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Zbl 0625.14001
Muñoz Porras, J.M.; Sancho de Salas, J.B.
The last coefficient of the Samuel polynomial. (English)
[J] Compos. Math. 61, 171-179 (1987). ISSN 0010-437X; ISSN 1570-5846/e

Let X be a Noetherian scheme, proper over an Artinian ring, and let I be a coherent ideal of ${\cal O}\sb X$. Let $\pi: \bar X\to X$ be the blowing up of X along I. Then it is well known that the Samuel function $S\sb I=\chi (X,{\cal O}\sb X/I\sp n)$ 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 {\it A. Grothendieck}] and SGA 6 $[=$ Sém. Géom. algébr. 1966/67, Lect. Notes Math. 225 (1971)] the authors compute the last coefficient of $S\sb I(n)$, for $n\gg 0$, as the difference $\chi(X,{\cal O}\sb X) - \chi(\bar X,{\cal O}\sb{\bar X})$ in the Euler characteristics; see theorem 2.4, 3.2) and also theorem 2.6 where I is supposed to be an ${\frak m}\sb x$-primary ideal in (${\cal O}\sb{X,x},{\frak m}\sb x)$ for a closed point $x\in X$.
[M.Herrmann]

MSC 2000:: *14A10 Varieties
13H15 Multiplicity theory and related topics
14E15 Global theory of singularities

Keywords: Samuel polynomial; Hilbert polynomial; blowing up

Citations: Zbl 0218.14001; Zbl 0118.362; Zbl 0122.161

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