Title：Syzygy modules and simplicial resolutions

Reporter：Aliaksandr Hancharuk

Work Unit：University Lyon 1

Time：2023/11/13-16，21:00-22:00

Address：Zoom id：904 645 6677 Password:2023

**Summary of the report: **

Hilbert in 1890s famously proved that a graded finitely generated module over a polynomial ring admits a free resolution of finite length. Since then a lot of progress has been done on the structure and properties of finite free resolutions. We will study those underlying some abstract simplicial complex. Lastly, we relate these finite resolutions to the infinite ones and discuss its applications. Based on "Combinatorial commutative algebra" by Miller and Sturmfels, "Graded Syzygies" by Peeva and some new results by the author.

**Four subtitles of the minicourse:**

1. Graded free resolutions. Koszul complex. Hilbert Syzygy theorem.

2. Simpicial resolutions, Taylor complex, Scarf complex.

3. Duality for resolutions, Alexander duality.

4. Infinite graded resolutions. Tate resolutions.

**Introduction of the Reporter: **

Aliaksandr Hancharuk did his Ph.D. in 2023 with Prof.Strobl in University Lyon 1. His research interests and work is concentrated in the intersection of mathematical physics and homological algebra, namely in the algebraic aspects of gauge theories.