BibTeX
@MISC{Sierra_geometricidealizers,
author = {S. J. Sierra},
title = {Geometric idealizers},
year = {}
}
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Abstract
Abstract. Let X be a projective variety, σ an automorphism of X, L a σ-ample invertible sheaf on X, and Z a closed subscheme of X. Inside the twisted homogeneous coordinate ring B = B(X, L, σ), let I be the right ideal of sections vanishing at Z. We study the subring R = k + I of B. Under mild conditions on Z and σ, R is the idealizer of I in B: the maximal subring of B in which I is a two-sided ideal. We give geometric conditions on Z and σ that determine the algebraic properties of R, and show that if Z and σ are sufficiently general, in a sense we make precise, then R is left and right noetherian, has finite left and right cohomological dimension, is strongly right noetherian but not strongly left noetherian, and satisfies right χd (where d = codim Z) but fails left χ1. We also give an example of a right noetherian ring with infinite right cohomological dimension, partially answering a question of Stafford and Van den Bergh. This
Keyphrases
geometric idealizers right noetherian right ideal twisted homogeneous coordinate ring geometric condition algebraic property ample invertible sheaf closed subscheme projective variety infinite right cohomological dimension two-sided ideal maximal subring right cohomological dimension mild condition
