Mathematics – Rings and Algebras
Scientific paper
2008-09-23
Trans. Amer. Math. Soc. 363 (2011), 457-500
Mathematics
Rings and Algebras
43 pages; comments welcome
Scientific paper
10.1090/S0002-9947-2010-05110-4
Let X be a projective variety, $\sigma$ an automorphism of X, L a $\sigma$-ample invertible sheaf on X, and Z a closed subscheme of X. Inside the twisted homogeneous coordinate ring $B = B(X, L, \sigma)$, 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 $\sigma$, 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 $\sigma$ that determine the algebraic properties of R, and show that if Z and $\sigma$ 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 $\chi_d$ (where d = \codim Z) but fails left $\chi_1$. 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 generalizes results of Rogalski in the case that Z is a point in $\mathbb{P}^d$.
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