I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived algebraic geometry"-type setting. First recall the
Standard geometric way to think about filtrations: Set $F = \oplus_n F_n$. Then $F$ is a $\mathbb S[t]$-module, where the action of $t$ shifts along the filtration. So we may view $F$ as a quasicoherent sheaf over $\mathbb A^1 = Spec \mathbb S[t]$. Note that the natural grading of $\mathbb S[t]$ corresponds to the natural action of $\mathbb G_m = Spec \mathbb S[t^{\pm}]$ on $\mathbb A^1$, and the natural grading of $F$ can be interpreted as a $\mathbb G_m$-equivariant structure on $F$ viewed as a sheaf over $\mathbb A^1$. The map $\mathbb S[t] \to \mathbb S$ sending $t \mapsto 0$ corresponds to the origin of $\mathbb A^1$, a fixed point for the $\mathbb G_m$ action. So the pullback of $F$ to this point, which is the associated graded of the filtration we started with, has a $\mathbb G_m$ action, corresponding to its natural grading.
So the fiber of $F$ over $0$ is where the $E_1$ page of the spectral sequence comes from. Now, consider the maps $Spec \mathbb S \to Spec \mathbb S[t]/t^2 \xrightarrow{j} Spec \mathbb S[t]/t^3 \to \dots$. These correspond to closed infinitesimal, $\mathbb G_m$-equivariant neighborhoods of the origin, and the pullbacks of $F$ to theses neighborhoods correspond to the later pages of the spectral sequence. The way the spectral sequence is constructed also has a geometric interpretation: the exact couple allowing you to compute the $E_3$ page $F^{(3)}$ from the $E_2$ page $F^{(2)} = j^\ast F^{(3)}$, for instance, comes from the fact that $F^{(3)}$ fits into an exact sequence $j^! F^{(2)} \to F^{(3)} \to j^\ast F^{(2)}$ and moreover that $j^! F^{(2)}$ and $j^\ast F^{(2)}$ differ by a shift, being the fiber and cofiber of the map $F^{(3)} \xrightarrow{t^2} F^{(3)}$. I'm not sure how closely related the $r$th page of the spectral sequence is to the restriction of $F$ to $Spec \mathbb S[t]/t^{r}$; although the latter can certainly be computed as the cofiber of $t^{r}: F \to F$, and in the case $r=1$ this gives the exact couple leading to the spectral sequence, I think maybe the spectral sequence and the deformation theory diverge after this. To obtain the $r+1$th page from the $r$th, one needs to resolve $\mathbb S[t]/t^r$ as a $\mathbb S[t]/t^{r+1}$-module; I'm getting a cyclic resolution $\cdots \to \mathbb S[t]/t^{r+1} \xrightarrow{t^r} \mathbb S[t]/t^{r+1} \xrightarrow{t} \mathbb S[t]/t^{r+1} \xrightarrow {t^r} \mathbb S[t]/t^{r+1}$. So understanding the spectral sequence seems to bring us back into more familiar territory where we can use algebraic geometry ideas to think about the homotopy groups of $F$, but we're not really thinking of $F$ itself algebro-geometrically any more.
The $E_\infty$ page is going to hopefully tell you something about $F$ over the big $\mathbb G_m$-equivariant, but still infinitesimal neighborhood of the origin given by $Spec \mathbb S[[t]]$, which you use to approximate the fiber of $F$ over the point $t=1$, namely $\varinjlim F_n$.
This perspctive seems to beg generalization to things more general than filtrations, corresponding to fancier group actions than $\mathbb G_m$ acting on $\mathbb A^1$. What these "generalized filtrations" look like, I'm not sure!
Questions:
Is there anywhere a perspective like this is developed in more detail?
What kinds of things can one say about convergence of the spectral sequence from this perspective?
For other groups $G$ and $G$-spaces $X$, are there algebraic interpretations of $G$-equivariant sheaves on $X$ of a similar flavor to the "filtration" interpretation of a $\mathbb G_m$-equivariant sheaf over $\mathbb A^1$? (And if so, what do their associated "spectral sequences" look like in this picture...)
For (2), I'm thinking in particular that there should be standard algebraic geometry concepts allowing one to relate information over an infinitesimal neighborhood like $Spec k[[t]]$ to information over points which are are "nearby, but not infinitesimally so" like the point at $t=1$. I think flatness has something to do with this? But I'm not sure. Also, the relationship between the $E_\infty$ page of the spectral sequence and the pullback of $F$ to $Spec k[[t]]$ is likely a bit delicate.
