Emily Berghofer: Constructing Koszul Filtrations

Abstract: A standard graded algebra is called Koszul if its residue field has a linear free resolution. These resolutions are infinite, and hence studying them directly is often impractical. Fortunately, there are other ways to show that an algebra is Koszul. One common approach is to show that it has a quadratic Gröbner basis, such algebras are called G-quadratic. Another approach is to find a Koszul filtration. Both properties are sufficient but not necessary conditions for being Koszul. There are known examples of algebras that have a Koszul filtration but are not G-quadratic. This talk is motivated by the question of when the reverse implication holds. I will discuss situations where it does hold and also present an example of a G-quadratic algebra that does not admit a Koszul filtration.