ( ) ∶= ( , ). We assumed that ( −1) ∧ ( ) ∉ +1, , hence it holds that ( −1) 1 ≤ and thus ( −1) ∧ ( ) = ( −1) ∧̃( ) . It follows that Now we are able to prove Theorem 5.3:

Proof of Theorem 5.3. (a) ⇒ (b):
Let be a finite torsionfree -module. After shifting the degrees, we may assume that all homogeneous components of have nonnegative degrees. Let ∈ ℤ 2 be a multidegree. By Theorem 5.6 we can find a decomposition = 1 + 2 , such that all coefficients of 2 in degrees less or equal than vanish. Hence ( ⋅ , ) .
This and (c) imply that ) .
Note that the coefficient of equals ( ), hence the latter is nonnegative. So satisfies (ST). (d) ⇒ (a): First, choose a decomposition = + ′ , such that is a polynomial with nonnegative coefficients and ′ has a Hilbert decomposition with no polynomial part. Let̃∈ ℤ 2 such that all non-zero coefficients of lie in degrees below . Again, using Theorem 5.6 there is ≥̃and a decomposition ′ = 1 + 2 , such that 1 and 2 satisfy the conditions mentioned above.
We are going to construct a Hilbert decomposition without polynomial part of 3 ∶= + 1 . As 2 already has such a Hilbert decomposition, this is enough to prove the claim. For this, we first need to show that 3 still satisfies (ST). Let = ( ( ) ) =0 ∈  be a declining sequence. Then ′ ∶= ( ( ) ∧ ) =0 is a weakly declining sequence. If ∈ ′ , then it is easy to see that Otherwise, our choice of and Theorem 5.6 imply that 3 for all ∈ ℤ 2 . Hence it follows that We obtain a Hilbert decomposition of 3 without polynomial part by repeatedly applying Theorem 5.7. □ With a little more work, one can show directly that condition (d) implies (c). Indeed, we already showed that every declining sequence gives rise to a fractional ideal. On the other hand, to every fractional ideal one can associate a declining sequence in a natural way, thus proving the equivalence directly.

THE NON-STANDARD ℤ-GRADED CASE
with the grading given by deg = and deg = for two coprime numbers , ∈ ℕ. In [10], the second and third author characterized the Hilbert series of modules of positive depth over this ring. In this section, we give a reformulation of this result along the lines of the previous results.
Denote by  , the set of all fundamental couples. The characterization of positive Hilbert depth over can be stated as follows:  . Then = 1∕ ( 1 − 1 ) clearly has Hilbert depth 1, but (b) In the setting of Theorem 6.1, let = 2, = 3 and consider the Hilbert series = 1 + 3 . Any module with this Hilbert series has finite length and therefore has Hilbert depth 0, but for any torsionfree -module it holds where for the last inequality, we use that multiplication by gives an injection (−2) → .
. Using that is torsionfree, it easily follows that Equation (6.1) holds for each summand and therefore for . It is easy to see that̃is the preimage of under the projection → . In particular, note that − ∈̃, because 0 , ∈̃. Hence ≅̃∕ ( − ) and thus =̃− . By considering the minimal free resolution of̃, one sees that its syzygies are generated in the degrees −1 + for 1 ≤ ≤ . So we can compute the Hilbert series of as follows: .
Together with Equation (6.1), we obtain the following:

E N D N O T E
1 Here we follow the terminology of [15]. In the literature concerned with the Stanley depth, the term "Hilbert depth" refers to a different invariant, see for example Bruns, Krattenthaler and Uliczka [4]. In the standard-ℤ-graded setup, these two notions coincide, but in general they are different.