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I wanted to understand or at least to know if what follows make sense. Given a connected toplogical space $X$, I want to associate a scheme. In the following way.

For a space $X$ and $A(X)$ the Sullivan minimal model, first I associate the differential graded commutative ring $A(X)$. Then I associate to $A(X)$ the corresponding symmetric monoidal triangulated category of differential graded $A(X)$-modules. Finaly, we associate to symmetric monoidal triangulated category of differential graded $A(X)$-modules the scheme $\mathbf{X}$ using the $spec$ functor defined by paul Balmer. My question is the following. What can we say about $X$ using $\mathbf{X}$ ? Can we reconstruct (partially) $X$ from $\mathbf{X}$ ?

Thank you very much for any help.

Edit: I replaced the rational cohomology ring $H^{\ast}(X, \mathbf{Q})$ by $A(X)$.

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    $\begingroup$ I don't really understand your construction, but it seems to depend only on $H^*(X,\mathbb Q)$, so it seems like the answer is "you can reconstruct $H^*(X,\mathbb Q)$". $\endgroup$
    – Allen Knutson
    Commented Sep 5, 2015 at 20:35
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    $\begingroup$ If X is finite-dimensional then both the cohomology and the Sullivan algebras have a unique prime ideal, so I wouldn't hope too much. $\endgroup$
    – Fernando Muro
    Commented Sep 5, 2015 at 21:36
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    $\begingroup$ You can canonically associate a derived scheme, namely $\text{Spec } C^{\bullet}(X, \mathbb{Q})$. This knows something about the rational homotopy type of $X$. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 6, 2015 at 3:26
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    $\begingroup$ @AllenKnutson I'm using Balmer's construction. To any Tensor triangulated category we associate a honnest scheme. $\endgroup$
    – lilia
    Commented Sep 6, 2015 at 6:52
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    $\begingroup$ Whatever it is, this construction still seems to only depend on $C^\bullet(X,\mathbb Q)$, so apparently the answer is somewhere between "it recovers the rational homotopy type" and "it recovers less information than the rational homotopy type"... right? $\endgroup$
    – Allen Knutson
    Commented Sep 7, 2015 at 1:47

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If $X$ has mild finiteness conditions (its even-dimensional cohomology is noetherian and its odd-dimensional cohomology is finitely generated as a module over it), then it is possible to completely determine the spectrum (as defined by Balmer) of the category of (perfect) modules over the cochain algebra $C^*(X; \mathbb{Q})$. The points of the spectrum correspond to homogeneous prime ideals of the even-dimensional cohomology in the natural sense, so perfect modules over $C^*(X; \mathbb{Q})$ are "stratified" by their cohomology as a module over $H^*(X; \mathbb{Q})$ (and the associated support). For example, if $X$ is a finite CW complex and is connected, then the spectrum is just a point.

This result holds more generally for $E_\infty$-ring spectra over $\mathbb{Q}$ that satisfy this type of noetherianness condition, and is contained in my preprint "Residue fields for a class of rational $E_\infty$-rings and applications." As indicated in the title, the idea is to construct "residue fields" via a transfinite iterated cell attachment procedure and then to prove an analog of the Devinatz-Hopkins-Smith nilpotence theorem (from which these types of thick subcategory theorems can be derived). Incidentally, these methods do not (at least so far as I am aware) lead to a classification of localizing subcategories, which has been carried in some other instances (notably for stable module categories by Benson-Iyengar-Krause).

In any event, the conclusion is that one won't see very much of $X$ that one didn't have before.

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