The geometry of filtrations

We display a symmetric monoidal equivalence between the stable $\mathrm{\infty }$-category of filtered spectra, and quasi-coherent sheaves on ${\mathbb{A}}^1\mathrm{/}{\mathbb{G}}_m$, 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.