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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Geometric idealizer rings


Author: Susan J. Sierra
Journal: Trans. Amer. Math. Soc. 363 (2011), 457-500
MSC (2000): Primary 16S38; Secondary 14L30
Published electronically: August 23, 2010
MathSciNet review: 2719690
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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

$\displaystyle R = k + I$

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

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

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.


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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
Email: sjsierra@math.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05110-4
PII: S 0002-9947(2010)05110-4
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.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.