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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

##### MSC:
 14A10 Varieties and morphisms 13H15 Multiplicity theory and related topics 14E15 Global theory and resolution of singularities (algebro-geometric aspects)
##### Keywords:
Samuel polynomial; Hilbert polynomial; blowing up
##### Citations:
Zbl 0218.14001; Zbl 0118.362; Zbl 0122.161
Full Text:
##### References:
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