Fractional ideals and integration with respect to the generalised Euler characteristic

Let $b$ be a fractional ideal of a one-dimensional Cohen-Macaulay local ring $O$ containing a perfect field $k$. This paper is devoted to the study some $O$-modules associated with $b$. In addition, different motivic Poincar\'e series are introduced by considering ideal filtrations associated with $b$; the corresponding functional equations of these Poincar\'e series are also described.


Introduction
Canonical ideals over one-dimensional Cohen-Macaulay rings were profusely studied by Kunz, Herzog, et al., giving nice characterisations of the Gorenstein property: the formula bringing conductor and deltainvariant together, or relating it with symmetry properties of the value semigroup of the ring (cf. e.g. [10]; [8]). This idea was developed by Delgado for several branches of complex curve singularities (see [6]), and by Campillo, Delgado and Kiyek in a more general context (see [5]); they also introduced a Poincaré series P (t), deducing its functional equations when the ring is Gorenstein. Later, 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. [2], [3]). In fact, a motivic approach to P (t) was also introduced just by taking in the integral the generalised Euler characteristic instead. Recently, some connections with number-theoretical local zeta functions were founded by specialising to the case of finite fields (cf. [7]). In that paper, the authors introduce integrals with respect to the generalised Euler characteristic over fractional ideals of the ring. The aim of the actual work is to discuss more in detail such motivic integrals, which turn out to be Poincaré series of fractional ideals, and related objects; in particular, we extend some of the results concerning the "classical" series P (t) and the Gorenstein property contained in [8] and [5]. We will also use techniques of motivic integration.
The paper goes as follows. Section 2 is an introduction to the notion of canonical ideal. We define the dual of an ideal, characterise the self-dual ideals and show that it is indeed an extension of well-known properties of Gorenstein rings (Theorem (2.12)). Section 3 is devoted to study an analogous 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 will give a notion of the symmetry of S(b); the statements of Section 2 allow us to characterise the symmetry of S(b) (cf. Proposition (3.8)). In Section 4 we define a Poincaré series of motivic nature for the fractional ideal b (Definition (4.10)); we prove its rationality and also its functional equations in absence of the Gorenstein condition (Theorem (4.19)). Finally, in Section 5 we investigate the analogous of the extended semigroup S(O) of the ring O for a fractional ideal-it has again structure of S(O)-module-, as well as an alternative motivic Poincaré series associated with b, giving its functional equations as well (see Proposition (5.5), Theorem (5.8)). (2.2) Recall that a fractional ideal of a ring R is a R-submodule a = (0) of the total ring of quotients of R such that aa ⊆ R for some a ∈ O\{0}. Let us take a fractional ideal a of O. It is easy to see that a can be written uniquely as a product a = m v 1 1 · . . . · m vr r for v := (v 1 , . . . , v r ) ∈ Z r . We will denote m v := m v 1 1 · . . . · m vr r and v(a) := v 1 + . . . + v r .

Duality and fractional ideals
Notice that γ O = γ.
(2.9) Theorem: (Gorenstein; Apéry; Samuel; Herzog, Kunz) We have: Moreover, the equality holds if and only if the ring O is Gorenstein.
(2.10) The rest of the section is devoted to generalise Theorem (2.9) to any fractional ideal of O. The obvious task is to find a candidate to substitute the conductor ideal in the formula preserving such dimensions. It is easy now: . First of all note the following fact (cf. [12]): (2.11) Lemma: We have (2.12) Theorem: Let b be a fractional ideal of O. We have

Moreover, the equality holds if and only if b is self-dual.
Proof. Without loss of generality, let us assume that and plugging the last equality into (3) shows λ(b * /b) = 0, i.e., b * = b, hence the ideal b is self-dual.
(2.13) Remark: From Lemma (2.11) and Theorem (2.12) it follows: The ideal b is self-dual if and only if (2.14) Let v := (v 1 , . . . , v r ), 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 ∈ {1, . . . , r}, 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 or 0 if i ∈ I or i / ∈ I respectively. For any v ∈ Z r and any fractional ideal b in O, we define the set For every i ∈ {1, . . . , r}, let us define for every v ∈ Z r ; also the filtration is finitely determined, i.e., for any v ∈ Z r there exists N ∈ Z such that J b (v) ⊃ m N ; that means that every subspace J b (v) of O has finite codimension ℓ b (v) (cf. [4, p. 194]). Notice that, for every i ∈ {1, . . . , r} one has 0 ≤ c . . , r} (The proof of these facts follows much more [5]).
(2.15) The Gorenstein condition on the ring O was proven to be equivalent to the following equality (cf.
Our purpose now is to state the analogue of this result for the case of a fractional ideal. The proof is adapted from [5]. We show first: This map is clearly injective, so we have to prove that φ is not an epimorphism. Let i ∈ {1, . . . , r}. It is easily seen that which proves the non-surjectivity of φ i .
Proof. For every v ∈ Z r and every i ∈ {1, . . . , r}, 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 (2.17) holds if and only if b is self-dual. This follows by the same method as in the proof of [5,Theorem (3.6)], just by applying the characterisation of the self-dual fractional ideals provided by Theorem (2.12) instead of using Theorem (2.9).
and then By Proposition (2.17) this expression is smaller than or equal to and by Theorem (2.12), and the assumption b · O = O, we are done. The converse follows in the same manner as in the proof of [5, (3.6)], part (d).
The following statements are equivalent: Proof. If the equality * holds for every v ∈ Z r and for every i ∈ {1, . . . , r}, then one can choose a strictly increasing sequence as in Lemma (2.18), and we obtain that which by Theorem (2.17) implies the statement.

