Tame group actions on central simple algebras

We study actions of linear algebraic groups on finite-dimensional central simple algebras. We describe the fixed algebra for a broad class of such actions.Comment: 19 pages, LaTeX; slightly revised; final version will appear in Journal of Algebr

Nesting maps of Grassmannians

Let F be a field and i < j be integers between 1 and n. A map of Grassmannians f : Gr(i, F^n) --> Gr(j, F^n) is called nesting, if l is contained in f(l) for every l in Gr(i, F^n). We show that there are no continuous nesting maps over C and no algebraic nesting maps over any algebraically closed field F, except for a few obvious ones. The continuous case is due to Stong and Grover-Homer-Stong; the algebraic case in characteristic zero can also be deduced from their results. In this paper we give new proofs that work in arbitrary characteristic. As a corollary, we give a description of the algebraic subbundles of the tangent bundle to the projective space P^n over F. Another application can be found in a recent paper math.AC/0306126 of George Bergman

Trace forms of Galois extensions in the presence of a fourth root of unity

We study quadratic forms that can occur as trace forms of Galois field extensions L/K, under the assumption that K contains a primitive 4th root of unity. M. Epkenhans conjectured that any such form is a scaled Pfister form. We prove this conjecture and classify the finite groups G which admit a G-Galois extension L/K with a non-hyperbolic trace form. We also give several applications of these results.Comment: 19 pages, to appear in International Math Research Notice

Group actions on central simple algebras: a geometric approach

We study actions of linear algebraic groups on central simple algebras using algebro-geometric techniques. Suppose an algebraic group G acts on a central simple algebra A of degree n. We are interested in questions of the following type: (a) Do the G-fixed elements form a central simple subalgebra of A of degree n? (b) Does A have a G-invariant maximal subfield? (c) Does A have a splitting field with a G-action, extending the G-action on the center of A? Somewhat surprisingly, we find that under mild assumptions on A and the actions, one can answer these questions by using techniques from birational invariant theory (i.e., the study of group actions on algebraic varieties, up to equivariant birational isomorphisms). In fact, group actions on central simple algebras turn out to be related to some of the central problems in birational invariant theory, such as the existence of sections, stabilizers in general position, affine models, etc. In this paper we explain these connections and explore them to give partial answers to questions (a)-(c).Comment: 33 pages. Final version, to appear in Journal of Algebra. Includes a short new section on Brauer-Severi varietie

Resolving G-torsors by abelian base extensions

Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions on k and G, there exists a finite k-subgroup S of G such that the natural map H^1(K, S) -> H^1(K, G) is surjective for every field extension K/k. We give several applications of this result in the case where k an algebraically closed field of characteristic zero and K/k is finitely generated. In particular, we prove that for every z in H^1(K, G) there exists an abelian field extension L/K such that z_L \in H^1(L, G) is represented by a G-torsor over a projective variety. From this we deduce that z_L has trivial point obstruction. We also show that a (strong) variant of the algebraic form of Hilbert's 13th problem implies that the maximal abelian extension of K has cohomological dimension =< 1. The last assertion, if true, would prove conjectures of Bogomolov and Koenigsmann, answer a question of Tits and establish an important case of Serre's Conjecture II for the group E_8.Comment: New material added on the no-name lemma in Section 4 and on Hilbert's 13th problem in Section 9. A mistake in the proof of Proposition 2.3 is correcte

Projectively simple rings

We introduce the notion of a projectively simple ring, which is an infinite-dimensional graded k-algebra A such that every 2-sided ideal has finite codimension in A (over the base field k). Under some (relatively mild) additional assumptions on A, we reduce the problem of classifying such rings (in the sense explained in the paper) to the following geometric question, which we believe to be of independent interest. Let X is a smooth irreducible projective variety. An automorphism f: X -> X is called wild if it X has no proper f-invariant subvarieties. We conjecture that if X admits a wild automorphism then X is an abelian variety. We prove several results in support of this conjecture; in particular, we show that the conjecture is true if X is a curve or a surface. In the case where X is an abelian variety, we describe all wild automorphisms of X. In the last two sections we show that if A is projectively simple and admits a balanced dualizing complex, then Proj(A) is Cohen-Macaulay and Gorenstein.Comment: Some new material has been added in Section 1; to appear in Advances in Mathematic

An Embedding Property of Universal Division Algebras

AbstractLet A be a central simple algebra of degree n and let k be a subfield of its center. We show that A contains a copy of the universal division algebra Dm, n(k) generated by m generic n × n matrices if and only if trdegkA ≥ trdegkDm, n(k) = (m − 1)n2 + 1. Moreover, if in addition the center of A is finitely and separately generated over k then "almost all" division subalgebras of A generated by m elements are isomorphic to Dm, n(k). In the last section we give an application of our main result to the question of embedding free groups in division algebras

Polynomial identity rings as rings of functions

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGL_n-action, regular and rational "functions" on them are matrix-valued, "coordinate rings" are prime polynomial identity algebras, and "function fields" are central simple algebras of degree n. In particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the "coordinate ring" of a "variety" in this setting. For n = 1 our definitions and results reduce to those of classical affine algebraic geometry.Comment: 24 pages. This is the final version of the article, to appear in J. Algebra. Several proofs have been streamlined, and a new section on Brauer-Severi varieties has been adde