  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 $\mathbb{P}^d$.
