scholarly journals FRAMED CORRESPONDENCES AND THE MILNOR–WITT -THEORY

2016 ◽  
Vol 17 (4) ◽  
pp. 823-852 ◽  
Author(s):  
Alexander Neshitov
Keyword(s):  
Characteristic Zero ◽  
Triangulated Category ◽  
Algebraic Varieties ◽  
Base Field ◽  
Witt Theory

Given a field $k$ of characteristic zero and $n\geqslant 0$, we prove that $H_{0}(\mathbb{Z}F(\unicode[STIX]{x1D6E5}_{k}^{\bullet },\mathbb{G}_{m}^{\wedge n}))=K_{n}^{MW}(k)$, where $\mathbb{Z}F_{\ast }(k)$ is the category of linear framed correspondences of algebraic $k$-varieties, introduced by Garkusha and Panin [The triangulated category of linear framed motives $DM_{fr}^{eff}(k)$, in preparation] (see [Garkusha and Panin, Framed motives of algebraic varieties (after V. Voevodsky), Preprint, 2014, arXiv:1409.4372], § 7 as well), and $K_{\ast }^{MW}(k)$ is the Milnor–Witt $K$-theory of the base field $k$.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

Properties of the Invariants of Solvable Lie Algebras

2000 ◽  
Vol 43 (4) ◽  
pp. 459-471 ◽  
Author(s):  
J. C. Ndogmo
Keyword(s):  
Lie Algebras ◽  
Upper Bound ◽  
Characteristic Zero ◽  
Complex Numbers ◽  
Coadjoint Representation ◽  
Base Field ◽  
Invariant Functions ◽  
Solvable Lie Algebras ◽  
Restricted Type

AbstractWe generalize to a field of characteristic zero certain properties of the invariant functions of the coadjoint representation of solvable Lie algebras with abelian nilradicals, previously obtained over the base field ℂ of complex numbers. In particular we determine their number and the restricted type of variables on which they depend. We also determine an upper bound on the maximal number of functionally independent invariants for certain families of solvable Lie algebras with arbitrary nilradicals.

scholarly journals Strongly Incompressible Curves

2016 ◽  
Vol 68 (3) ◽  
pp. 541-570 ◽  
Author(s):  
Mario Garcia-Armas
Keyword(s):  
Finite Group ◽  
Characteristic Zero ◽  
Base Field ◽  
Group A ◽  
A Finite Group

AbstractLet G be a finite group. A faithful G-variety X is called strongly incompressible if every dominant G-equivariant rationalmap of X onto another faithful G-variety Y is birational. We settle the problem of existence of strongly incompressible G-curves for any finite group G and any base field k of characteristic zero.

Model Theoretic Algebra

1976 ◽  
Vol 41 (2) ◽  
pp. 537-545
Author(s):  
G. L. Cherlin
Keyword(s):  
Algebraic Geometry ◽  
Algebra I ◽  
Arbitrary Field ◽  
Algebraic Varieties ◽  
Base Field ◽  
Prime Field ◽  
Lefschetz Principle ◽  
Universal Domain ◽  
Image Position ◽  
S Model

Since the late 1940's model theory has found numerous applications to algebra. I would like to indicate some of the points of contact between model theoretic methods and strictly algebraic concerns by means of a few concrete examples and typical applications.§1. The Lefschetz principle. Algebraic geometry has proved to be a fruitful source of model theoretic ideas. What exactly is algebraic geometry? We consider a field K, and let Kn be the set of n-tuples (a1 … an) with coordinates ai in K. Kn is called affine n-space over K. Fix polynomials p1 …, pk in K[x1, …, xn] and definethat is V(p1 …, pk) is the locus of common zeroes of the pi in Kn. We call V(Pi …, Pk) the algebraic variety determined by p1, …, pk. With this terminology we may say:Algebraic geometry is the study of algebraic varieties defined over an arbitrary field K. This definition lacks both rigor and accuracy, and we will indicate below how it may be improved.So far we have placed no restrictions on the base field K. Following Weil [4] it is convenient to start with a so-called “universal domain”; in other words take K to be algebraically closed and of infinite transcendence degree over the prime field. Any particular field can of course be embedded in such a universal domain.

scholarly journals THE p-COHOMOLOGY OF ALGEBRAIC VARIETIES AND SPECIAL VALUES OF ZETA FUNCTIONS

2014 ◽  
Vol 14 (4) ◽  
pp. 801-835 ◽  
Author(s):  
James S. Milne ◽  
Niranjan Ramachandran
Keyword(s):  
Algebraic Variety ◽  
Homological Algebra ◽  
Zeta Functions ◽  
Compelling Evidence ◽  
Algebraic Varieties ◽  
Cohomology Groups ◽  
Base Field ◽  
Graded Modules ◽  
Special Values ◽  
Correct Target

The $p$-cohomology of an algebraic variety in characteristic $p$ lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl, Illusie, Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc., and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for $p$-cohomology in characteristic $p$.

scholarly journals Spinors and essential dimension

2017 ◽  
Vol 153 (3) ◽  
pp. 535-556 ◽  
Author(s):  
Skip Garibaldi ◽  
Robert M. Guralnick
Keyword(s):  
Number Theory ◽  
Characteristic Zero ◽  
Quadratic Forms ◽  
Low Rank ◽  
Essential Dimension ◽  
Base Field ◽  
Spin Groups ◽  
Group Schemes ◽  
Clifford Groups

We prove that spin groups act generically freely on various spinor modules, in the sense of group schemes and in a way that does not depend on the characteristic of the base field. As a consequence, we extend the surprising calculation of the essential dimension of spin groups and half-spin groups in characteristic zero by Brosnan et al. [Essential dimension, spinor groups, and quadratic forms, Ann. of Math. (2) 171 (2010), 533–544], and Chernousov and Merkurjev [Essential dimension of spinor and Clifford groups, Algebra Number Theory 8 (2014), 457–472] to fields of characteristic different from two. We also complete the determination of generic stabilizers in spin and half-spin groups of low rank.

Classifying the Minimal Varieties of Polynomial Growth

2014 ◽  
Vol 66 (3) ◽  
pp. 625-640 ◽  
Author(s):  
Antonio Giambruno ◽  
Daniela La Mattina ◽  
Mikhail Zaicev
Keyword(s):  
Finite Number ◽  
Characteristic Zero ◽  
Polynomial Growth ◽  
Building Blocks ◽  
Base Field ◽  
Associative Algebras ◽  
Proper Subvariety ◽  
Variety Of Associative Algebras

AbstractLet ν be a variety of associative algebras generated by an algebra with 1 over a field of characteristic zero. This paper is devoted to the classification of the varieties ν that are minimal of polynomial growth (i.e., their sequence of codimensions grows like nk, but any proper subvariety grows like nt with t < k). These varieties are the building blocks of general varieties of polynomial growth.It turns out that for k ≤ 4 there are only a finite number of varieties of polynomial growth nk, but for each k > 4, the number of minimal varieties is at least |F|, the cardinality of the base field, and we give a recipe for their construction.

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