scholarly journals Semistable reduction for overconvergent F-isocrystals, IV: local semistable reduction at nonmonomial valuations

2011 ◽  
Vol 147 (2) ◽  
pp. 467-523 ◽  
Author(s):  
Kiran S. Kedlaya
Keyword(s):  
Positive Characteristic ◽  
Degree Zero ◽  
Local Problem ◽  
Finite Cover ◽  
Semistable Reduction ◽  
Numerical Invariants ◽  
Rational Rank ◽  
Logarithmic Singularities ◽  
Meromorphic Connection ◽  
Field Of Positive Characteristic

AbstractWe complete our proof that given an overconvergent F-isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic singularities and nilpotent residues, to some complete variety. We also establish an analogue for F-isocrystals overconvergent inside a partial compactification. By previous results, this reduces to solving a local problem in a neighborhood of a valuation of height 1 and residual transcendence degree zero. We do this by studying the variation of some numerical invariants attached to p-adic differential modules, analogous to the irregularity of a complex meromorphic connection. This allows for an induction on the transcendence defect of the valuation, i.e., the discrepancy between the dimension of the variety and the rational rank of the valuation.

THE RAMIFICATION GROUPS AND DIFFERENT OF A COMPOSITUM OF ARTIN–SCHREIER EXTENSIONS

2010 ◽  
Vol 06 (07) ◽  
pp. 1541-1564 ◽  
Author(s):  
QINGQUAN WU ◽  
RENATE SCHEIDLER
Keyword(s):  
Class Number ◽  
Function Field ◽  
Positive Characteristic ◽  
Field Extension ◽  
Constant Field ◽  
Divisor Class ◽  
Characteristic P ◽  
Similar Nature ◽  
Field Of Positive Characteristic ◽  
The Ideal

Let K be a function field over a perfect constant field of positive characteristic p, and L the compositum of n (degree p) Artin–Schreier extensions of K. Then much of the behavior of the degree pn extension L/K is determined by the behavior of the degree p intermediate extensions M/K. For example, we prove that a place of K totally ramifies/is inert/splits completely in L if and only if it totally ramifies/is inert/splits completely in every M. Examples are provided to show that all possible decompositions are in fact possible; in particular, a place can be inert in a non-cyclic Galois function field extension, which is impossible in the case of a number field. Moreover, we give an explicit closed form description of all the different exponents in L/K in terms of those in all the M/K. Results of a similar nature are given for the genus, the regulator, the ideal class number and the divisor class number. In addition, for the case n = 2, we provide an explicit description of the ramification group filtration of L/K.

The Hilbert series of the irreducible quotient of the polynomial representation of the rational Cherednik algebra of type An−1 in characteristic p for p|n − 1

2021 ◽  
pp. 2250191
Author(s):  
Merrick Cai ◽  
Daniil Kalinov
Keyword(s):  
Hilbert Series ◽  
Positive Characteristic ◽  
Closed Field ◽  
Simple Criterion ◽  
Characteristic P ◽  
Polynomial Representation ◽  
Cherednik Algebra ◽  
Rational Cherednik Algebra ◽  
Polynomial Formula ◽  
Field Of Positive Characteristic

In this paper, we study the irreducible quotient [Formula: see text] of the polynomial representation of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] over an algebraically closed field of positive characteristic [Formula: see text] where [Formula: see text]. In the [Formula: see text] case, for all [Formula: see text] we give a complete description of the polynomials in the maximal proper graded submodule [Formula: see text], the kernel of the contravariant form [Formula: see text], and subsequently find the Hilbert series of the irreducible quotient [Formula: see text]. In the [Formula: see text] case, we give a complete description of the polynomials in [Formula: see text] when the characteristic [Formula: see text] and [Formula: see text] is transcendental over [Formula: see text], and compute the Hilbert series of the irreducible quotient [Formula: see text]. In doing so, we prove a conjecture due to Etingof and Rains completely for [Formula: see text], and also for any [Formula: see text] and [Formula: see text]. Furthermore, for [Formula: see text], we prove a simple criterion to determine whether a given polynomial [Formula: see text] lies in [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text] fixed.

scholarly journals Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

2008 ◽  
Vol 191 ◽  
pp. 111-134 ◽  
Author(s):  
Christian Liedtke
Keyword(s):  
Characteristic Zero ◽  
General Type ◽  
Positive Characteristic ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Arbitrary Characteristic ◽  
Numerical Invariants ◽  
Characteristic 2

