215 research outputs found
Protoadditive functors, derived torsion theories and homology
Protoadditive functors are designed to replace additive functors in a
non-abelian setting. Their properties are studied, in particular in
relationship with torsion theories, Galois theory, homology and factorisation
systems. It is shown how a protoadditive torsion-free reflector induces a chain
of derived torsion theories in the categories of higher extensions, similar to
the Galois structures of higher central extensions previously considered in
semi-abelian homological algebra. Such higher central extensions are also
studied, with respect to Birkhoff subcategories whose reflector is
protoadditive or, more generally, factors through a protoadditive reflector. In
this way we obtain simple descriptions of the non-abelian derived functors of
the reflectors via higher Hopf formulae. Various examples are considered in the
categories of groups, compact groups, internal groupoids in a semi-abelian
category, and other ones
Unbounded quasitraces, stable finiteness and pure infiniteness
We prove that if A is a \sigma-unital exact C*-algebra of real rank zero,
then every state on K_0(A) is induced by a 2-quasitrace on A. This yields a
generalisation of Rainone's work on pure infiniteness and stable finiteness of
crossed products to the non-unital case. It also applies to k-graph algebras
associated to row-finite k-graphs with no sources. We show that for any k-graph
whose C*-algebra is unital and simple, either every twisted C*-algebra
associated to that k-graph is stably finite, or every twisted C*-algebra
associated to that k-graph is purely infinite. Finally we provide sufficient
and necessary conditions for a unital simple k-graph algebra to be purely
infinite in terms of the underlying k-graph.Comment: 38 page
Factorizations of Elements in Noncommutative Rings: A Survey
We survey results on factorizations of non zero-divisors into atoms
(irreducible elements) in noncommutative rings. The point of view in this
survey is motivated by the commutative theory of non-unique factorizations.
Topics covered include unique factorization up to order and similarity, 2-firs,
and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and
Jordan and generalizations thereof. We recall arithmetical invariants for the
study of non-unique factorizations, and give transfer results for arithmetical
invariants in matrix rings, rings of triangular matrices, and classical maximal
orders as well as classical hereditary orders in central simple algebras over
global fields.Comment: 50 pages, comments welcom
Torsion pairs in a triangulated category generated by a spherical object
We extend Ng's characterisation of torsion pairs in the 2-Calabi-Yau
triangulated category generated by a 2-spherical object to the characterisation
of torsion pairs in the w-Calabi-Yau triangulated category, , generated by
a w-spherical object for any integer w. Inspired by the combinatorics of
for w < 0, we also characterise the torsion pairs in a certain w-Calabi-Yau
orbit category of the bounded derived category of the path algebra of Dynkin
type A.Comment: v2: 36 pages, 11 figures, added Section 4 which deals with extensions
whose outer terms are decomposable, minor changes in presentation, accepted
in J. Algebr
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