In [this paper](https://core.ac.uk/download/pdf/82458627.pdf), 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 $\mathbb{Z}$?**



A generalization of integral Poincaré duality
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 $R\to \Bbb Z$).