AbstractWe establish Noether’s inequality for surfaces of general type in positive characteristic. Then we extend Enriques’ and Horikawa’s classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

scholarly journals The Square-Zero Basis of Matrix Lie Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1032
Author(s):  
Raúl Durán Díaz ◽  
Víctor Gayoso Martínez ◽  
Luis Hernández Encinas ◽  
Jaime Muñoz Masqué
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Characteristic ◽  
Linear Algebraic Group ◽  
Invariant Functions ◽  
Arbitrary Characteristic ◽  
Independent Invariant ◽  
Field Of Positive Characteristic

A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of positive characteristic. The class of such Lie algebras is studied in the framework of the classical Lie algebras of arbitrary characteristic. Some examples and applications are also given.

scholarly journals Indecomposable orthogonal invariants of several matrices over a field of positive characteristic

2020 ◽  
pp. 1-11
Author(s):  
Artem Lopatin
Keyword(s):  
Characteristic Polynomial ◽  
Orthogonal Group ◽  
General Linear Group ◽  
Positive Characteristic ◽  
The Other ◽  
Symmetric Matrices ◽  
Maximal Degree ◽  
Infinite Field ◽  
The Given ◽  
Field Of Positive Characteristic

We consider the algebra of invariants of [Formula: see text]-tuples of [Formula: see text] matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic [Formula: see text] different from two. It is well known that this algebra is generated by the coefficients of the characteristic polynomial of all products of generic and transpose generic [Formula: see text] matrices. We establish that in case [Formula: see text] the maximal degree of indecomposable invariants tends to infinity as [Formula: see text] tends to infinity. In other words, there does not exist a constant [Formula: see text] such that it only depends on [Formula: see text] and the considered algebra of invariants is generated by elements of degree less than [Formula: see text] for any [Formula: see text]. This result is well-known in case of the action of the general linear group. On the other hand, for the rest of [Formula: see text] the given phenomenon does not hold. We investigate the same problem for the cases of symmetric and skew-symmetric matrices.

scholarly journals The realization of the degree zero part of the motivic polylogarithm on abelian schemes in Deligne–Beilinson cohomology

2017 ◽  
Vol 13 (09) ◽  
pp. 2471-2485 ◽  
Author(s):  
Danny Scarponi
Keyword(s):  
Chow Groups ◽  
Degree Zero ◽  
Deligne Cohomology ◽  
Abelian Scheme ◽  
Logarithmic Singularities ◽  
Abelian Schemes

In 2014, Kings and Rössler showed that the realization of the degree zero part of the abelian polylogarithm in analytic Deligne cohomology can be described in terms of a class of currents which was previously defined by Maillot and Rössler and strongly related to the Bismut–Köhler higher torsion form of the Poincaré bundle. In this paper we show that, if the base of the abelian scheme is proper, Kings and Rössler’s result can be refined to hold already in Deligne–Beilinson cohomology. More precisely, by means of Burgos’ theory of arithmetic Chow groups, we prove that the class of currents defined by Maillot and Rössler has a representative with logarithmic singularities at the boundary and therefore defines an element in Deligne–Beilinson cohomology. This element coincides with the realization of the degree zero part of the motivic polylogarithm on abelian schemes in Deligne–Beilinson cohomology.

scholarly journals On the algebraic K-theory of truncated polynomial algebras in several variables

2014 ◽  
Vol 13 (1) ◽  
pp. 57-81 ◽  
Author(s):  
Vigleik Angeltveit ◽  
Teena Gerhardt ◽  
Michael A. Hill ◽  
Ayelet Lindenstrauss
Keyword(s):  
Positive Characteristic ◽  
Polynomial Algebra ◽  
Cyclic Homology ◽  
Perfect Field ◽  
Polynomial Algebras ◽  
Trace Map ◽  
Witt Vectors ◽  
Several Variables ◽  
Topological Cyclic Homology ◽  
Field Of Positive Characteristic

AbstractWe consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, . This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field of positive characteristic we describe the K-theory computation in terms of a cube of these Witt vectors on ℕn. If the characteristic of k does not divide any of the ai we compute the K-groups explicitly. We also compute the K-groups modulo torsion for k = ℤ.To understand this K-theory spectrum we use the cyclotomic trace map to topological cyclic homology, and write as the iterated homotopy cofiber of an n-cube of spectra, each of which is easier to understand.

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