5,338 research outputs found
Sheaves that fail to represent matrix rings
There are two fundamental obstructions to representing noncommutative rings
via sheaves. First, there is no subcanonical coverage on the opposite of the
category of rings that includes all covering families in the big Zariski site.
Second, there is no contravariant functor F from the category of rings to the
category of ringed categories whose composite with the global sections functor
is naturally isomorphic to the identity, such that F restricts to the Zariski
spectrum functor Spec on the category of commutative rings (in a compatible way
with the natural isomorphism). Both of these no-go results are proved by
restricting attention to matrix rings.Comment: 13 pages; final versio
A prime ideal principle for two-sided ideals
Many classical ring-theoretic results state that an ideal that is maximal
with respect to satisfying a special property must be prime. We present a
"Prime Ideal Principle" that gives a uniform method of proving such facts,
generalizing the Prime Ideal Principle for commutative rings due to T.Y. Lam
and the author. Old and new "maximal implies prime" results are presented, with
results touching on annihilator ideals, polynomial identity rings, the
Artin-Rees property, Dedekind-finite rings, principal ideals generated by
normal elements, strongly noetherian algebras, and just infinite algebras.Comment: 22 page
On discretization of C*-algebras
The C*-algebra of bounded operators on the separable infinite-dimensional
Hilbert space cannot be mapped to a W*-algebra in such a way that each unital
commutative C*-subalgebra C(X) factors normally through .
Consequently, there is no faithful functor discretizing C*-algebras to
AW*-algebras, including von Neumann algebras, in this way.Comment: 5 pages. Please note that arXiv:1607.03376 supersedes this paper. It
significantly strengthens the main results and includes positive results on
discretization of C*-algebra
Skew Calabi-Yau triangulated categories and Frobenius Ext-algebras
We investigate the conditions that are sufficient to make the Ext-algebra of
an object in a (triangulated) category into a Frobenius algebra and compute the
corresponding Nakayama automorphism. As an application, we prove the conjecture
that hdet() = 1 for any noetherian Artin-Schelter regular (hence skew
Calabi-Yau) algebra A.Comment: 31 page
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