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 S(O)-module and only depends on the ideal class of b in the ideal class semigroup of O. The elements of S(b) are connected to filtrations {J b (v)} in the following sense: The next two lemmas show that γ b plays the role of a kind of conductor of the set S(b).
(3.4) Lemma: We have: (3.5) In the rest of the section, the ring O will be assumed to be residually rational, i.e., k = k i for every i ∈ {1, . . . , r}.
where t v := t v 1 1 · . . . · t vr 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:  We can also choose other measures than χ, for instance the socalled 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: and taking the fibred product as multiplication:   (4.7) As in [7], 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 zX 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 In particular, χ g (m p+1 ) = L −d(p) .
(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.
We define now the generalised Poincaré series of the projectivisation of the fractional ideal b: then we get (4.13) Lemma: Proof. The result may be proved in much the same way as in [4,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 (see [12] and [13]). We include the proofs by the sack of completeness.
(4.14) Lemma: Let a, b be fractional ideals of O such that a ⊇ b.
We have Proof. By definition of the ideal c we have c :  The fractional ideals a of O are of the form t −n · b, where t n := t n 1 1 · . . . · t nr r for some n = (n 1 , . . . , n r ) ∈ Z r and being b a Proof. (a) Since a is a fractional ideal of O, aO is a fractional ideal of O. By Remark 2, we have that aO must be of the form m n 1 1 · . . . · m nr r for some n ∈ Z r . That is, i.e., t −n · a · O = O for some n ∈ Z r . Then, it suffices to take b = t −n · a and the claim follows. If we multiply the ideal c by a convenient element of K, then we may assume c : Next proposition relates the degree of the ideal J b * (v) and the value γ b .
(4.16) Proposition: For every v ∈ Z r , we have Proof. The second equality holds by Lemma (2.11). Moreover, since for v ∈ Z r , by Lemma (4.14) and Lemma (4.15) it follows that The definition of γ b allows us now to conclude.
Proof. It is just to apply Proposition  Proposition (4.16) allows us to describe the functional equations for the generalised Poincaré series: (4.19) Theorem: We apply now Proposition (4.16) to obtain Moreover, taking the inverse of t, we have (4.20) Corollary: Furthermore, if O is Gorenstein, then we obtain Proof. It is a straight consequence of Corollary (4.20) and Corollary (4.17).

Extended generalised value ideal Poincaré series
Let O be a one-dimensional Cohen-Macaulay local Noetherian ring having a perfect coefficient field K. Campillo, Delgado and Gusein-Zade introduced in [1] the notion of extended semigroup of a germ of complex plane curve singularity. We want now to define the concept of extended value ideal of a fractional ideal b. Let us preserve notations as in (2.1). Remember that the ideal m j V j is regular maximal of V j and V j = O m j for every j ∈ {1, . . . , r}.
If K j is a coefficient field of V j and t j is an indeterminate over K j , then one can identify and v j with the order function respect to Since V j is an O-module of finite type, the field extensions O/m ֒→ O/m j are finite for every j ∈ {1, . . . r}. Furthermore, as O/m is assumed to be perfect, every such a extension is separable and therefore, for every coefficient field K of O there exists a unique coefficient field K j of V j with K ⊂ K j which is isomorphic to O/m j for every j ∈ {1, . . . , r}.
Let us consider the vector spaces . . , r} and the map: We can identify Im Proof. It is enough to define an isomorphism ϕ v : with a 1 (z) ∈ K * 1 , thus ϕ v can be defined by z → a 1 (z).
(5.2) For every v ∈ Z r we write i.e., F b v is the complement to an arrangement of vector subspaces in a vector space (it is not a vector subspace itself). Notice that this is precisely the space (2) in (4.2). (5.4) The group K * of non-zero elements of K acts freely on Z r ×(K * 1 × . . . × K * r ) (by multiplication of all coordinates in K * 1 × . . . × K * r ). The is invariant with respect to the K * -action. The factor space P S(b) = S(b)/K * is called the projectivisation of S(b) (it is also a graded S(O)-module in a natural sense).
From the previous definitions, the projectivisation of S(b) can be described as Set the Laurent series On the other hand, we also have the extended generalised value ideal Poincaré series of a filtration {J b (v)} defined by v(z) = (v 1 (z), . . . , v r (z)), for z ∈ b: t v(z) dχ g (we will use P g (b, t, L) instead of P g (b, t, L, {v i }) when the filtration is clear from the context). Notice that if b = O, then P g (O, t, L) coincides with the generalised semigroup Poincaré series defined in [4, p. 507].
All projectivisations PF b v of the fibres F b v (i.e., all connected components of P S(b)) are complements to arrangements of projective subspaces in finite dimensional projective spaces. We define (5.5) Proposition: Proof. Let w ∈ Z r and set L I := {(a 1 , . . . , a r ) ∈ K r | a i = 0 for i ∈ I}. Then The coefficient of t v in the polynomial   v∈Z r is equal to and the latter formula coincides with ( * ).
(5.6) Remark: Since L − 1 is invertible in K 0 (ν k ) (L) , the extended generalised value ideal Poincaré series can be rewritten as We generalise this functional equation by using the following result: (5.7) Proposition: For every v ∈ Z r , we have Proof. By means of Proposition (4.16) we deduce the equality: We want to describe functional equations for the series L g (b, t, L) and the Poincaré series P g (b, t, L).