by Gruenberg, K. W.

Given a finite group G and a G-free resolution F
_* of Z, then d

_G
(Im(F

_m+1→F

_m
))−∑(−1)
^m−i

d

_G
(F

_i
) is almost always an invariant of G.

DOI: 10.1023/A:1009961813539
Print publication date: 4/1/2001
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by Kemer, Alexander

We describe the multilinear components of the prime subvarieties of the variety Var(M
₂(F)) generated by the matrix algebra of order 2 over a field of characteristic p>0.

DOI: 10.1023/A:1009957729468
Print publication date: 4/1/2001
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by Ringel, Claus Michael

Let R be a ring. Any R-module M which is Artinian or Noetherian can be written as the direct sum of a finite number of indecomposable R-modules. The theorem of Krull–Remak–Schmidt asserts that in the case where M is of finite length, such a decomposition is unique up to isomorphism. On the other hand, examples of Noetherian R-modules which have essentially different decompositions have been known for a long time. The first examples of Artinian R-modules with essentially different decompositions were published only in 1995 by Facchini, Herbera, Levy and Vámos. In order to construct such examples, one needs to deal with suitable rings R. Note that for R Noetherian or commutative, all the Artinian modules have the Krull–Remak–Schmidt property. In 1998, Facchini raised the problem of whether the same is true in the case where R is a local ring. The aim of this note is to show that this is not so: we are going to present a local ring R and Artinian R-modules M with essentially different direct decompositions into indecomposables. The military importance of these results has been discussed during the NATO meeting at Constantia (August 2000) which was organized by K. W. Roggenkamp.

DOI: 10.1023/A:1009905728560
Print publication date: 4/1/2001
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by de Chela, Delia Flores; Green, James A.

The ‘plus part’ U
⁺ of a quantum group U

_q
() has been identified by M. Rosso with a subalgebra G
_sym of an algebra G which is a quantized version of R. Ree’s shuffle algebra. Rosso has shown that G
_sym and G – and hence also Hopf algebras which are analogues of quantum groups – can be defined in a much wider context. In this paper we study one of Rosso’s quantizations, which depends on a family of parameters t

_ij
. G
_sym is determined by a family of matrices Ω^α whose coefficients are polynomials in the t

_ij
. The determinants of the Ω^α factorize into a number of irreducible polynomials, and our main Theorem 5.2a gives strong information on these factors. This can be regarded as a first step towards the (still very distant!) goal, the classification of the symmetric algebras G
_sym which can be obtained by giving special values to the parameters t

_ij
.

DOI: 10.1023/A:1009953611721
Print publication date: 4/1/2001
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by Marcus, Andrei

We study Rickard equivalences between p-blocks of twisted group algebras and their local structure, in connection with Dade’s conjectures. We prove that an extended form of Broué’s conjecture implies Dade’s Inductive Conjecture in the Abelian defect group case; this is a consequence of the fact that Rickard equivalences induced by complexes of graded bimodules preserve the relevant Clifford theoretical invariants. As an application, we show that these conjectures hold for p′-extensions of blocks with cyclic defect groups.

DOI: 10.1023/A:1009901627651
Print publication date: 4/1/2001
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by Reiten, Idun; Van den Bergh, Michel

Let C be a connected Noetherian hereditary Abelian category with a Serre functor over an algebraically closed field k, with finite-dimensional homomorphism and extension spaces. Using the classification of such categories from our 1999 preprint, we prove that if C has some object of infinite length, then the Grothendieck group of C is finitely generated if and only if C has a tilting object.

DOI: 10.1023/A:1009902810813
Print publication date: 4/1/2001
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