Central differential operators on split semi-simple groups over fields of positive characteristic

1980 ◽  
pp. 35-85 ◽  
Author(s):  
W. J. Haboush
Keyword(s):  
Differential Operators ◽  
Positive Characteristic ◽  
Simple Groups

scholarly journals Length of Local Cohomology in Positive Characteristic and Ordinarity

2018 ◽  
Vol 2020 (7) ◽  
pp. 1921-1932 ◽  
Author(s):  
Thomas Bitoun
Keyword(s):  
Characteristic Zero ◽  
Local Cohomology ◽  
Differential Operators ◽  
Positive Characteristic ◽  
Isolated Singularity ◽  
Local Cohomology Module ◽  
Perfect Field ◽  
Tight Closure ◽  
Characteristic P

Abstract Let D be the ring of Grothendieck differential operators of the ring R of polynomials in d ≥ 3 variables with coefficients in a perfect field of characteristic p. We compute the D-module length of the 1st local cohomology module ${H^{1}_{f}}(R)$ with respect to a polynomial f with an isolated singularity, for p large enough. The expression we give is in terms of the Frobenius action on the top coherent cohomology of the exceptional fibre of a resolution of the singularity. Our proof rests on a tight closure computation of Hara. Since the above length is quite different from that of the corresponding local cohomology module in characteristic zero, we also consider a characteristic zero D-module whose length is expected to equal that above, for ordinary primes.

scholarly journals Local solutions to positive characteristic non-Archimedean differential equations

2007 ◽  
Vol 143 (6) ◽  
pp. 1465-1477
Author(s):  
João Pedro Pinto dos Santos
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Analytic Functions ◽  
Differential Operators ◽  
Positive Characteristic ◽  
Existence Of Solutions ◽  
Base Field ◽  
Absolute Value ◽  
Monodromy Groups ◽  
Local Solutions

AbstractIn the complex domain, one can integrate (solve) holomorphic ordinary differential equations (ODEs) near a non-singular point. We study the existence of solutions in the case of a positive characteristic base field k which is complete with respect to a non-Archimedean absolute value. ODEs are substituted by modules over a ring of analytic functions endowed with an action of all differential operators. The monodromy groups associated to the corresponding category are computed.

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.

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