Essential dimension of quadrics

Let X be an anisotropic projective quadric over a field F of characteristic not 2. The essential dimension dim_es(X) of X, as defined by Oleg Izhboldin, is dim_es(X)=dim(X)-i(X) +1, where i(X) is the first Witt index of X (i.e., the Witt index of X over its function field).Let Y be a complete (possibly singular) algebraic variety over F with all closed points of even degree and such that Y has a closed point of odd degree over F(X). Our main theorem states that dim_es(X)dim(Y) and that in the case dim_es(X)=dim(Y) the quadric X is isotropic over F(Y).Applying the main theorem to a projective quadric Y, we get a proof of Izhboldins conjecture stated as follows: if an anisotropic quadric Y becomes isotropic over F(X), then dim_es(X)dim_es(Y), and the equality holds if and only if X is isotropic over F(Y). We also solve Knebuschs problem by proving that the smallest transcendence degree of a generic splitting field of a quadric X is equal to dim_es(X). To the memory of Oleg Izhboldin     

On abelian subalgebras and ideals of maximal dimension in supersolvable Lie algebras

In this paper, the main objective is to compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian subalgebras of solvable Lie algebras and study solvable Lie algebras containing an abelian subalgebra of codimension 2. Finally, we prove that nilpotent Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not 2. Throughout the paper, we also give several examples to clarify some results.     

Essential dimension of Hermitian spaces

Given an hermitian space we compute its essential dimension, Chow motive and prove its incompressibility in certain dimensions.     

The minimal dimension of maximal commutative subalgebras of full matrix algebras

Let F be a field and A a maximal commutative subalgebra of the full matrix algebra M_n(F). It is shown that dim A > (2n)^{$\text{2}\text{3}$} ? 1. It is also shown that if the radical of A has cube zero, then dim A ? [3n^{$\text{2}\text{3}$} ? 4], and that this result is best possible for infinitely many natural numbers n.     

Essential dimension: a survey

In the paper we survey research on the essential dimension. The highlights of the survey are the computations of the essential dimensions of finite groups, groups of multiplicative type and the spinor groups. We present self-contained proofs of these cases and give applications in the theory of simple algebras and quadratic forms.     

Essential dimension of central simple algebras

Let p be a prime integer, 1≤s≤r be integers and F be a field of characteristic different from p. We find upper and lower bounds for the essential p-dimension ed _p (\( Al{{g}_{{{{p}^r},{{p}^s}}}} \)) of the class \( Al{{g}_{{{{p}^r},{{p}^s}}}} \) of central simple algebras of degree p ^r and exponent dividing p ^s . In particular, we show that ed(Alg _8,2)=ed₂(Alg _8,2)=8 and ed _p (\( Al{{g}_{{{{p}^2},p}}} \))=p ²+p for p odd.     

Essential dimension of finite groups in prime characteristic

Let F be a field of characteristic $p>0$ and G be a smooth finite algebraic group over F. We compute the essential dimension ${\mathrm{ed}}_F(G;p)$ of G at p. That is, we show that

${\mathrm{ed}}_F(G;p)=\{\begin{array}{cc}1,\hfill & \text{if}p\text{divides}|G|,\text{and}\hfill \\ 0,\hfill & \text{otherwise.}\hfill \end{array}$

     

Essential dimension of Abelian varieties over number fields

We show that the essential dimension of a non-trivial Abelian variety over a number field is infinite. To cite this article: P. Brosnan, R. Sreekantan, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Essential extensions, excellent extensions and finitely presented dimension

In Section 1 of this paper, we investigate the finitely presented dimension of an essential extension for a module, and obtain results concerning an essential extension of a torsion-free module. We partially answer the question: When is an essential extension of a finitely presented module (an almost finitely presented module) also finitely presented (almost finitely presented)? In Section 2, we study theC-excellent extensions and the finitely presented dimensions. We obtain some results on the homological dimensions of matrix rings and group rings.     

Pfister involutions

The question of the existence of an analogue, in the framework of central simple algebras with involution, of the notion of Pfister form is raised. In particular, algebras with orthogonal involution which split as a tensor product of quaternion algebras with involution are studied. It is proven that, up to degree 16, over any extension over which the algebra splits, the involution is adjoint to a Pfister form. Moreover, cohomological invariants of those algebras with involution are discussed.     

Essential dimension of algebraic groups,including bad characteristic

Let A and G be finite groups of relatively prime orders and assume that A acts on G via automorphisms. We study how certain conditions on G imply its solvability when we assume the existence of a unique A-invariant Sylow p-subgroup for p equal to 2 or 3.     

Essential dimension and the second serre conjecture for exceptional groups

We study the essential dimension of exceptional connected simply connected algebraic groups over algebraically closed fields. In the present paper, we find upper estimates for the essential dimensions of the groupsF ₄ ,E ₆ , andE ₇ . For the groupF ₄ , the upper estimate, thus obtained coincides with the known lower estimate. We also prove the second Serre conjecture for the groupE ₆ and for a function field. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 539–547, October, 2000.