Schematization of a topological space 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 $A(X)$.
 A: 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. 
