Fridays at 11:00
Room: LeConte 317R
Organizer: Blake Farman
A brief overview of Serre finiteness in the commutative case.
Noncommutative projective schemes were introduced by M. Artin and J. J. Zhang in their 1994 paper, Noncommutative Projective Schemes, as a generaliztion of projective schemes. In this talk, we’ll discuss the method of forming the quotient of an abelian category by an epaisse subcategory introduced by Grothendieck in his famous “Tohoku” paper, Sur quelques points d’algèbre homologique. We will use this to define the projective scheme of a noncommutative graded algebra due to Artin and Zhang, and present some results from their 1994 paper.
Test modules are modules that detect finite projective dimension. We will see some examples of these modules and determine (in some cases) the set of all test modules. In addition, we will see the connection to totally reflexive modules via the canonical module.
In this talk we will look at the 1988 paper of Eisenbud and Herzog classifying homogeneous rings generated in degree 1 of finite representation type. The proof relies on known classifications and two important lemmas. During the talk we will focus on the lemma which states that a homogeneous ring generated in degree one that has finite representation type must be "stretched," which I will define in the talk. The converse of the lemma is false and we will go over a counterexample.