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Semicontinuity for the local Hilbert function. (English) Zbl 0595.13014
Let X be a noetherian scheme, and let ($${\mathfrak O,m)}$$ be the local ring of X at a point x. The Samuel and Hilbert functions are defined by $$S_{X,x}(n)=length {\mathfrak O}/{\mathfrak m}^{n+1}$$, $$H_{X,x}(n)=length {\mathfrak m}^ n/{\mathfrak m}^{n+1}$$. For every function $$f:\quad {\mathbb{Z}}^+\to {\mathbb{Z}}$$ one defines: $$(\Delta f)(n)=f(n)-f(n-1)$$ for $$n\geq 1$$, $$(\Delta f)(0)=f(0)$$. Our purpose it to prove the following semicontinuity theorem: Let $${\mathfrak O}$$ be a local excellent ring, $$X=Spec {\mathfrak O}$$ and let x be the closed point of X. If y is the generic point of an r-dimensional integral subscheme of X, then $$S_{X,y}(n)\leq \Delta^ rS_{X,x}(n)$$. - Moreover, we prove that the Hilbert function stabilizes in any sequence of permissible blowing-ups.

##### MSC:
 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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##### References:
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