An [old chestnut](https://mathoverflow.net/questions/25873/geometric-interpretation-of-filtered-rings-and-modules/25888#25888) is that filtered objects are the same as sheaves over $\mathbb A^1 / \mathbb G\_m$.


**Question:** Is there a similar description of chain complexes?


More precisely, if $\mathcal C$ is a category, then define the category of *filtered objects* in $\mathcal C$ to be the functor category $Fil(\mathcal C) = Fun(\mathbb Z, \mathcal C)$, where $\mathbb Z$ is the integers regarded as a poset. Then if $X$ is a scheme, there is a canonical equivalence $Fil(QCoh(X)) \simeq QCoh(X \times \mathbb A^1 / \mathbb G\_m)$, where $\mathbb G\_m$ acts on $\mathbb A^1$ in the usual way. What this says is that $\mathbb G\_m$-equivariant sheaves on $\mathbb A^1 \times X$ are the same as filtered sheaves on $X$. As $\mathbb G\_m$-actions are the same as gradings, this says in other words that a graded object equipped with an an endomorphism of degree 1 is the same as a filtered object.


I'd like a similar description of the category of chain complexes $Ch(QCoh(X)) \simeq QCoh(X \times S)$, where $S$ is some fixed stack, probably a quotient $S = T / G$ for some scheme $T$ and some action by a group scheme $G$.


**Note:** I believe that if $\mathcal C$ is stable, then $Fil(\mathcal C) \simeq Ch(\mathcal C)$ via some sort of $\infty$-categorical Dold-Kan correspondence (at any rate, I'm quite sure this is true if we talk about nonnegatively-graded chain complexes and nonnegative filtrations). So the stack $S$ will have to be derived-equivalent to $\mathbb A^1 / \mathbb G\_m$, but perhaps not equivalent in an underived sense.



Is there a Hopf algebra-style description of chain complexes?
Shouldn't it just be $\operatorname{Spec}(\Lambda)/\mathbb{G}\_m$, where $\Lambda = \mathbb{Z}[d]/d^2$ and the $\mathbb{G}\_m$ action encodes the grading with $d$ in degree $-1$? This is just the observation that chain complexes are the same as graded modules over an exterior algebra in a degree $-1$ generator, similar to how the $\mathbb{A}^1/\mathbb{G}\_m$ description of filtered objects relates to their description as graded modules over a polynomial algebra $\mathbb{Z} [\tau]$.


