scholarly journals GEOMETRIC LOCAL -FACTORS IN HIGHER DIMENSIONS

2021 ◽  
pp. 1-27
Author(s):  
Quentin Guignard
Keyword(s):  
Positive Characteristic ◽  
Higher Dimensions ◽  
Product Formula ◽  
Perfect Field ◽  
Local Factors ◽  
Higher Dimensional ◽  
Vanishing Cycle ◽  
Field Of Positive Characteristic

Abstract We prove a product formula for the determinant of the cohomology of an étale sheaf with $\ell $ -adic coefficients over an arbitrary proper scheme over a perfect field of positive characteristic p distinct from $\ell $ . The local contributions are constructed by iterating vanishing cycle functors as well as certain exact additive functors that can be considered as linearised versions of Artin conductors and local $\varepsilon $ -factors. We provide several applications of our higher dimensional product formula, such as twist formulas for global $\varepsilon $ -factors.

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.

A note on the cancellation property of k[X, Y]

2015 ◽  
Vol 14 (09) ◽  
pp. 1540007 ◽  
Author(s):  
S. M. Bhatwadekar ◽  
Neena Gupta
Keyword(s):  
Polynomial Ring ◽  
Positive Characteristic ◽  
Perfect Field ◽  
Cancellation Property ◽  
Rational Surfaces ◽  
Field Of Positive Characteristic

In [On affine-ruled rational surfaces, Math. Ann.255(3) (1981) 287–302], Russell had proved that when k is a perfect field of positive characteristic, the polynomial ring k[X, Y] is cancellative. In this note, we shall show that this cancellation property holds even without the hypothesis that k is perfect.

scholarly journals Semiample perturbations for log canonical varieties over an F-finite field containing an infinite perfect field

2017 ◽  
Vol 28 (05) ◽  
pp. 1750030 ◽  
Author(s):  
Hiromu Tanaka
Keyword(s):  
Finite Field ◽  
Positive Characteristic ◽  
Ample Divisor ◽  
Perfect Field ◽  
Log Canonical ◽  
Canonical Pair ◽  
Field Of Positive Characteristic

Let [Formula: see text] be an [Formula: see text]-finite field containing an infinite perfect field of positive characteristic. Let [Formula: see text] be a projective log canonical pair over [Formula: see text]. In this note, we show that, for a semi-ample divisor [Formula: see text] on [Formula: see text], there exists an effective [Formula: see text]-divisor [Formula: see text] such that [Formula: see text] is log canonical if there exists a log resolution of [Formula: see text].

Elimination theory for addition and the frobenius map in polynomial rings

2004 ◽  
Vol 69 (4) ◽  
pp. 1006-1026 ◽  
Author(s):  
Thanases Pheidas ◽  
Karim Zahidi
Keyword(s):  
Positive Characteristic ◽  
Polynomial Rings ◽  
Elimination Theory ◽  
Perfect Field ◽  
Decidable Theory ◽  
Frobenius Map ◽  
Ring Of Polynomials ◽  
Field Of Positive Characteristic

Abstract.We develop an elimination theory for addition and the Frobenius map over rings of polynomials. As a consequence we show that if F is a countable, recursive and perfect field of positive characteristic p, with decidable theory, then the structure of addition, the Frobenius map x → xp and the property ‘x ∈ F1, over the ring of polynomials F[T], has a decidable theory.

scholarly journals Swan conductors for p-adic differential modules. II Global variation

2010 ◽  
Vol 10 (1) ◽  
pp. 191-224 ◽  
Author(s):  
Kiran S. Kedlaya
Keyword(s):  
Fundamental Group ◽  
Positive Characteristic ◽  
Perfect Field ◽  
Numerical Invariant ◽  
Swan Conductor ◽  
Global Variation ◽  
Local Construction ◽  
Differential Modules ◽  
Field Of Positive Characteristic

AbstractUsing a local construction from a previous paper, we exhibit a numerical invariant, the differential Swan conductor, for an isocrystal on a variety over a perfect field of positive characteristic overconvergent along a boundary divisor; this leads to an analogous construction for certain p-adic and l-adic representations of the étale fundamental group of a variety. We then demonstrate some variational properties of this definition for overconvergent isocrystals, paying special attention to the case of surfaces.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

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.

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