In [this paper][1], Felix, Halperin and Thomas define the notion of a Gorenstein space over a field $\mathbb{k}$:

> An augmented differential graded algebra $R$ over $\mathbb{k}$ is *Gorenstein* if $\text{Ext}_R(\mathbb{k},R)$ is concentrated in a single degree and has $\mathbb{k}$-dimension one.
>
> $X$ is *Gorenstein over $\mathbb{k}$* if the cochain algebra
> $C^*(X,\mathbb{k})$ is Gorenstein.

This definition is motivated by their subsequent results on this being a generalization of the notion of a Poincaré duality space.

Does there exist a parallel notion of a Gorenstein space over $\mathbb{Z}$ which similarly generalizes Poincaré duality over $\mathbb{Z}$?

**EDIT: Alternatively, is it thought that no such generalization is available, so that one needs to use the machinery of  symmetric spectra to get such a generalization over $\mathbb{Z}$?**

  [1]: https://core.ac.uk/download/pdf/82458627.pdf