Twisted tensor product, smooth DG algebras, and noncommutative resolutions of singular curves

TL;DR

This paper introduces new algebraic and DG algebra constructions using twisted tensor products, demonstrating their smoothness and finite global dimension, and showing they can serve as noncommutative resolutions of singular curves.

Contribution

It develops a novel approach to constructing smooth DG algebras with finite global dimension via twisted tensor products, applied to resolving singular rational curves.

Findings

New families of algebras with two simple modules are constructed.

These algebras have finite global dimension and are smooth DG algebras.

Some DG algebras provide smooth noncommutative resolutions of singular rational curves.

Abstract

New families of algebras and DG algebras with two simple modules are introduced and described. Using the twisted tensor product operation, we prove that such algebras have finite global dimension, and the resulting DG algebras are smooth. This description allows us to show that some of these DG algebras determine smooth proper noncommutative curves that provide smooth minimal noncommutative resolutions for singular rational curves.