Geometric idealizer rings

Let $B = B(X, \mathcal{L}, \sigma)$ be the twisted homogeneous coordinate ring of a projective variety $X$ over an algebraically closed field $\Bbbk$. We construct and investigate a large class of interesting and highly noncommutative noetherian subrings of $B$. Specifically, let $Z$ be a closed subscheme of $X$ and let $I \subseteq B$ be the corresponding right ideal of $B$. We study the subalgebra

$$R = k + I$$

of $B$. Under mild conditions on $Z$ and $\sigma \in \operatorname{Aut}_{\Bbbk}(X)$, $R$ is the idealizer of $I$ in $B$: the maximal subring of $B$ in which $I$ is a two-sided ideal.

Our main result gives geometric conditions on $Z$ and $\sigma$ that determine the algebraic properties of $R$. We say that

$$\{\sigma^n(Z)\}$$

is critically transverse if for any closed subscheme $Y$ of $Z$, for $\vert n\vert \gg 0$ the subschemes $Y$ and $\sigma^n(Z)$ are homologically transverse. We show that if $\{\sigma^n(Z)\}$ is critically transverse, 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 $d = \operatorname{codim} Z$) but fails left $\chi_1$. This generalizes results of Rogalski in the case that $Z$ is a point in $\mathbb{P}^d$. 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.

Further, we study the geometry of critical transversality and show that it is often generic behavior, in a sense that we make precise.