Geometric idealizers

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 ${\mathbb{P}}^d$.

43 pages; comments welcome

References in corpus (1)

• The dynamical Mordell-Lang problem for etale maps

Cited by in corpus (5)

• Some noncommutative projective surfaces of GK-dimension 4
• On a Dynamical Mordell-Lang Conjecture for Coherent Sheaves
• A Skolem-Mahler-Lech theorem for iterated automorphisms of $K$-algebras
• Idealisers in Skew Group Rings
• Generalised Witt algebras and idealizers