On spectra and affine strict polynomial functors

TL;DR

This paper explores the relationship between categories of strict polynomial functors and endofunctors, introducing intermediate categories that clarify the role of Frobenius morphism in cohomology comparisons.

Contribution

It introduces two new intermediate categories, $ abla$--affine strict polynomial functors and spectra of strict polynomial functors, to better understand their derived categories.

Findings

Provides a conceptual framework for computational theorems.

Clarifies the role of inverting Frobenius morphism.

Connects rational and discrete cohomology theories.

Abstract

We compare derived categories of the category of strict polynomial functors over a finite field and the category of ordinary endofunctors on the category of vector spaces. We introduce two intermediate categories: the category of $\infty$--affine strict polynomial functors and the category of spectra of strict polynomial functors. They provide a conceptual framework for compuational theorems of Franjou--Friedlander--Scorichenko--Suslin and clarify the role of inverting Frobenius morphism in comparing rational and discrete cohomology.