Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T05:37:11.640Z Has data issue: false hasContentIssue false
  • English
  • Français

On the center of the ring of differential operators on a smooth variety over ℤ/pnℤ

Part of: (Co)homology theory Rings and algebras arising under various constructions Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  31 October 2012

Allen Stewart  and
Vadim Vologodsky
Allen Stewart
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA (email: [email protected])
Vadim Vologodsky
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We compute the center of the ring of PD differential operators on a smooth variety over ℤ/pnℤ, confirming a conjecture of Kaledin (private communication). More generally, given an associative algebra A0 over ℤp and its flat deformation An over ℤ/pn+1ℤ, we prove that under a certain non-degeneracy condition, the center of An is isomorphic to the ring of length-(n+1) Witt vectors over the center of A0.

Keywords

deformations of algebrasPoisson algebrasWitt vectors

MSC classification

Secondary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14G17: Positive characteristic ground fields 16S34: Group rings , Laurent polynomial rings 16S80: Deformations of rings
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 1 , January 2013 , pp. 63 - 80
Copyright
Copyright © The Author(s) 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[BK05]Belov-Kanel, A. and Kontsevich, M., Automorphisms of the Weyl algebra, Lett. Math. Phys. 74 (2005), 181–199.CrossRefGoogle Scholar
[BO78]Berthelot, P. and Ogus, A., Notes on crystalline cohomology (Princeton University Press, Princeton, NJ and University of Tokyo Press, Tokyo, 1978).Google Scholar
[BK08]Bezrukavnikov, R. and Kaledin, D., Fedosov quantization in positive characteristic, J. Amer. Math. Soc. 21 (2008), 409–438.CrossRefGoogle Scholar
[BMR08]Bezrukavnikov, R., Mirkovic, I. and Rumynin, D., Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. of Math. (2) 167 (2008), 945–991.CrossRefGoogle Scholar
[Ill79]Illusie, L., Complexe de de Rham–Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Sup. (4) 12 (1979), 501–661.CrossRefGoogle Scholar
[Mum66]Mumford, D., Lectures on curves on an algebraic surface, Annals of Mathematics Studies, vol. 59 (Princeton University Press, Princeton, NJ, 1966).CrossRefGoogle Scholar
[OV07]Ogus, A. and Vologodsky, V., Nonabelian Hodge theory in characteristic p, Publ. Math. Inst. Hautes Études Sci. 106, 2007, 1–138.Google Scholar