Fractional ideals and integration with respect to the generalised Euler characteristic

Let b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {b}$$\end{document} be a fractional ideal of a one-dimensional Cohen–Macaulay local ring containing a perfect field. This paper is devoted to the study of the motivic Poincaré series defined by different filtrations associated with b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {b}$$\end{document} in the form of Euler integrals with respect to the generalised Euler characteristic; in particular, the functional equations of the Poincaré series are also described.

together, or the relation between being Gorenstein and the symmetry property of the semigroup of values of the ring. This idea was developed by Delgado [8] for complex curve singularities with several branches, and in a more general framework by Campillo, Delgado and Kiyek [7]; they also introduced a Poincaré series P(t) for certain ideal filtration associated to the singularity, and deduced its functional equation for Gorenstein rings. Later on, Campillo, Delgado and Gusein-Zade showed that this Poincaré series coincides with the integral over the projectivisation of the ring with respect to the Euler characteristic (cf. [4,5]). A motivic approach to P(t) was also introduced just by taking in the integral the generalised Euler characteristic instead, see e.g., Campillo, Delgado and Gusein-Zade [6], or the recent results by the author [13].
(1.2) Recently, some connections with number-theoretical local zeta functions were found by Delgado and the author [9]; see also Zúñiga and the author [14]. The paper [9] introduces integrals with respect to the generalised Euler characteristic over fractional ideals of the ring; this goes back to some constructions involving fractional ideals by Stöhr [15,16]. The aim of the present work is to investigate those motivic integrals over fractional ideals: they turn out to be Poincaré series related to the "classical" series P(t) mentioned above. We also formulate their functional equations in the absence of the Gorenstein condition; in particular we obtain functional equations for the generalised Poincaré series defined in [6]. (1. 3) The paper goes as follows. Section 2 is an introduction to the notion of canonical ideal. We define the dual of a fractional ideal and characterise the self-dual ideals; in fact, this phenomenon is a generalisation of well-known properties of Gorenstein rings, cf. Theorem (2.11). Section 3 is devoted to the study of an analogue of the value semigroup S(O) of a one-dimensional Cohen-Macaulay local ring O for a fractional ideal b: the resulting set S(b) is no longer a semigroup, but it has structure of module over S (O). We also give a notion of the symmetry of S(b), and we characterise it by means of the statements in Sect. (2), cf. Proposition (3.8). In Sect. 4 we define a Poincaré series of motivic nature for the fractional ideal b, cf. Definition (4.10); we prove its rationality and also its functional equations in absence of the Gorenstein condition, see Theorem (4.19). Finally, in Sect. 5 we investigate the setŜ(b), i.e., the analogue of the extended semigroupŜ(O) of the ring O for a fractional ideal b-it has again structure ofŜ(O)-module-, as well as an alternative motivic Poincaré series associated with b, and its corresponding functional equations, cf. Proposition (5.5), Theorem (5.8).
(1.4) Set N 0 := {0, 1, 2, . . .} and I 0 := {1, 2, . . . , r } N 0 . We will denote by R × the set of units of a ring R. In particular, if R is a field, we have R × = R\{0}. Recall that a fractional ideal of R is an R-submodule a = (0) of the total ring of fractions of R such that aa ⊆ R for some regular a ∈ R. Notice also that the fractional ideals we are going to deal with are always regular. For a general reference on the topics discussed here we refer the reader to Campillo, Delgado, and Gusein-Zade [3], Campillo, Delgado, and Kiyek [7], Herzog and Kunz [10], and the book of Kiyek and Vicente [11].
If m(V i ) denotes the maximal ideal of V i for every i ∈ I 0 , then the ideals m i := m(V i ) ∩ O are principal, regular and maximal (cf. [11,Theorem II.(2.11) In case of d i = 1 for all i ∈ I 0 , the ring O is said to be residually rational. Write also d := d 1 + · · · + d r .
(2.2) Let a be a regular fractional ideal of O. It is easily seen that a can be written uniquely as a product a = m v 1 1 · · · m v r r for v := (v 1 , . . . , v r ) ∈ Z r . Set m v := m v 1 1 · · · m v r r and v(a) := v 1 + · · · + v r .

