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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.

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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[1] Bennet, B.M.: On the characteristic function of a local ring. Ann. Math.91, 25-87 (1970) · Zbl 0198.06101 · doi:10.2307/1970601
[2] Grothendieck, A., Dieudonné, J.: E.G.A. IV, Publ. Math. IHES no 32, (1967)
[3] Hartshorne, R.: Connectedness of the Hilbert scheme. Publ. Math. IHES no 29 (1966) · Zbl 0171.41502
[4] Hironaka, H.: Resolution of singularities of an algebraic variety of characteristic zero. Ann. Math.79, 109-236 (1964) · Zbl 0122.38603 · doi:10.2307/1970486
[5] Singh, B.: Effect of a permissible blowing-up on the local Hilbert functions. Invent. Math.26, 201-212 (1974) · Zbl 0277.14004 · doi:10.1007/BF01418949
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