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 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 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 SUBALGEBRAS OF THE POLYNOMIAL ALGEBRA IN POSITIVE CHARACTERISTIC AND THE JACOBIAN

2011 ◽  
Vol 10 (04) ◽  
pp. 605-613
Author(s):  
ALEXEY V. GAVRILOV
Keyword(s):  
Principal Ideal ◽  
Positive Characteristic ◽  
Polynomial Algebra ◽  
Characteristic P ◽  
The Ideal

Let 𝕜 be a field of characteristic p > 0 and R be a subalgebra of 𝕜[X] = 𝕜[x1, …, xn]. Let J(R) be the ideal in 𝕜[X] defined by [Formula: see text]. It is shown that if it is a principal ideal then [Formula: see text], where q = pn(p - 1)/2.

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