Moreover, the equality holds if and only if the ring O is Gorenstein.
(2.9) The rest of the section is devoted to generalise Theorem (2.8) to any fractional ideal of O. The first obvious task is to search for a candidate to substitute the conductor ideal in the formula preserving the dimensions above. This is not difficult: [16,Prop. 4.1]). The following fact is also remarkable (see again [16]): (2.11) Theorem Let b be a fractional ideal of O. The equality holds if and only if b is self-dual.
Proof Without loss of generality, let us assume by Lemma (2.10), and a substitution above let us finish. Conversely, assuming the following equalities hold: again looking at ( †) we get Substituting and plugging the last equality into (3) one gets (2.13) We need to introduce here some notations to be used in the paper. Let v := (v 1 , . . . , v r ) and w := (w 1 , . . . , w r ) be vectors in Z r . We will write v ≥ w if and only if v i ≥ w i for every i ∈ I 0 , 0 := (0, 0, . . . , 0) ∈ Z r and 1 := (1, 1, . . . , 1) ∈ Z r . Moreover, for every subset I ⊆ I 0 = {1, . . . , r }, let I be the number of elements in I , and let 1 I be the element of Z r whose ith component is equal to 1 resp. 0 if i ∈ I resp. i / ∈ I . For any v ∈ Z r and any fractional ideal b in O, we define the set thus the filtration is finitely determined, i.e., for any v ∈ Z r there exists [6, p. 194]). Notice that, for every i = 1, . . . , r one has The proof of these facts follows much more [7]).
(2.14) The Gorenstein condition on the ring O was proven to be equivalent to the following equality (cf. [7, Corollary (3.7)]): Our purpose now is to state the analogue of this result for the case of a fractional ideal. The proof is adapted from [7]. We show first: This map is clearly injective, so we have to prove that φ is not an epimorphism. It is easily seen that which proves the non-surjectivity of φ i .

(2.16) Proposition
Let b be a fractional ideal. For every v ∈ Z r and for every i ∈ I 0 one has: Proof For every v ∈ Z r and every i ∈ I 0 , the map η v,i : Notice also that for every n, m ∈ Z r the following inclusion holds: .
It remains to show that the equality of Proposition (2.16) holds if and only if b is self-dual. This follows by the same method as in the proof of [7, Theorem (3.6)], just by applying the characterisation of the self-dual fractional ideals provided by Theorem (2.11) instead of using Theorem (2.8).
and then By Proposition (2.16) this expression is smaller than or equal to where the latter two identities follow from the Chinese Remainder Theorem (see also the book of Campillo The application of Theorem (2.11) allows us to conclude. The converse follows in the same manner as in the proof of [7, (3.6)], part (d).
The following statements are equivalent: Proof If the equality ( * ) holds for every v ∈ Z r and for every i ∈ I 0 , then one can choose a strictly increasing sequence as in Lemma (2.17), and we obtain that , which by Theorem (2.11) implies the statement.
Analogously as in [7, (3.7)] one shows: Moreover, b is self-dual if and only if the equality holds for every v ∈ Z r .

The value ideal of a fractional ideal
Let b be a fractional ideal of O. Consider the set This is a subsemigroup of Z r which is in fact a module over the semigroup S(O)for this reason we will speak about semimodule-and it only depends on the ideal class of b in the ideal class semigroup of O.
(3.1) For every v = (v 1 , . . . , v r ) ∈ Z r and for every i ∈ I 0 , we define = 0 for every i ∈ I 0 , and choose an element Since k is infinite and k = k i for every i ∈ I 0 , there exist elements a 1 , . . . , a r ∈ O such that v i (a 1 z 1 + · · · + a r z r ) = v i for every i ∈ I 0 , i.e., v(a 1 z 1 + · · · + a r z r ) = v. For the last assertion, consider a, b This implies c b (v, i) ≤ 1 and by the first assertion this must be exactly 1.
The next two lemmas show that γ b plays the role of the conductor of the set S(b).

(3.5) Lemma
We have: (3.6) In the rest of the section, the ring O is assumed to be residually rational.

(3.8) Proposition
Let O be residually rational. We have There exists a vector w = (w 1 , . . . , w r ) ∈ Z r with (w) = ∅, w i = v i and w j < v j for every j ∈ I 0 , j = i. Since S(b) is symmetric, where t v := t v 1 1 · · · t v r r . We will write L(b, t) instead of L(b, t 1 , . . . , t r ) if the number of variables is clear from the context.

(4.2)
There is a priori no fixed way to choose a suitable coefficient in L(b, t). We may consider the following spaces: (1)

(4.3)
We can also choose other measures than χ , for instance the so-called generalised Euler characteristic χ g . It is a sort of motivic Euler characteristic which makes use of the notion of Grothendieck ring. The Grothendieck ring K 0 (ν k ) is defined to be the free Abelian group on isomorphism classes [X ] of quasi-projective schemes X of finite type over k subject to the following relations: for a closed subscheme Z of X ∈ ν k ; and taking the fibred product as multiplication:

