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Fractional ideals and integration with respect to the generalised Euler characteristic. (English) Zbl 1321.14026

Let \((\mathcal O,\mathfrak m)\) be a \(1\)-dimensional local analytically unramified CM-ring which contains a perfect field \(k\), let \(\overline{\mathcal O}\) be the integral closure of \(\mathcal O\), a finitely generated \(\mathcal O\)-module, and assume that \(\rho:=[\mathcal O/\mathfrak m:k]\) is finite. Let \(\mathfrak f=(\mathcal O:\overline{\mathcal O})\) be the conductor, and let \(\mathfrak c\) be a canonical ideal of \(\mathcal O\); for any fractionary ideal \(\mathfrak a\) of \(\mathcal O\) call \(\mathfrak a^*:=(\mathfrak c:\mathfrak a)\) the dual of \(\mathfrak a\), and call \(\mathfrak a\) self-dual if \(\mathfrak a^{**}=\mathfrak a\). Since \(\mathcal O\) is Gorenstein iff \(\mathcal O\) is a canonical module, \(\mathcal O\) is Gorenstein iff \(\mathcal O\) is self-dual.
It is well known that \(2\text{dim}_k(\mathcal O/\mathfrak f)= \text{dim}_k(\overline{\mathcal O}/\mathfrak f)\) iff \(\mathcal O\) is Gorenstein. Generalizing this result, the author proves that for any fractionary ideal \(\mathfrak b\) of \(\mathcal O\) one has \(2\text{dim}_k(\mathfrak b/\mathfrak b:\overline{\mathcal O})=\text{dim}_k(\mathfrak b\cdot \overline{\mathcal O}/\mathfrak b:\overline{\mathcal O})\) iff \(\mathfrak b\) is selfdual.
Using the Manis valuations \(v_1,\ldots,v_r\) of \(\mathcal O\), the author proves another criterion for a fractionary ideal to be self-dual [Theorem (2.18)].
Let \(\mathcal O\) be residually rational and \(k\) be an infinite field. In [A. Campillo et al., Manuscr. Math. 83, No. 3–4, 405–423 (1994; Zbl 0822.13011)] it was shown that the semigroup \(S(\mathcal O)\subset\mathbb{Z}^r\) is symmetric iff \(\mathcal O\) is Gorenstein. A fractionary \(\mathcal O\)-ideal gives rise to an \(S(\mathcal O)\)-module \(S(\mathfrak b)\); in Prop.(3.8) it is shown that \(S(\mathfrak b)\) is symmetric iff \(\mathfrak b\) is self-dual.
Using the valuations \(v_1,\ldots,v_r\), any fractionary \(\mathcal O\)-deal \(\mathfrak b\) gives rise to a filtration \(\{J^{\mathfrak b}(v_1,\ldots,v_r)\}\) of \(\mathcal O\). Using this, the author associates to \(\mathfrak b\) a Poincaré series of motivic nature; he proves its rationality and also a functional equation. The notion of extended semigroup \(\widehat S(\mathcal O)\) was intoduced in [A. Campillo et al., Duke Math. J. 117, No. 1, 125–156 (2003; Zbl 1028.32013)]; in section 5 the author introduces for a fractionary ideal \(\mathfrak b\) similarly an \(\widehat S(\mathcal O)\)-module \(\widehat S(\mathfrak b)\), associates to it a motivic Poincaré series and proves a functional equation. These Poincaré series are generalizations of the Poincaré series introduced in first paper referred to above. If \(\mathfrak b=\mathcal O\), then this series is the Poincaré series introduced in [A. Campillo et al., Monatsh. Math. 150, No. 3, 193–209 (2007; Zbl 1111.14020)].

MSC:

14H20 Singularities of curves, local rings
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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References:

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