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Stephen Zito

Publications and source records attributed to Stephen Zito.

11 recordsLinked to original sources
arXiv2021

Three Results Concerning Auslander Algebras

Our first result provides a new characterization of Auslander algebras using a property of hereditary torsion pairs. The second result shows an Auslander algebra $\Lambda$ is left or right glued if and only if $\Lambda$ is representation-finite. Finally, our third result shows the module category of any Auslander algebra contains a tilting module with a particular property, which we call the hereditary property. Applications of this property are investigated.

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arXiv2020

Several Results Concerning Convex Subcategories

We apply the notion of a full convex subcategory to a wide range of algebras including tilted, quasi-tilted, shod, weakly shod, left and right glued, laura, simply connected, strongly simply connected, left supported, and cluster-tilted. In particular, given an algebra $\Lambda$ from one of the aforementioned classes, we investigate certain factor algebras $\Lambda/I$ where $I$ is an ideal generated by a suitable idempotent.

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arXiv2018

A New Proof Concerning Quasi-Tilted Algebras

Let $\Lambda$ be a quasi-tilted algebra. If $\Lambda$ is representation-finite, it was shown by Happel, Reiten, and Smal{\o} that $\Lambda$ is tilted. We provide a new, short proof of this result.

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arXiv2018

Auslander algebras which are tilted

Let $A$ be an Auslander algebra of global dimension equal to 2. We provide a necessary and sufficient condition for $A$ to be a tilted algebra. In particular, $A$ is tilted if and only if pd$(\tau_{A}\Omega_{A}DA)\leq1$.

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arXiv2017

Bongartz ${\tau}$-Complements Over Split-By-Nilpotent Extensions

Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E, and let M be a ${\tau}$-rigid C-module with U its Bongartz ${\tau}$-complement. If the induced module, $M{\otimes_C}B$, is ${\tau}$-rigid as a B-module, we give a necessary and sufficient condition for $U{\otimes_C}B$ to be its Bongartz ${\tau}$-complement in mod B. If M is ${\tau}$-rigid in mod B, we again provide a necessary and sufficient condition for $U{\otimes_C}B$ to be its Bongartz ${\tau}$-complement in mod B.

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arXiv2016

Projective Dimensions and Extensions of Modules from Titled to Cluster-Tilted Algebras

We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. We investigate how various properties of a C-module are affected when considered in the module category of B. We give a complete classification of the projective dimension of a C-module inside the module category of B. If a C-module M is rigid, we show two sufficient conditions for M to be a rigid B-module. In particular, if M is an indecomposable and rigid C-module, we prove M is always a rigid B-module.

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