DEPTH OF HIGHER ASSOCIATED GRADED RINGS
Published online by Cambridge University Press: 23 July 2004
Abstract
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 $I^n$ is asymptotically constant when $n$ tends to infinity, and this value is characterized in terms of Valabrega–Valla conditions of $I^m$ for some large integer $m\ge 0$. 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 ${\bf Spec}(R)$ along the closed sub-scheme defined by $I$.
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- The London Mathematical Society 2004
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