Geometric idealizer rings

Abstract

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 $|n| \gg 0$ 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.