(4.4) Let k[T ]
be the polynomial ring in one indeterminate T over the field k. The affine scheme Spec(k[T ]) over k is the affine line over k, which will be denoted by A 1 k . The class of the affine line in K 0 (ν k ), denoted by L, is called the Lefschetz class of K 0 (ν k ).
(4.6) The generalised Euler characteristic χ g (X ) of a cylindric subset (4.7) As in the paper of Delgado and the author [9], we can extend these definitions to subsets of K (in particular to fractional ideals): A subset X ⊆ K is called cylindric if there exists a non-zero divisor element z ∈ O such that the set z X is a subset of O and is cylindric. In this situation, the generalised Euler characteristic is Let a ⊆ O be an ideal of O. Since a is m-primary, we have m p+1 ⊆ a. Let a be the ideal a/m p+1 of O/m p+1 so that π −1 p (a) = a. As O/m p+1 is a finite-dimensional k-vector space, the ideal a is constructible. Then a is cylindric and we get were deg(a) denotes the degree of a; the degree of a fractional O-ideal is defined by the following two properties: (1) (4.8) Let G be an abelian group with countable many values. Let X be a cylindric subset of K. A function ψ : X → G is called cylindric if the set ψ −1 (a) ⊆ K is cylindric for all a ∈ G\{0}. The integral of ψ over X with respect to the generalised Euler characteristic is if this sum makes sense in K 0 (ν k ) (L) ⊗ Z G; in such a case, the function ψ is said to be integrable.
is considered as a (cylindric) function on PO with values in Z[[t 1 , . . . , t r ]] (the entry v i (z) is supposed to be 0 as soon as v i (z) = ∞ for i ∈ I 0 ).
then we get (4.13) Lemma Proof The result may be proved in much the same way as in [6, Proposition 2].
We describe now the functional equations for the series P g (b, t, L). First of all, we state the following two results, due to Stöhr [15,16].
The next proposition relates the degree of the ideal J b * (v) and the value γ b .
Proof The second equality holds by Lemma (2.10). Moreover, since for v ∈ Z r , by Lemmas (4.14) and (4.15) it follows that The definition of γ b allows us to conclude.
. As a consequence of Proposition (4.16) we obtain the following result due to Zúñiga and the author [14, Lemma 9]: Proof It is just to apply Proposition (4.16) to b = O. Notice that O is Gorenstein (cf. (2.6)).
Proposition (4.16) allows us to describe the functional equations for the generalised Poincaré series: Using Proposition (4.16) it is easily seen that Moreover, taking the inverse t −1 of t, we deduce the equality and therefore we obtain (4.20) Corollary · P g (O, t −1 , L).

(4.22) Corollary
If O is both residually rational and Gorenstein, then we have  (a 1 , . . . , a r ) whose kernel is just the subspace J b (v + 1).

(5.4) For every v ∈ Z r and every
for every i ∈ I 0 ; in this way we obtain Let v ∈ Z r . The purpose is to consider the inital forms of those elements in O having value exactly v, and to this aim it is useful to study the vector space C b (v) by removing the coordinate hyperplanes. In doing so we define This is in fact equivalent to (5.5) If we attach F b v to each element of the semimodule S(b) we obtain the extended semimodule associated to b:Ŝ The spaces F b v are called fibres of the extended semimoduleŜ(b). For a fixed value v, the extended semimodule measures which initial forms reach that level. Note that, if b = O, thenŜ(O) is the extended semigroup of the ring O introduced by Campillo, Delgado and Gusein-Zade in [3].
in particular, F v is not a vector subspace itself.
(5.6) The fibre F b v admits a free K × -action-namely, multiplication by a nonzero element of K -and therefore the projective arrangement PF b v := F b v /K × can be defined. This allows us to consider the projectivisation of the extended semimodule as which is also a graded semimodule in a natural sense. For v ∈Ŝ(b), the space PF b v is the complement to an arrangement of projective hyperplanes in a projective space . So it makes sense to consider the Laurent series On the other hand, one can also define the extended generalised semimodule Poincaré series of a filtration All projectivisations PF b v of the fibres F b v (i.e., all connected components of PŜ(b)) are complements to arrangements of projective subspaces in finite dimensional projective spaces. We definê

(5.7) Proposition
Proof Let I ⊆ I 0 and set L I := {(a 1 , . . . , a r ) ∈ K r | a i = 0 for i ∈ I }. Then Therefore (t 1 · · · t r − 1)χ g (PF b v ) = v∈Z r I ⊆I 0 The coefficient of t v in the polynomial is equal to and the latter formula coincides with ( * ).

(5.8) Remark
Since L − 1 is invertible in K 0 (ν k ) (L) , the extended generalised semimodule Poincaré series can be rewritten aŝ Observe the following equality: (5.9) Proposition For every v ∈ Z r , we have Proof From Proposition (4.16) we deduce the equalities Proposition (5.9) allows us to describe functional equations for the seriesL g (b, t, L) and the Poincaré seriesP g (b, t, L), which constitutes the closing result of the paper.