Projectivity of modules for infinitesimal unipotent group schemes

In this paper, it is shown that the projectivity of a rational module for an infinitesimal unipotent group scheme over an algebraically closed field of positive characteristic can be detected on a family of closed subgroups.     

The tame infinitesimal groups of odd characteristic

Given an infinitesimal group G over an algebraically closed field k of characteristic p?3, we provide criteria for the principal block B₀(G) of its algebra of distributions to be of tame representation type. These are employed in conjunction with Galois coverings to determine the structure of G modulo its multiplicative center as well as the quiver and the relations of the algebra B₀(G).     

Galois subfields of tame division algebras

We show that a finite-dimensional tame division algebra D over a Henselian field F has a maximal subfield Galois over F if and only if its residue division algebra \(\overline D \) has a maximal subfield Galois over the residue field \(\overline F \).     

Infinitesimal group schemes as iterative differential Galois groups

This article is concerned with Galois theory for iterative differential fields (ID-fields) in positive characteristic. More precisely, we consider purely inseparable Picard-Vessiot extensions, because these are the ones having an infinitesimal group scheme as iterative differential Galois group. In this article we prove a necessary and sufficient condition to decide whether an infinitesimal group scheme occurs as Galois group scheme of a Picard-Vessiot extension over a given ID-field or not. In particular, this solves the inverse ID-Galois problem for infinitesimal group schemes. Furthermore, this gives a tool to tell whether all purely inseparable ID-extensions are in fact Picard-Vessiot extensions.     

A Galois correspondence for compact group actions on C-algebras

In this paper, we prove a Galois correspondence for compact group actions on C^?-algebras in the presence of a commuting minimal action. Namely, we show that there is a one-to-one correspondence between the C^?-subalgebras that are globally invariant under the compact action and the commuting minimal action, that in addition contain the fixed point algebra of the compact action and the closed, normal subgroups of the compact group.     

Galois groups of quantum group actions and regularity of fixed-point algebras

It is shown that, for a minimal and integrable action of a locally compact quantum group on a factor, the group of automorphisms of the factor leaving the fixed-point algebra pointwise invariant is identified with the intrinsic group of the dual quantum group. It is proven also that, for such an action, the regularity of the fixed-point algebra is equivalent to the cocommutativity of the quantum group.

     

Partial actions and Galois theory

In this article, among other results, we develop a Galois theory of commutative rings under partial actions of finite groups, extending the well-known results by S.U. Chase, D.K. Harrison and A. Rosenberg.     

Galois actions on Q-curves and winding quotients

We prove two ``large images' results for the Galois representations attached to a degree d Q-curve E over a quadratic field K: if K is arbitrary, we prove maximality of the image for every prime p>13 not dividing d, provided that d is divisible by q (but d≠q) with q=2 or 3 or 5 or 7 or 13. If K is real we prove maximality of the image for every odd prime p not dividing d D, where D= disc(K), provided that E is a semistable Q-curve. In both cases we make the (standard) assumptions that E does not have potentially good reduction at all primes p∤6 and that d is square free. The first author is supported by BFM2003-06092.     

The Valentiner group as Galois group

We obtain the complete set of solutions to the Galois embedding problem given by the Valentiner group as a triple cover of the alternating group .

     

Invariant group orderings and Galois conjugates

This paper investigates conditions under which a given automorphism of a residually torsion-free nilpotent group respects some ordering of the group. For free groups and surface groups, this has relevance to ordering the fundamental groups of three-dimensional manifolds which fiber over the circle.     

The generic fiber of the universal deformation space associated to a tame Galois representation

We will study the generic fiber over of the universal deformation ring R _Q , as defined by Mazur, for deformations unramified outside a finite set of primes Q of a given Galois representation , E a number field, k a finite field of characteristic l. The main result will be that, if ˉρ is tame and absolutely irreducible, and if one assumes the Leopoldt conjecture for the splitting field E ₀ of , then defines a smooth l-adic analytic variety, near the trivial lift ρ₀ of ˉρ, whose dimension is given by cohomological constraints and as predicted by Mazur. As a corollary it follows that, in the cases considered here, R _Q is a quotient of by an ideal I generated by exactly m equations, where and . Under the above assumptions for and ˉρ odd, using ideas of Coleman, Gouvêa and Mazur it should now be possible to show that modular points are Zariski-dense in the component of , that contains the trivial lift ρ₀, provided this lift satisfies the Artin conjecture and E ₀ satisfies the Leopoldt conjecture. Furthermore, in the Borel case, we show that the Krull dimension of R _Q can exceed any given number, provided Q is chosen appropriately. At the same time, we present some evidence that despite this fact, one might however expect that the dimension of the generic fiber is given by the same cohomological formula as in the tame case. Received: 12 December 1997 / Revised version: 5 February 1998     

Galois actions on torsion points of one-dimensional formal modules

Let F be a local non-Archimedean field with ring of integers o and uniformizer ?, and fix an algebraically closed extension k of the residue field of o. Let X be a one-dimensional formal o-module of F-height n over k. By the work of Drinfeld, the universal deformation X of X is a formal group over a power series ring R₀ in n−1 variables over Wo(k). For h∈{0,…,n−1} let Uh⊂Spec(R₀) be the locus where the connected part of the associated ?-divisible module X[?^∞] has height h. Using the theory of Drinfeld level structures we show that the representation of π₁(Uh) on the Tate module of the étale quotient is surjective.     

Classifying tame blocks and related algebras up to stable equivalences of Morita type

We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of Erdmann’s algebras of dihedral, semi-dihedral and quaternion type and obtain as byproduct the validity of the Auslander-Reiten conjecture for stable equivalences of Morita type between two algebras, one of which is of dihedral, semi-dihedral or quaternion type.