Starting with Peskine and Szpiro's intersection theorem, we will introduce several homological conjectures/theorems and the connections between them. Peskine Szpiro's intersection theorem tells us about the codimension of the support of a coherent sheave of finite projective dimension. It also settled some earlier homological conjectures including Auslander’s Zero Divisor Conjecture and Bass’s Conjecture. We will introduce several later homological conjectures which are meant to generalize Peskine Szpiro intersection theorem, including the New Improved Intersection Conjecture.
We followed the notes by Paul Roberts here: https://www.math.utah.edu/vigre/minicourses/algebra/roberts.pdf