11 research outputs found
Twisting the Infinitesimal Site
We classify twistings of Grothendieck's differential operators on a smooth
variety in prime characteristic . We prove isomorphism classes of
twistings are in bijection with , the degree 2, weight
1 syntomic cohomology of . We also discuss the relationship between
twistings of crystalline and Grothendieck differential operators. Twistings in
mixed characteristic are also analyzed.Comment: 20 pages. Comments welcom
A module-theoretic approach to matroids
Speyer recognized that matroids encode the same data as a special class of
tropical linear spaces and Shaw interpreted tropically certain basic matroid
constructions; additionally, Frenk developed the perspective of tropical linear
spaces as modules over an idempotent semifield. All together, this provides
bridges between the combinatorics of matroids, the algebra of idempotent
modules, and the geometry of tropical linear spaces. The goal of this paper is
to strengthen and expand these bridges by systematically developing the
idempotent module theory of matroids. Applications include a geometric
interpretation of strong matroid maps and the factorization theorem; a
generalized notion of strong matroid maps, via an embedding of the category of
matroids into a category of module homomorphisms; a monotonicity property for
the stable sum and stable intersection of tropical linear spaces; a novel
perspective of fundamental transversal matroids; and a tropical analogue of
reduced row echelon form.Comment: 22 pages; v3 minor corrections/clarifications; to appear in JPA
Quantization of restricted Lagrangian subvarieties in positive characteristic
Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties X
in positive characteristic which endow the Poisson bracket on X with the
structure of a restricted Lie algebra. We consider deformation quantization of
line bundles on Lagrangian subvarieties Y of X to modules over such
quantizations. If the ideal sheaf of Y is a restricted Lie subalgebra of the
structure sheaf of X, we show that there is a certain cohomology class which
vanishes if and only if a line bundle on Y admits a quantization.Comment: Comments welcome! 22 page
Almost all wreath product character values are divisible by given primes
For a finite group with integer-valued character table and a prime ,
we show that almost every entry in the character table of is
divisible by as . This result generalizes the work of Peluse
and Soundararajan on the character table of .Comment: v2: minor edits. 15 pages, 4 figures. Comments welcom
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On Deformation Quantization and Differential Operators in Positive Characteristic
This thesis consists of two papers studying noncommutative rings in positive characteristic closely related to differential operators. Their abstracts are as follows:1. Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties X in positive characteristic which endow the Poisson bracket on X with the structure of a restricted Lie algebra. We consider deformation quantization of line bundles on Lagrangian subvarieties Y of X to modules over such quantizations. If the ideal sheaf of Y is a restricted Lie subalgebra of the structure sheaf of X, we show that there is a certain cohomology class which vanishes if and only if a line bundle on Y admits a quantization.
2. For k a field of positive characteristic and X a smooth variety over k, we compute the Hochschild cohomology of Grothendieck’s differential operators on X. The answer involves the derived inverse limit of the Frobenius acting on the cohomology of the structure sheaf of X