# The geometry of filtrations

## TL;DR

This paper establishes a deep connection between filtered spectra and quasi-coherent sheaves on a specific quotient stack in spectral algebraic geometry, revealing new insights into their categorical and geometric structures.

## Contribution

It introduces a symmetric monoidal equivalence between filtered spectra and sheaves on a quotient stack, using Tannaka duality to identify key functors with geometric pull-backs.

## Key findings

- Equivalence between filtered spectra and sheaves on a quotient stack
- Identification of spectrum and graded functors with geometric pull-backs
- Application of Tannaka duality in spectral algebraic geometry

## Abstract

We display a symmetric monoidal equivalence between the stable $\infty$-category of filtered spectra, and quasi-coherent sheaves on $\mathbb{A}^1 / \mathbb{G}_m$, the quotient in the setting of spectral algebraic geometry, of the flat affine line by the canonical action of the flat multiplicative group scheme. Via a Tannaka duality argument, we identify the underlying spectrum and associated graded functors with pull-backs of quasi-coherent sheaves along certain morphisms of stacks.