## Geometric idealizer rings

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- by Susan J. Sierra PDF
- Trans. Amer. Math. Soc.
**363**(2011), 457-500 Request permission

## 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.

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## Additional Information

**Susan J. Sierra**- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- Address at time of publication: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350
- MR Author ID: 860198
- Email: sjsierra@math.washington.edu
- Received by editor(s): October 22, 2008
- Received by editor(s) in revised form: May 6, 2009
- Published electronically: August 23, 2010
- Additional Notes: This paper was completed as part of the author’s Ph.D. thesis at the University of Michigan, under the supervision of J. T. Stafford. The author was partially supported by NSF grants DMS-0802935, DMS-0555750, and DMS-0502170, and by a Rackham Pre-Doctoral Fellowship from the University of Michigan.
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**363**(2011), 457-500 - MSC (2000): Primary 16S38; Secondary 14L30
- DOI: https://doi.org/10.1090/S0002-9947-2010-05110-4
- MathSciNet review: 2719690