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The last coefficient of the Samuel polynomial. (English) Zbl 0625.14001
Let X be a Noetherian scheme, proper over an Artinian ring, and let I be a coherent ideal of \({\mathcal O}_ X\). Let \(\pi: \bar X\to X\) be the blowing up of X along I. Then it is well known that the Samuel function \(S_ I=\chi (X,{\mathcal O}_ X/I^ n)\) is a polynomial in n for \(n\gg 0\) and that every coefficient of this polynomial, except the last one, can be expressed in terms of the exceptional divisor of \(\pi\). Using standard methods from EGA III \([=\) Éléments de géométrie algébrique. III, Publ. Math., Inst. Hautes Étud. Sci. 11 (1962; Zbl 0118.362) and 17 (1963; Zbl 0122.161) by A. Grothendieck] and SGA 6 \([=\) Sém. Géom. algébr. 1966/67, Lect. Notes Math. 225 (1971)] the authors compute the last coefficient of \(S_ I(n)\), for \(n\gg 0\), as the difference \(\chi(X,{\mathcal O}_ X) - \chi(\bar X,{\mathcal O}_{\bar X})\) in the Euler characteristics; see theorem 2.4, 3.2) and also theorem 2.6 where I is supposed to be an \({\mathfrak m}_ x\)-primary ideal in (\({\mathcal O}_{X,x},{\mathfrak m}_ x)\) for a closed point \(x\in X\).
Reviewer: M.Herrmann

14A10 Varieties and morphisms
13H15 Multiplicity theory and related topics
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
Full Text: Numdam EuDML
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