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Journal of the London Mathematical Society 2004 70(1):41-58; doi:10.1112/S0024610704005332
© 2004 by London Mathematical Society
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© The London Mathematical Society

Depth of Higher Associated Graded Rings

Juan Elias

Departament d'Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona Gran Via 585, 08007 Barcelona, Spain, elias{at}mat.ub.es

Received 19 May 2003.

The depth of the associated graded ring of the powers of an ideal I of a local ring R is studied. It is proved that the depth of the associated graded ring of In is asymptotically constant when n tends to infinity, and this value is characterized in terms of Valabrega–Valla conditions of Im for some large integer m ≥ 0. As a corollary, a generalization is obtained of the 2-dimensional algebraic version of the Grauert–Riemenschneider vanishing theorem (due to Huckaba and Huneke) to ideals satisfying the second Valabrega–Valla condition. The positiveness of Hilbert coefficients is also studied, and Valabrega–Valla conditions are linked to the vanishing of the cohomology groups of the closed fiber of the blowing up of Spec(R) along the closed sub-scheme defined by I.


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