Results 1  10
of
11
COHERENT ALGEBRAS AND NONCOMMUTATIVE PROJECTIVE LINES
, 2007
"... Abstract. A wellknown conjecture says that every onerelator group is coherent. We state and partly prove a similar statement for graded associative algebras. In particular, we show that every Gorenstein algebra A of global dimension 2 is graded coherent. This allows us to define a noncommutative a ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
(Show Context)
Abstract. A wellknown conjecture says that every onerelator group is coherent. We state and partly prove a similar statement for graded associative algebras. In particular, we show that every Gorenstein algebra A of global dimension 2 is graded coherent. This allows us to define a noncommutative analogue of the projective line P1 as a noncommutative scheme based on the coherent noncommutative spectrum
Ampleness of twosided tilting complexes
, 2012
"... In this paper we define the notion of ampleness for twosided tilting complexes over finite dimensional algebras and prove its basic properties. We call a finite dimensional kalgebra A of finite global dimension Fano if (A¤[−d])¡1 is ample for some d ≥ 0. For example geometric algebras in the sense ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
In this paper we define the notion of ampleness for twosided tilting complexes over finite dimensional algebras and prove its basic properties. We call a finite dimensional kalgebra A of finite global dimension Fano if (A¤[−d])¡1 is ample for some d ≥ 0. For example geometric algebras in the sense of BondalPolishchuk are Fano. We give a characterization of representation type of a quiver from a noncommutative algebrogeometric view point, that is, a finite acyclic quiver has finite representation type if and only if its path algebra is fractional CalabiYau, and a finite acyclic quiver has infinite representation type if and only if its path algebra is Fano. 0
A 3CalabiYau algebra with G2 symmetry constructed from the Octonions, arXiv:1104.3824v1
"... ar ..."
On degenerations and deformations of Sklyanin algebras
"... For my family and friends. ii ACKNOWLEDGEMENTS It is difficult to picture my graduate career without the encouragement and influence of Professor Karen Smith. She recruited me to the University of Michigan’s graduate mathematics program. We met regularly throughout my time at Michigan. She introduc ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
For my family and friends. ii ACKNOWLEDGEMENTS It is difficult to picture my graduate career without the encouragement and influence of Professor Karen Smith. She recruited me to the University of Michigan’s graduate mathematics program. We met regularly throughout my time at Michigan. She introduced me to algebraic geometry. She even became my ‘foster ’ adviser when my adviser transferred from Michigan’s math department. This list goes on and on, and I could honestly fill pages with examples of her generosity and mentorship. In short, I cannot thank her enough! Professor Jeanne Wald has been my mentor since my sophomore year at Michigan State University. I enrolled in four of her courses at MSU, one of which she taught me how to write proofs, and another of which sparked my interest in noncommutative algebra. Needless to say, she has been a great professional influence. More importantly, it is our candid conversations about navigating academia (and life!) that has
Lecture notes on noncommutative algebraic geometry and noncommutative tori
, 2008
"... ..."
(Show Context)
AMPLENESS OF TWOSIDED TILTING COMPLEXES AND FANO ALGEBRAS
"... Abstract. From the view point of noncommutative algebraic geometry (NCAG), a twosided tilting complex is an analog of a line bundle. In this paper we define the notion of ampleness for twosided tilting complexes over finite dimensional algebras of finite global dimension, and prove its basic prope ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. From the view point of noncommutative algebraic geometry (NCAG), a twosided tilting complex is an analog of a line bundle. In this paper we define the notion of ampleness for twosided tilting complexes over finite dimensional algebras of finite global dimension, and prove its basic properties, which justify the name ”ampleness”. From the view point of NCAG, Serre functors are considered to be shifted canonical bundles. A finite dimensional algebra A of finite global dimension is called Fano if the shifted Serre functor A∗[¡d] is antiample. Some classes of algebras studies before are Fano. We show by an example that the property of A∗[¡d] from the view point of NCAG captures some representation theoretic property of the algebra A. From our view point, we give a structure theorem of ASregular algebras. ASregular algebras are defined to extract a good homological property of polynomial algebras. Our theorem shows that ASregular algebra is polynomial algebra in some sense. 1.
NONCOMMUTATIVE ALGEBRAIC GEOMETRY AND THE STUDY OF NONCOMMUTATIVE TORI
, 2008
"... I would like to thank all the organizers of the International Workshop on Noncommutative Geometry, 2005 for giving me this opportunity to speak. In section 1 we shall browse through some interesting definitions and constructions which will be referred to later on. In section 2 we shall deal with non ..."
Abstract
 Add to MetaCart
(Show Context)
I would like to thank all the organizers of the International Workshop on Noncommutative Geometry, 2005 for giving me this opportunity to speak. In section 1 we shall browse through some interesting definitions and constructions which will be referred to later on. In section 2 we shall deal with noncommutative