3,325 research outputs found
Generic Noncommutative Surfaces
We study a class of noncommutative surfaces and their higher dimensional
analogues which provide answers to several open questions in noncommutative
projective geometry. Specifically, we give the first known graded algebras
which are noetherian but not strongly noetherian, answering a question of
Artin, Small, and Zhang. In addition, these examples are maximal orders and
satisfy the condition but not for , answering a
questions of Stafford and Zhang and a question of Stafford and Van den Bergh.
Finally, we show that these algebras have finite cohomological dimension.Comment: 43 pages, Latex, to appear in Advances in Math. Result on finite
global dimension added, other minor change
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
Specialization orders on atom spectra of Grothendieck categories
We introduce systematic methods to construct Grothendieck categories from
colored quivers and develop a theory of the specialization orders on the atom
spectra of Grothendieck categories. We show that any partially ordered set is
realized as the atom spectrum of some Grothendieck category, which is an analog
of Hochster's result in commutative ring theory. We also show that there exists
a Grothendieck category which has empty atom spectrum but has nonempty
injective spectrum.Comment: 39 page
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