Geometric idealizers

TL;DR

This paper investigates the algebraic and geometric properties of a subring R derived from a twisted homogeneous coordinate ring, focusing on conditions that influence its noetherianity, cohomological dimensions, and other algebraic features.

Contribution

It provides new geometric criteria for the algebraic properties of idealizer subrings in twisted homogeneous coordinate rings, extending previous results to more general subschemes Z.

Findings

R is often left and right noetherian under general conditions.

R has finite cohomological dimension and is strongly right noetherian.

Counterexamples show R can have infinite right cohomological dimension.

Abstract

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 $\mathbb{P}^d$.