# Adam Erlandsson: Spectral sequences for composite functors

## Masters thesis seminar

Time: Tue 2023-03-21 13.00 - 14.00

Location: 3418

Language: English

Participating: Adam Erlandsson, Tilman Bauer

Abstract:

Spectral sequences were developed during the mid-twentieth century as a way of computing (co)homology, and have wide uses in both algebraic topology and algebraic geometry.

Grothendieck introduced in his Tôhoku paper the Grothendieck spectral sequence, which given left exact functors \$F\$ and \$G\$ between abelian categories, uses the right-derived functors of \$F\$ and \$G\$ as initial data and converges to the right-derived functors of the composition \$G\circ F.\$

This thesis focuses on instead constructing a spectral sequence that uses the derived functors of \$G\$ and \$G\circ F\$ as initial data and converges to the derived functors of \$F.\$ Our approach takes inspiration from the construction of the Eilenberg-Moore spectral sequence, which given a fibration of topological spaces can calculate the singular cohomology of the fiber from the singular cohomology of the base space and total space. The Eilenberg-Moore spectral sequence can be constructed through the use of differential graded algebras and their bar construction, since this defines a double complex for which the column-wise filtration of the corresponding total complex induces the spectral sequence.

The correct analogue of this with respect to composite functors is the bar construction for monads. Specifically, we let \$G\$ have an exact left adjoint \$H\$, which makes \$G\circ H\$ into a monad. Then, we extend our adjunction so that the derived functor \$RG\$ has left adjoint \$RH\$ in the corresponding derived categories, making \$RG\circ RH\$ into a monad. This allows us to apply the bar construction in the derived category, but we show that there emerge issues in obtaining a double complex and subsequent total complex from this construction.

Additionally, we present the essential theory of spectral sequences in general, and of the Serre, Eilenberg-Moore and Grothendieck spectral sequences in particular.