This is the first of a series of three talks on the spectral sequence of a filtered complex.
This material is by now classical and is an important part of homological algebra.
The main difficulty in dealing with spectral sequences is that there are a lot of indexes involved and this is a considerable obstacle to understanding what is going on. The goal of these talks is to present this material, including most proofs, in an accessible manner.

This is the third of a series of three talks on the spectral sequence of a filtered complex. This material is by now classical and is an important part of homological algebra. The main difficulty in dealing with spectral sequences is that there are a lot of indexes involved and this is a considerable obstacle to understanding what is going on. The goal of these talks is to present this material, including most proofs, in an accessible manner.

I will talk about the two papers Eisenbud-Fløystad-Schreyer-2003 and Eisenbud-Erman-Schreyer-2015 in which they introduce Tate resolutions for projective spaces and products of projective spaces. As an application, I will talk about Horrocks' criterion for vector bundles in those settings.

A more generalized form of factorization, called $\tau$-factorization, was introduced in 2011 by D.D. Anderson and J. Reinkoester. In $\tau$-factorization, all factors of a factorization must belong to the same equivalence class modulo a fixed ideal. We discuss $\tau$-factorization in small settings and $\tau$-elasticity in a more general setting.