Is there a Hopf algebra-style description of chain complexes?

Question

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.

Answer 1

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 $\mathbb{Z} [\tau]$.

Answer 2

$\newcommand{\AA}{\mathbf{A}}\newcommand{\GG}{\mathbf{G}}\newcommand{\Mod}{\mathrm{Mod}}\newcommand{\fil}{\mathrm{fil}}\newcommand{\QCoh}{\mathrm{QCoh}} \newcommand{\spec}{\mathrm{Spec}}$Let $R$ be a commutative ring, and consider $\hat{\GG}_a/\GG_m$ over $R$, where the $\GG_m$-action on $\GG_a = \spec(R[t])$ has weight $1$. Then the ($\infty$-)category of complete filtered $R$-modules is equivalent to $\QCoh(\hat{\GG}_a/\GG_m)$, i.e., the category of quasicoherent sheaves over $\GG_a/\GG_m$ whose pullback to $\GG_a$ is $t$-complete. Let $\GG_a^\sharp$ denote the PD-hull $\spec(R[\frac{x^n}{n!}])$. Then the pairing $\GG_a^\sharp \times \hat{\GG}_a \to \GG_m$ sending $(x, t) \mapsto e^{xt}$ exhibits $\GG_a^\sharp$ as the Cartier dual of $\hat{\GG}_a$. This pairing is $\GG_m$-equivariant as long as the coordinate $x$ on $\GG_a^\sharp$ has weight $-1$. General principles of Cartier duality now imply that $\QCoh(\hat{\GG}_a/\GG_m)$ is equivalent to $\QCoh((B\GG_a^\sharp)/\GG_m)$, where this equivalence swaps the usual tensor product with convolution. (This is from my comment.) Now, $B\GG_a^\sharp$ is affine (as a derived stack), and $R\Gamma(B\GG_a^\sharp; \mathcal{O}) \cong R[d]/d^2$ where $d$ is in degree $-1$. So $(B\GG_a^\sharp)/\GG_m = \mathrm{Spec}(R[d]/d^2)/\GG_m$, and one recovers Achim Krause's answer.

As for the relation to tangent spaces: indeed, the appearance of $B\GG_a^\sharp$ in both places is not a coincidence. Namely, if $X$ is a (smooth, say) $R$-scheme, the mapping stack $\mathrm{Map}(B\GG_a^\sharp, X)$ can be identified with $\spec_{\mathcal{O}_X}\left(\bigoplus_{j\geq 0} \Omega^j_{X/R}[j]\right) = BT_X^\sharp$, where $T_X^\sharp$ is the PD-hull of the zero section in the tangent bundle of $X$. See, e.g., Section 2.5 of https://www.math.ias.edu/~bhatt/teaching/mat549f22/lectures.pdf . This mapping stack has a natural $B(\GG_a^\sharp\rtimes \GG_m)$-action, placing $\Omega^j_{X/R}[j]$ in weight $j$; writing $B\GG_a^\sharp = \mathrm{Spec}(R[d]/d^2)$ with $d$ in weight $-1$ and homological degree $-1$, the resulting coaction of $R[d]/d^2$ corresponds to the de Rham differential. Under these Koszul/Cartier duality identifications, this object corresponds to the Hodge filtration $\Omega^{\geq \star}_{X/R}$ on the algebraic de Rham complex. (I just want to mention that this perspective is useful in that it can be generalized to other situations, where one replaces $\hat{\GG}_a$ by a general $1$-dimensional formal group over $R$. The resulting category of filtered spectra with its symmetric monoidal structure encodes the Leibniz rule on the $q$-de Rham complex in the case of the rescaled multiplicative group law $x + y + (q-1) xy$, and more generally complexes studied in my joint work https://arxiv.org/abs/2304.04739 .)