243 research outputs found
Categories of lattices, and their global structure in terms of almost split sequences
A major part of Iyama’s characterization of
Auslander-Reiten quivers of representation-finite orders Λ consists
of an induction via rejective subcategories of Λ-lattices, which
amounts to a resolution of Λ as an isolated singularity. Despite
of its useful applications (proof of Solomon’s second conjecture
and the finiteness of representation dimension of any artinian algebra), rejective induction cannot be generalized to higher dimensional Cohen-Macaulay orders Λ. Our previous characterization
of finite Auslander-Reiten quivers of Λ in terms of additive functions [22] was proved by means of L-functors, but we still had to
rely on rejective induction. In the present article, this dependence
will be eliminated
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
The Auslander bijections: How morphisms are determined by modules
Let A be an artin algebra. In his seminal Philadelphia Notes published in
1978, M. Auslander introduced the concept of morphisms being determined by
modules. Auslander was very passionate about these ivestigations (they also
form part of the final chapter of the Auslander-Reiten-Smaloe book and could
and should be seen as its culmination), but the feedback until now seems to be
somewhat meager. The theory presented by Auslander has to be considered as an
exciting frame for working with the category of A-modules, incorporating all
what is known about irreducible maps (the usual Auslander-Reiten theory), but
the frame is much wider and allows for example to take into account families of
modules - an important feature of module categories. What Auslander has
achieved is a clear description of the poset structure of the category of
A-modules as well as a blueprint for interrelating individual modules and
families of modules. Auslander has subsumed his considerations under the
heading of "morphisms being determined by modules". Unfortunately, the wording
in itself seems to be somewhat misleading, and the basic definition may look
quite technical and unattractive, at least at first sight. This could be the
reason that for over 30 years, Auslander's powerful results did not gain the
attention they deserve. The aim of this survey is to outline the general
setting for Auslander's ideas and to show the wealth of these ideas by
exhibiting many examples
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