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A note on Frobenius divided modules in mixed characteristics

Pierre Berthelot (2012)

Bulletin de la Société Mathématique de France

If X is a smooth scheme over a perfect field of characteristic p , and if 𝒟 X ( ) is the sheaf of differential operators on X [7], it is well known that giving an action of 𝒟 X ( ) on an 𝒪 X -module is equivalent to giving an infinite sequence of 𝒪 X -modules descending via the iterates of the Frobenius endomorphism of X [5]. We show that this result can be generalized to any infinitesimal deformation f : X S of a smooth morphism in characteristic p , endowed with Frobenius liftings. We also show that it extends to adic...

Codimension B-W d’un idéal à droite non nul de A 1 ( )

Mathias Konan Kouakou (2005)

Bulletin de la Société Mathématique de France

Soit A 1 ( ) la première algèbre de Weyl sur . La codimension B-W d’un idéal à droite non nul I de A 1 ( ) a été introduite par Yuri Berest et George Wilson. Nous montrons d’une part que cette codimension est invariante par la relation de Stafford : si x Q 1 = Frac ( A 1 ( ) ) , le corps de fractions de A 1 ( ) , et si σ Aut ( A 1 ( ) ) , le groupe des -automorphismes de A 1 ( ) , sont tels que J = x σ ( I ) soit un idéal à droite de A 1 ( ) , alors codim I = codim x σ ( I ) . Nous relions d’autre part la codimension d’un idéal I à la codimension de Gail Letzter-Makar Limanov, de End ( I ) , l’anneau des endomorphismes...

Gröbner δ-bases and Gröbner bases for differential operators

Francisco J. Castro-Jiménez, M. Angeles Moreno-Frías (2002)

Banach Center Publications

This paper deals with the notion of Gröbner δ-base for some rings of linear differential operators by adapting the works of W. Trinks, A. Assi, M. Insa and F. Pauer. We compare this notion with the one of Gröbner base for such rings. As an application we give some results on finiteness and on flatness of finitely generated left modules over these rings.

Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products

Pavel Etingof, Wee Liang Gan, Victor Ginzburg, Alexei Oblomkov (2007)

Publications Mathématiques de l'IHÉS

The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras...

Hochschild homology and cohomology of generalized Weyl algebras

Marco A. Farinati, Andrea L. Solotar, Mariano Suárez-Álvarez (2003)

Annales de l’institut Fourier

We compute Hochschild homology and cohomology of a class of generalized Weyl algebras, introduced by V. V. Bavula in St. Petersbourg Math. Journal, 4 (1) (1999), 71-90. Examples of such algebras are the n-th Weyl algebras, 𝒰 ( 𝔰 𝔩 2 ) , primitive quotients of 𝒰 ( 𝔰 𝔩 2 ) , and subalgebras of invariants of these algebras under finite cyclic groups of automorphisms. We answer a question of Bavula–Jordan (Trans. A.M.S., 353 (2) (2001), 769-794) concerning the generators of the group of automorphisms of a generalized Weyl...

Invariant differential operators on the tangent space of some symmetric spaces

Thierry Levasseur, J. Toby Stafford (1999)

Annales de l'institut Fourier

Let 𝔤 be a complex, semisimple Lie algebra, with an involutive automorphism ϑ and set 𝔨 = Ker ( ϑ - I ) , 𝔭 = Ker ( ϑ + I ) . We consider the differential operators, 𝒟 ( 𝔭 ) K , on 𝔭 that are invariant under the action of the adjoint group K of 𝔨 . Write τ : 𝔨 Der 𝒪 ( 𝔭 ) for the differential of this action. Then we prove, for the class of symmetric pairs ( 𝔤 , 𝔨 ) considered by Sekiguchi, that d 𝒟 ( 𝔭 ) : d 𝒪 ( 𝔭 ) K = 0 = 𝒟 ( 𝔭 ) τ ( 𝔨 ) . An immediate consequence of this equality is the following result of Sekiguchi: Let ( 𝔤 0 , 𝔨 0 ) be a real form of one of these symmetric pairs ( 𝔤 , 𝔨 ) , and suppose that T is a K 0 -invariant...

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