Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p
Abstract “Even more so is the word ‘crystalline’, a glacial and impersonal concept of his which disdains viewing existence from a single portion of time and space” Eileen Myles, “The Importance of Being Iceland” For a smooth variety $X$ over an algebraically closed field of characteristic $p$ to a d...
Published in: | International Mathematics Research Notices |
---|---|
Main Authors: | , |
Format: | Article in Journal/Newspaper |
Language: | English |
Published: |
Oxford University Press (OUP)
2019
|
Subjects: | |
Online Access: | http://dx.doi.org/10.1093/imrn/rnz169 https://academic.oup.com/imrn/article-pdf/2021/19/14629/42269080/rnz169.pdf |
id |
croxfordunivpr:10.1093/imrn/rnz169 |
---|---|
record_format |
openpolar |
spelling |
croxfordunivpr:10.1093/imrn/rnz169 2023-10-09T21:52:50+02:00 Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p Kubrak, Dmitry Travkin, Roman 2019 http://dx.doi.org/10.1093/imrn/rnz169 https://academic.oup.com/imrn/article-pdf/2021/19/14629/42269080/rnz169.pdf en eng Oxford University Press (OUP) https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model International Mathematics Research Notices volume 2021, issue 19, page 14629-14719 ISSN 1073-7928 1687-0247 General Mathematics journal-article 2019 croxfordunivpr https://doi.org/10.1093/imrn/rnz169 2023-09-22T11:17:30Z Abstract “Even more so is the word ‘crystalline’, a glacial and impersonal concept of his which disdains viewing existence from a single portion of time and space” Eileen Myles, “The Importance of Being Iceland” For a smooth variety $X$ over an algebraically closed field of characteristic $p$ to a differential 1-form $\alpha $ on the Frobenius twist $X^{\textrm{(1)}}$ one can associate an Azumaya algebra ${{\mathcal{D}}}_{X,\alpha }$, defined as a certain central reduction of the algebra ${{\mathcal{D}}}_X$ of “crystalline differential operators” on $X$. For a resolution of singularities $\pi :X\to Y$ of an affine variety $Y$, we study for which $\alpha $ the class $[{{\mathcal{D}}}_{X,\alpha }]$ in the Brauer group $\textrm{Br}(X^{\textrm{(1)}})$ descends to $Y^{\textrm{(1)}}$. In the case when $X$ is symplectic, this question is related to Fedosov quantizations in characteristic $p$ and the construction of noncommutative resolutions of $Y$. We prove that the classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend étale locally for all $\alpha $ if ${{\mathcal{O}}}_Y\widetilde{\rightarrow }\pi _\ast{{\mathcal{O}}}_X$ and $R^{1}\pi _*\mathcal O_X = R^2\pi _*\mathcal O_X =0$. We also define a certain class of resolutions, which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic $0$ to an algebraically closed field of characteristic $p$ classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend to $Y^{\textrm{(1)}}$ globally for all $\alpha $. Finally we give some examples; in particular, we show that Slodowy slices, Nakajima quiver varieties, and hypertoric varieties are resolutions with conical slices. Article in Journal/Newspaper Iceland Oxford University Press (via Crossref) International Mathematics Research Notices |
institution |
Open Polar |
collection |
Oxford University Press (via Crossref) |
op_collection_id |
croxfordunivpr |
language |
English |
topic |
General Mathematics |
spellingShingle |
General Mathematics Kubrak, Dmitry Travkin, Roman Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p |
topic_facet |
General Mathematics |
description |
Abstract “Even more so is the word ‘crystalline’, a glacial and impersonal concept of his which disdains viewing existence from a single portion of time and space” Eileen Myles, “The Importance of Being Iceland” For a smooth variety $X$ over an algebraically closed field of characteristic $p$ to a differential 1-form $\alpha $ on the Frobenius twist $X^{\textrm{(1)}}$ one can associate an Azumaya algebra ${{\mathcal{D}}}_{X,\alpha }$, defined as a certain central reduction of the algebra ${{\mathcal{D}}}_X$ of “crystalline differential operators” on $X$. For a resolution of singularities $\pi :X\to Y$ of an affine variety $Y$, we study for which $\alpha $ the class $[{{\mathcal{D}}}_{X,\alpha }]$ in the Brauer group $\textrm{Br}(X^{\textrm{(1)}})$ descends to $Y^{\textrm{(1)}}$. In the case when $X$ is symplectic, this question is related to Fedosov quantizations in characteristic $p$ and the construction of noncommutative resolutions of $Y$. We prove that the classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend étale locally for all $\alpha $ if ${{\mathcal{O}}}_Y\widetilde{\rightarrow }\pi _\ast{{\mathcal{O}}}_X$ and $R^{1}\pi _*\mathcal O_X = R^2\pi _*\mathcal O_X =0$. We also define a certain class of resolutions, which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic $0$ to an algebraically closed field of characteristic $p$ classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend to $Y^{\textrm{(1)}}$ globally for all $\alpha $. Finally we give some examples; in particular, we show that Slodowy slices, Nakajima quiver varieties, and hypertoric varieties are resolutions with conical slices. |
format |
Article in Journal/Newspaper |
author |
Kubrak, Dmitry Travkin, Roman |
author_facet |
Kubrak, Dmitry Travkin, Roman |
author_sort |
Kubrak, Dmitry |
title |
Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p |
title_short |
Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p |
title_full |
Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p |
title_fullStr |
Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p |
title_full_unstemmed |
Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p |
title_sort |
resolutions with conical slices and descent for the brauer group classes of certain central reductions of differential operators in characteristic p |
publisher |
Oxford University Press (OUP) |
publishDate |
2019 |
url |
http://dx.doi.org/10.1093/imrn/rnz169 https://academic.oup.com/imrn/article-pdf/2021/19/14629/42269080/rnz169.pdf |
genre |
Iceland |
genre_facet |
Iceland |
op_source |
International Mathematics Research Notices volume 2021, issue 19, page 14629-14719 ISSN 1073-7928 1687-0247 |
op_rights |
https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model |
op_doi |
https://doi.org/10.1093/imrn/rnz169 |
container_title |
International Mathematics Research Notices |
_version_ |
1779316019233292288 |