# The homotopy Leray spectral sequence

## TL;DR

This paper develops a motivic homotopy spectral sequence analogous to classical topological and geometric spectral sequences, establishing foundational theory and describing its E2-page using homology of local systems.

## Contribution

It introduces a new spectral sequence in motivic homotopy theory, providing foundational constructions and a description of the E2-page in terms of homology of local systems.

## Key findings

- Established the foundations for the homotopy Leray spectral sequence.
- Provided a description of the E2-page using homology of local systems.
- Presented initial applications and future directions.

## Abstract

In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its $E_2$-page. Our description of the $E_2$-page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost's cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work.