by Berger, Roland

Any quadratic algebra endowed with an ordered set of generators can bedescribed by some linear map called a reduction operator. The general lineargroup naturally acts on reduction operators, which allows us to introduceweak and strong confluence. With respect to these notions, a completeclassification for two generators and complex coefficients is obtainedshowing that weak confluence is equivalent to Koszulity in this case. Bycontrast, some Sklyanin algebras with three generators fail to be weaklyconfluent. For an arbitrary number of generators and under some assumptionson the first terms of the Hilbert series, a weak confluence hypothesis isequivalent to some rather drastic conditions which determine the whole ofthe Hilbert series.

DOI: 10.1023/A:1009918131382
Print publication date: 9/1/1998
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by Donkin, Stephen

This is the second part of our study of the existence ofAuslander–Reiten sequences of group representations. In Part I weconsidered representations of group schemes in characteristic 0; in thispart we consider representations of group schemes in characteristicp; and in Part III we give applications to representations ofabstract groups and Lie algebras.

DOI: 10.1023/A:1009936706866
Print publication date: 9/1/1998
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by Picavet, Gabriel

We define decent schemes and canonically decent projective schemes. Forsuch schemes, total quotient schemes exist, allowing to get thenormalization, seminormalization and t-closure of a decent scheme as ascheme. We exhibit the seminormalization and t-closure of a filtration on aring. If A is a decent ring and F a regular filtration onA, the associated Rees ring R is decent andProj(R) is canonically decent. The seminormalization and t-closureof R are Rees rings and the seminormalization and t-closure ofProj(R) are gotten by using projective morphisms.

DOI: 10.1023/A:1009921701627
Print publication date: 9/1/1998
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