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

2012 ◽  
Vol 149 (1) ◽  
pp. 63-80 ◽  
Author(s):  
Allen Stewart ◽  
Vadim Vologodsky
Keyword(s):  
Associative Algebra ◽  
Differential Operators ◽  
Private Communication ◽  
Smooth Variety ◽  
Witt Vectors ◽  
Degeneracy Condition

AbstractWe 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 algebraA0over ℤpand its flat deformationAnover ℤ/pn+1ℤ, we prove that under a certain non-degeneracy condition, the center ofAnis isomorphic to the ring of length-(n+1) Witt vectors over the center ofA0.

scholarly journals Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p

2019 ◽  
Author(s):  
Dmitry Kubrak ◽  
Roman Travkin
Keyword(s):  
Differential Operators ◽  
Closed Field ◽  
Brauer Group ◽  
Affine Variety ◽  
Smooth Variety ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Quiver Varieties ◽  
Hypertoric Varieties ◽  
General Reduction

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.

scholarly journals On ζn-Weyl algebra Wr(ζn, Z)

1989 ◽  
Vol 113 ◽  
pp. 153-159 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Associative Algebra ◽  
Weyl Algebra ◽  
Differential Operators

Weyl algebra is an associative algebra generated by two elements â and a over R such that the generating relation is given byâa — aâ = 1,which is isomorphic to the algebra of differential operators

scholarly journals L p boundedness of rough bi-parameter Fourier integral operators

2018 ◽  
Vol 30 (1) ◽  
pp. 87-107 ◽  
Author(s):  
Qing Hong ◽  
Guozhen Lu ◽  
Lu Zhang
Keyword(s):  
Compact Support ◽  
Phase Function ◽  
Differential Operators ◽  
Integral Operators ◽  
Fourier Integral ◽  
Fourier Multipliers ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Degeneracy Condition ◽  
Phase Functions

Abstract In this paper, we will investigate the boundedness of the bi-parameter Fourier integral operators (or FIOs for short) of the following form: T(f\/)(x)=\frac{1}{(2\pi)^{2n}}\int_{\mathbb{R}^{2n}}e^{i\varphi(x,\xi,\eta)}% \cdot a(x,\xi,\eta)\cdot\widehat{f}(\xi,\eta)\,d\xi\,d\eta, where {x=(x_{1},x_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{n}} and {\xi,\eta\in\mathbb{R}^{n}\setminus\{0\}} , {a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} is the amplitude, and the phase function is of the form \varphi(x,\xi,\eta)=\varphi_{1}(x_{1},\xi\/)+\varphi_{2}(x_{2},\eta) , with \varphi_{1},\varphi_{2}\in L^{\infty}\Phi^{2}(\mathbb{R}^{n}\times\mathbb{R}^{% n}\setminus\{0\}) , and satisfies a certain rough non-degeneracy condition (see (2.2)). The study of these operators are motivated by the {L^{p}} estimates for one-parameter FIOs and bi-parameter Fourier multipliers and pseudo-differential operators. We will first define the bi-parameter FIOs and then study the {L^{p}} boundedness of such operators when their phase functions have compact support in frequency variables with certain necessary non-degeneracy conditions. We will then establish the {L^{p}} boundedness of the more general FIOs with amplitude {a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} and non-smooth phase function {\varphi(x,\xi,\eta)} on x satisfying a rough non-degeneracy condition.

scholarly journals On the strong homotopy associative algebra of a foliation

2015 ◽  
Vol 17 (02) ◽  
pp. 1450026 ◽  
Author(s):  
Luca Vitagliano
Keyword(s):  
Associative Algebra ◽  
Normal Bundle ◽  
Smooth Manifold ◽  
Vector Fields ◽  
Differential Operators ◽  
Lie Algebroid ◽  
Integral Manifolds ◽  
Homotopy Algebra

An involutive distribution C on a smooth manifold M is a Lie-algebroid acting on sections of the normal bundle TM/C. It is known that the Chevalley–Eilenberg complex associated to this representation of C possesses the structure 𝕏 of a strong homotopy Lie–Rinehart algebra. It is natural to interpret 𝕏 as the (derived) Lie–Rinehart algebra of vector fields on the space P of integral manifolds of C. In this paper, we show that 𝕏 is embedded in an A∞-algebra 𝔻 of (normal) differential operators. It is natural to interpret 𝔻 as the (derived) associative algebra of differential operators on P. Finally, we speculate about the interpretation of 𝔻 as the universal enveloping strong homotopy algebra of 𝕏.

A Device for Magnification Measurement

1972 ◽  
Vol 30 ◽  
pp. 622-623
Author(s):  
J.A. Eades ◽  
A. van Dun
Keyword(s):  
Electron Microscope ◽  
Small Displacement ◽  
Private Communication ◽  
Piezoelectric Crystal ◽  
Double Exposure ◽  
Standard Methods ◽  
Elegant Method ◽  
The Right ◽  
Piezo Electric

The measurement of magnification in the electron microscope is always troublesome especially when a goniometer stage is in use, since there can be wide variations from calibrated values. One elegant method (L.M.Brown, private communication) of avoiding the difficulties of standard methods would be to fit a device which displaces the specimen a small but known distance and recording the displacement by a double exposure. Such a device would obviate the need for changing the specimen and guarantee that the magnification was measured under precisely the conditions used.Such a small displacement could be produced by any suitable transducer mounted in one of the specimen translation mechanisms. In the present case a piezoelectric crystal was used. Modern synthetic piezo electric ceramics readily give reproducible displacements in the right range for quite modest voltages (for example: Joyce and Wilson, 1969).

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

Continuity of homotopies

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Open Subset ◽  
Small Neighborhood ◽  
Unit Vector ◽  
Zariski Topology ◽  
Simple Point ◽  
Smooth Variety ◽  
Additional Material ◽  
Generic Type ◽  
Zariski Open Subset

This chapter includes some additional material on homotopies. In particular, for a smooth variety V, there exists an “inflation” homotopy, taking a simple point to the generic type of a small neighborhood of that point. This homotopy has an image that is properly a subset of unit vector V, and cannot be understood directly in terms of definable subsets of V. The image of this homotopy retraction has the merit of being contained in unit vector U for any dense Zariski open subset U of V. The chapter also proves the continuity of functions and homotopies using continuity criteria and constructs inflation homotopies before proving GAGA type results for connectedness. Additional results regarding the Zariski topology are given.

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