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In this paper, 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 $\mathbb{Z}$?

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    $\begingroup$ A lot of work has been done (starting with Dwyer, Greenlees, and Iyengar) and furthered by Greenlees on when a morphism $f \colon R \to k$ of ring spectra is `Gorenstein', see arxiv.org/abs/math/0510247 for example. Taking $R=C^*(X;\mathbb{Q})$ and $f \colon R \to \mathbb{Q}$ the natural map gives you the above definition. Replacing $\mathbb{Q}$ with $\mathbb{Z}$ gives you a reasonable definition. (The theory tends to work best when $k$ is a field however). $\endgroup$ – Drew Heard May 29 at 17:03
  • $\begingroup$ Thank you for your comment. Are you aware of whether there is a definition of being Gorenstein over $\mathbb{Z}$ without using spectra, or are you saying that there is no such definition available, and the work of Dywer, Greenless, Iyengar addresses this deficiency? $\endgroup$ – Matt May 29 at 17:22
  • $\begingroup$ I'm not aware of anything else, but that could be more to do with my own ignorance! Looking into the references of D-G-I might help you find other sources (e.g., link.springer.com/article/10.1007/BF02776063) $\endgroup$ – Drew Heard May 30 at 6:46
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Prior to Dwyer-Greenlees-Iyengar, Dwyer and myself (independently) defined Gorenstein conditions for group rings over the sphere $S[G]$, i.e., the suspension spectrum of a topological group.

The definition easily extends to the case of a morphism $R\to k$.

In the orientable case the definition goes like this:

$R\to k$ is said to be Gorenstein of dimension $d$ if:

1) $k$ is finitely dominated as an $R$-module, i.e., $k$ is a retract up to homotopy of a finite $R$-module (this finiteness condition shouldn't be ignored!), and

2) the derived mapping spectrum $\hom_R(k,R)$ is weakly equivalent as an $R$-module to $k[-d] := \Sigma^{-d}k$.

If one wishes to have an unoriented Gorenstein condition, then it seems to me one needs to replace $k[-d]$ by a twisted version of it. In the special case when $R$ is an augmented $k$-algebra spectrum, we can simply require that the $k$-module $\hom_R(k,R)$ is equivalent to $k[-d]$ as a $k$-module (but not necessarily as an $R$-module).

If $R$ is isn't a $k$-algebra, we can fix another $R$-module structure on $k$, call it $k^\xi$, and require that $\hom_R(k,R)$ is equivalent to $k^\xi[-d]$ as an $R$-module.

Returning the the group ring case $S[G]$, we have:

Theorem. Assume in addition $\pi_0(G)$ is finitely presented. Then following are equivalent:

1) $S[G] \to S$ is Gorenstein in dimension $d$.

2) $BG$ is a (finitely dominated) Poincaré duality space in dimenson $d$.

And yes, there is a parallel story over $\Bbb Z$, but it is enough to work with dgas instead of ring spectra (for example, the derived $\hom_{R}(\Bbb Z,R)$ is a differential graded module for a discrete ring homomorphism $R\to \Bbb Z$).

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