by

DOI: 10.1023/A:1017137723680
Print publication date: 12/1/1998
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by Smith, S. Paul; Zhang, James J.

This paper concerns curves on noncommutative schemes, hereafter called quasi-schemes. Aquasi-scheme X is identified with the category
$$Mod{\text{ }}X$$
ofquasi-coherent sheaves on it. Let X be a quasi-scheme having a regularly embeddedhypersurface Y. Let C be a curve on X which is in ‘good position’ withrespect to Y (see Definition 5.1) – this definition includes a requirement that Xbe far from commutative in a certain sense. Then C is isomorphic to
$$\mathbb{V}_n^1 $$
, where n is the number of points of intersection of Cwith Y. Here
$$\mathbb{V}_n^1 $$
, or rather
$$Mod{\text{ }}\mathbb{V}_n^1 $$
, is the quotient category
$$GrModk[x_1 , \ldots ,x_n ]/\{ {\text{K}}\dim \leqslant n – 2\} {\text{ of }}\mathbb{Z}^n $$
-graded modules over the commutative polynomial ring, modulo the subcategory ofmodules having Krull dimension ≤ n − 2. This is a hereditary category whichbehaves rather like
$$Mod\mathbb{P}^1 $$
, the category of quasi-coherentsheaves on
$$\mathbb{P}^1 $$
.

DOI: 10.1023/A:1009984608942
Print publication date: 12/1/1998
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by Donkin, Stephen

This is the third and final part of our study of the existence of Auslander–Reiten sequences ofgroup representations. In Part I we considered representations of group schemes in characteristic 0.In Part II we considered representations of group schemes in characteristic p. In this partwe give applications to representations of abstract groups and Lie algebras.

DOI: 10.1023/A:1009969620734
Print publication date: 12/1/1998
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by Külshammer, Burkhard; Robinson, Geoffrey R.

We prove a new formula about local control of the number of p-regular conjugacyclasses of a finite group. We then relate the results to Alperin’s weight conjecture to obtain newresults describing the number of simple modules for a finite group in terms of weights of solvablesubgroups. Finally, we use the results to obtain new formulations of Alperin’s weight conjecture,and to obtain restrictions on the structure of a minimal counterexample.

DOI: 10.1023/A:1009961408896
Print publication date: 12/1/1998
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by Rumynin, Dmitriy

We set up a general framework to study representation theory of certain algebras whichusually appear in the study of restricted Lie algebras or various quantum objects at roots of unity.The object of the study is a Hopf–Galois extension with central invariants. It turns out that theseextensions possess some geometric properties which are close to those of principal bundles andFrobenius manifolds. We define Hopf–Galois extensions of not necessarily affine schemes andprove that the classification problem of such extensions leads to a stack.

DOI: 10.1023/A:1009944607078
Print publication date: 12/1/1998
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