@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