scholarly journals A lower bound on the essential dimension of simple algebras

2010 ◽  
Vol 4 (8) ◽  
pp. 1055-1076 ◽  
Author(s):  
Alexander Merkurjev
Keyword(s):  
Lower Bound ◽  
Essential Dimension ◽  
Simple Algebras

scholarly journals A lower bound on the essential dimension of PGL4 in characteristic 2

2017 ◽  
Vol 16 (04) ◽  
pp. 1750063
Author(s):  
Sanghoon Baek
Keyword(s):  
Lower Bound ◽  
Cohomology Group ◽  
Positive Characteristic ◽  
Essential Dimension ◽  
Trace Form ◽  
Simple Algebras ◽  
Characteristic 2 ◽  
Field Of Positive Characteristic

In the present paper, we provide a lower bound of the essential dimension over a field of positive characteristic via Kato’s cohomology group, defined by cokernel of a general Artin–Schreier operator. Combining this with Tignol’s result on the second trace form of simple algebras of degree [Formula: see text], we show that [Formula: see text] over a field of characteristic [Formula: see text].

scholarly journals ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC

2017 ◽  
Vol 5 ◽  
Author(s):  
KELLY MCKINNIE
Keyword(s):  
Lower Bound ◽  
Essential Dimension ◽  
Base Field ◽  
Witt Group ◽  
New Techniques ◽  
Characteristic 2 ◽  
Symbol Algebra ◽  
Bounded Below ◽  
Independent Parameters

In this article the $p$-essential dimension of generic symbols over fields of characteristic $p$ is studied. In particular, the $p$-essential dimension of the length $\ell$ generic $p$-symbol of degree $n+1$ is bounded below by $n+\ell$ when the base field is algebraically closed of characteristic $p$. The proof uses new techniques for working with residues in Milne–Kato $p$-cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on $p$-symbol algebras (i.e, degree 2 symbols) result from this work. The generic $p$-symbol algebra of length $\ell$ is shown to have $p$-essential dimension equal to $\ell +1$ as a $p$-torsion Brauer class. The second is a lower bound of $\ell +1$ on the $p$-essential dimension of the functor $\operatorname{Alg}_{p^{\ell },p}$. Roughly speaking this says that you will need at least $\ell +1$ independent parameters to be able to specify any given algebra of degree $p^{\ell }$ and exponent $p$ over a field of characteristic $p$ and improves on the previously established lower bound of 3.

scholarly journals A fiber dimension theorem for essential and canonical dimension

2012 ◽  
Vol 149 (1) ◽  
pp. 148-174 ◽  
Author(s):  
Roland Lötscher
Keyword(s):  
Algebraic Geometry ◽  
Lower Bound ◽  
Finite Type ◽  
Algebraic Groups ◽  
General Version ◽  
Essential Dimension ◽  
Algebraic Stacks ◽  
Algebraic Tori ◽  
Canonical Dimension

AbstractThe well-known fiber dimension theorem in algebraic geometry says that for every morphism f:X→Y of integral schemes of finite type the dimension of every fiber of f is at least dim X−dim Y. This has recently been generalized by Brosnan, Reichstein and Vistoli to certain morphisms of algebraic stacks f:𝒳→𝒴, where the usual dimension is replaced by essential dimension. We will prove a general version for morphisms of categories fibered in groupoids. Moreover, we will prove a variant of this theorem, where essential dimension and canonical dimension are linked. These results let us relate essential dimension to canonical dimension of algebraic groups. In particular, using the recent computation of the essential dimension of algebraic tori by MacDonald, Meyer, Reichstein and the author, we establish a lower bound on the canonical dimension of algebraic tori.

scholarly journals Essential dimension of central simple algebras

2012 ◽  
Vol 209 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Sanghoon Baek ◽  
Alexander S. Merkurjev
Keyword(s):  
Essential Dimension ◽  
Central Simple Algebras ◽  
Simple Algebras ◽  
Central Simple

FREE SPECTRA GAPS AND TAME CONGRUENCE TYPES

1995 ◽  
Vol 05 (06) ◽  
pp. 651-672 ◽  
Author(s):  
JOEL BERMAN
Keyword(s):  
Lower Bound ◽  
Finite Variety ◽  
Subdirectly Irreducible ◽  
Irreducible Algebra ◽  
Locally Finite ◽  
Finite Algebras ◽  
Locally Finite Variety ◽  
Simple Algebras ◽  
Type 3 ◽  
Free Spectra

Chapter 12 of "The Structure of Finite Algebras" by D. Hobby and R. McKenzie contains theorems revealing how the set of types appearing in a locally finite variety [Formula: see text] influences the size of the free algebra in [Formula: see text] freely generated by n elements. We provide more results in this vein. If A is a subdirectly irreducible algebra of size k, then a lower bound on the number of n-ary polynomials of A is obtained for each case that the monolith of A has type 3, 4, or 5. Examples for every k show that in each case the lower bound is the best possible. As an application of these results we show that for every finite k if all k-element simple algebras are partitioned into five classes according to their type, then algebras in each class have a sharply determined band of possible values for their free spectra. These five bands are disjoint except for some overlap on simple algebras of types 2 and 5.

scholarly journals On the convergence rate of the fraction of simple algebras

2020 ◽  
Vol 81 (4) ◽  
Author(s):  
Florian Aichinger
Keyword(s):  
Lower Bound ◽  
Convergence Rate ◽  
Simple Algebras

AbstractWe provide an improved lower bound for the convergence rate of the fraction of simple algebras using combinatorial arguments.

Sign in / Sign up

Export Citation Format

Share Document