Koszul Complexes, Local Cohomology, Universal Resolutions, and Cohomological Support Varieties

We will explore three fields which utilize the Koszul complex: local cohomology, Golodity, and the cohomological support variety. First we explore the support of local cohomology modules, showing in particular the Zariski closure of certain local cohomology modules of ideals generated by regular sequences. We then explore various universal resolutions. Within this, we first provide an overview of DG algebras and universal resolutions. We then provide a survey concerning the relationship between some relatively common conditions on module resolutions, namely Golodity and formality, and A-infinity algebras. We go over some of the definitions of formality and Golodity in terms of A-infinity algebras while providing some explanations regarding the bridges between them in the language of Massey products. We explicitly describe a relatively well-known relationship between Golodity and formality in a way slightly different from descriptions seen elsewhere using this framework. Finally, we discuss the cohomological support variety, and specifically its construction when resolving a quotient of a monomial ideal over a polynomial ring. We provide a construction for calculating support varieties of monomial ideals which is slightly more computationally manageable than any known construction. We use this to manually verify a computation of the cohomological support variety of the edge ideal on a cycle with 6 vertices. We also computationally verify a classification of all cohomological support varieties of equigenerated monomial ideals, as well as a description of the cohomological support varieties of edge ideals with ten and fourteen vertices.

Author dc:creator

• Michael Gintz (24399050)

• Commutative algebra
• Algebraic geometry

Statement dc:rights

• In Copyright

DOI dc:identifier

https://doi.org/10.25417/uic.32991938.v1

OAI identifier oai:identifier

oai:figshare.com:article/32991938