Speaker
Description
Syzygies are a fundamental topic of study in both commutative and non-commutative algebra. They are defined as submodules of projectives and appear as kernels of projective resolutions that are used to approximate every module. It is then natural to study the category of all syzygies, particularly when the entire module category cannot be explicitly understood.
In this series of talks, we will explore syzygy categories over Iwanaga-Gorenstein algebras of Gorenstein parameter 1, particularly focusing on the class of 2-Calabi-Yau tilted algebras. In this case, it is known that the syzygy category is triangulated and equivalent to the singularity category of an algebra as well as the category of maximal Cohen-Macauley modules. It is an open question to characterize algebras of finite CM type. As a first step in this direction, we will investigate the behavior of syzygy categories when passing from an algebra A to its quotient A/Ae_iA by an idempotent e_i, which corresponds to deleting a vertex from the quiver of A. Consequently, we describe various equivalent characterizations for these two algebras to have equivalent syzygy categories. We will then focus on the special class of Cohen-Macauley finite algebras, called dimer tree algebras and their skew group algebras, and provide a geometric model for their syzygy categories in terms of their associated checkerboard polygons.