A positive characteristic extension of a result of del busto on line bundles on an algebraic surface

This note gives a partial extension to positive characteristic of the results contained in the paper “A Matsusaka-type Theorem for surfaces” by G. Fernandez Del Busto.     

On pi-algebras in positive characteristic

We use the BEST theorem in Graph Theory to study a non-alternating version of the standard identity in prime characteristic. Using these re-sults and the linearized Hamilton-Cayley identity, we show the existence of an element with reduced trace 1.     

Fano threefolds in positive characteristic

We show that the birational classification in positive characteristicof smooth Fano threefolds X with Picard number 1 is the same as incharacteristic zero. In particular, there are no exotic such Fanos; asa consequence of the classification, X is shown to be liftable withoutramification to characteristic zero and to contain a line. The maintechniques employed are those of Ekedahl and of Mori and Takeuchi.     

PI non-equivalence in positive characteristic

The algebras A _a,b appeared in the study of the tensor products of verbally prime PI algebras. They are in-between the well known algebras M _n (E) and ${M_{a,b}(E)\otimes E}$ , see the definitions below. Here E is the Grassmann algebra. The main result of this note consists in showing that the algebras A _a,b and M _a+b (E) are not PI equivalent in characteristic p > 2.     

A vanishing theorem for differential operators in positive characteristic

Let X be a smooth variety over an algebraically closed field k of characteristic p, and let F: X → X be the Frobenius morphism. We prove that if X is an incidence variety (a partial flag variety in type A _n ) or a smooth quadric (in this case p is supposed to be odd) then Hⁱ( X,End( \sfF_*O_X ) ) = 0 {H^i}\left( {X,\mathcal{E}nd\left( {{\sf{F}_*}{\mathcal{O}_X}} \right)} \right) = 0 for i > 0. Using this vanishing result and the derived localization theorem for crystalline differential operators [3], we show that the Frobenius direct image \sfF_*O_X {\sf{F}_*}{\mathcal{O}_X} is a tilting bundle on these varieties provided that p > h, the Coxeter number of the corresponding group.     

Sub-principal homomorphisms in positive characteristic

 Let G be a reductive group over an algebraically closed field of characteristic p, and let uG be a unipotent element of order p. Suppose that p is a good prime for G. We show in this paper that there is a homomorphism φ:SL _2/k →G whose image contains u. This result was first obtained by D. Testerman (J. Algebra, 1995) using case considerations for each type of simple group (and using, in some cases, computer calculations with explicit representatives for the unipotent orbits). The proof we give is free of case considerations (except in its dependence on the Bala-Carter theorem). Our construction of φ generalizes the construction of a principal homomorphism made by J.-P. Serre in (Invent. Math. 1996); in particular, φ is obtained by reduction modulo 𝔭 from a homomorphism of group schemes over a valuation ring 𝒜 in a number field. This permits us to show moreover that the weight spaces of a maximal torus of φ(SL _2/k ) on Lie(G) are ``the same as in characteristic 0'; the existence of a φ with this property was previously obtained, again using case considerations, by Lawther and Testerman (Memoirs AMS, 1999) and has been applied in some recent work of G. Seitz (Invent. Math. 2000). Received: 1 February 2002; in final form: 17 June 2002 / Published online: 1 April 2003 The author was supported in part by a grant from the National Science Foundation.     

Fedosov quantization in positive characteristic

We study the problem of deformation quantization for (algebraic) symplectic manifolds over a base field of positive characteristic. We prove a reasonably complete classification theorem for one class of such quantizations; in the course of doing it, we also introduce a notion of a restricted Poisson algebra - the Poisson analog of the standard notion of a restricted Lie algebra - and we prove a version of the Darboux Theorem valid in the positive characteristic setting.

     

Absolute integral closure in positive characteristic

Let R be a local Noetherian domain of positive characteristic. A theorem of Hochster and Huneke [M. Hochster, C. Huneke, Infinite integral extensions and big Cohen–Macaulay algebras, Ann. of Math. 135 (1992) 53–89] states that if R is excellent, then the absolute integral closure of R is a big Cohen–Macaulay algebra. We prove that if R is the homomorphic image of a Gorenstein local ring, then all the local cohomology (below the dimension) of such a ring maps to zero in a finite extension of the ring. As a result there follow an extension of the original result of Hochster and Huneke to the case in which R is a homomorphic image of a Gorenstein local ring, and a considerably simpler proof of this result in the cases where the assumptions overlap, e.g., for complete Noetherian local domains.