An old chestnut 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.

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