by Altmann, Klaus; Hille, Lutz

Let Q be a finite quiver without oriented cycles. Denote by U → ℳ(Q) the fine moduli space of stable thin sincere representations of Q with respect to the canonical stability notion. We prove Ext_ℳ(Q)^l (U, U) = 0 for all l > 0 and compute the endomorphism algebra of the universal bundle U. Moreover, we obtain a necessary and sufficient condition for when this algebra is isomorphic to the path algebra of the quiver Q. If so, then the bounded derived categories of finitely generated right k Q-modules and that of coherent sheaves on ℳ(Q) are related via the full and faithful functor − ⊗_kQ^L U.

DOI: 10.1023/A:1009990727521
Print publication date: 3/1/1999
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by Daele, A. Van; Zhang, Y. H.

In this paper, we introduce a generalized Hopf Galois theory for regular multiplier Hopf algebras with integrals, which might be viewed as a generalization of the Hopf Galois theory of finite-dimensional Hopf algebras. We introduce the notion of a coaction of a multiplier Hopf algebra on an algebra. We show that there is a duality for actions and coactions of multiplier Hopf algebras with integrals. In order to study the Galois (co)action of a multiplier Hopf algebra with an integral, we construct a Morita context connecting the smash product and the coinvariants. A Galois (co)action can be characterized by certain surjectivity of a canonical map in the Morita context. Finally, we apply the Morita theory to obtain the duality theorems for actions and coactions of a co-Frobenius Hopf algebra.

DOI: 10.1023/A:1009938708033
Print publication date: 3/1/1999
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by Izhboldin, Oleg T.; Karpenko, Nikita A.

A field extension L / F is called excellent if, for any quadratic form φ over F, the anisotropic part (φL)an of φ over L is defined over F; L / F is called universally excellent if L ⋅ E / E is excellent for any field extension E / F. We study the excellence property for a generic splitting field of a central simple F-algebra. In particular, we show that it is universally excellent if and only if the Schur index of the algebra is not divisible by 4. We begin by studying the torsion in the second Chow group of products of Severi–Brauer varieties and its relationship with the relative Galois cohomology group H3(L / F) for a generic (common) splitting field L of the corresponding central simple F-algebras.

DOI: 10.1023/A:1009910324736
Print publication date: 3/1/1999
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by Green, James A.

The ‘discrete series’ characters of the finite general linear group GL(n, q) are expressed as uniquely defined integral linear combinations of characters induced from linear characters on certain subgroups Hd, n of GL(n, q). The coefficients in these linear combinations are determined (for all n, q) by a family of polynomials rλ(T) ∈ Z[T] indexed by the set of all partitions λ.

DOI: 10.1023/A:1009975523963
Print publication date: 3/1/1999
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