Realize as homology a given polynomial ring I am wondering if one can realize a polynomial ring as the homology of some chain complex in the same sense that the homology groups of a space with a cell complex structure is the homology of its chain complex. There are trivial solutions for this with a complex with only 0-chains, but I ask for something like a free resolution that is not exact. I hope there is a canonical thing for this, maybe something related to the dimension for algebraic varieties.
The easiest example to apply this, in my case, would be the Tjurina algebra of an isolated hypersurface singularity $(H,x)\subset (\mathbb{C}^{n+1},x)$:
$$T(f)=\frac{\mathcal{O}_{n+1}}{(f)+J(f)},$$
where $H=V(f)$ and $J(f)$ is the ideal generated by the partial derivatives of $f$.
Motivation:
There is a spectral sequence that has in the first page some chain complex of some objects $V_i$. In the second you find some easy to compute homologies of some easy objects, $H_*(V_i)$, which is what people use in practice. The spectral sequence is used to calculate the homology of something, say some $H_*(X)$ of $X$. This is very hard to do in general, hence the importance of the spectral sequence.
In turn, there is an associated algebraic object $A(X)$ to $X$ (and to every $V_i$ an associated $A(V_i)$). It seems that if you use the algebraic objects $A(V_i)$ instead of $H_*(V_i)$ in the SECOND page of a spectral sequence similar to the previous one (if it exists?) you would also compute $A(X)$. The problem is that the proof that works for homologies $H_*(X)$ uses that you have a double complex where you find the chain complexes of the $V_i$ (this is the FIRST page of the spectral sequence). I don't have such chain complexes for the algebraic setting, so I cannot prove that the spectral sequence converges to what I want.
(Maybe it is not the first and the second page, but the $0$-th and the first, that is not important).
 A: This seems related to the Steenrod problem (see below). As far as I know it's still open. Here are a couple of references; maybe the Anderson-Grodal one is closer to what you're looking for. N.B. I think there are some other problems known as Steenrod's problem so you may have to sift through what you get from Google or Mathscinet.
Smith, Justin R. Topological realizations of chain complexes. I. The general theory.
Topology Appl. 22 (1986), no. 3, 301–313.
Andersen, Kasper K. S.; Grodal, Jesper The Steenrod problem of realizing polynomial cohomology rings. J. Topol. 1 (2008), no. 4, 747–760.
Smith's paper is concerned with the non-simply connected case; his abstract gives some background:
Abstract: This paper studies the following question: Given a group $\pi$, and a projective $\pi$-chain complex $C$, does there exist a topological space with a fundamental group $\pi$ and with the property that the chain-complex of its universal cover is chain-homotopy equivalent to $C$? This is generalization of the Steenrod Problem. In the Steenrod Problem (proposed by Steenrod in 1960) the chain complex was a projective resolution of a
$\pi$-module. The present paper develops an obstruction theory for the existence of topological realizations of a chain-complex, algebraically classifies these realizations (if the obstructions vanish), and proves that rational chain-complexes are always stably realizable.
