3 added 2 characters in body

I have a different way to approach to this sort of question which appears in the paper:

Klein, John R. The dualizing spectrum of a topological group. Math. Ann. 319 (2001), no. 3, 421–456

The above was written using the language of naive $G$-spectra, but one can get the construction in the differential graded setting by means of a very simple idea.

The idea is this: if $R$ is an algebra augmented over a commutative ring $k$, then we have a composition pairing $$\hom_{R\text{-mod}}(k,R) \otimes_{R} \hom_{R\text{-mod}}(R,M) \to \hom_{R\text{-mod}}(k,M)$$ that can be defined for any left $R$-module $M$. We are using here the observation that $R$ is a bimodule over itself to give $\hom_{R\text{-mod}}(-,R)$ and $\hom_{R\text{-mod}}(R,-)$
the structure of $R$-modules (the first of these is a right module and the second one is a left module).

Notice that $M = \hom_{R\text{-mod}}(R,M)$ canonically, so we can rewrite the pairing as $$\hom_{R\text{-mod}}(k,R) \otimes_{R} M \to \hom_{R\text{-mod}}(k,M)$$

There is a derived version of this construction too: we can replace $k$ by a free resolution $\hat k$ over $R$, and $M$ by any chain complex over $R$ to get a pairing $$D_R \otimes_R M \to \hom_{R\text{-mod}}(\hat k,M)$$ where $D_R = \hom_{R\text{-mod}}(\hat k,R)$ and where $\otimes_R$ now means derived tensor product.

The target is the derived invariants of $R$ acting on $M$, and I tend to denote it by $M^{hR}$. So with these changes, we have a map $$D_R \otimes_R M \to M^{hR} .$$ I called this the norm map. Tate cohomology is defined to be homotopy cofiber of this map (algebraic mapping cone).

When $R = C_*(G)$ is the dga given by the chains on a $d$ dimensional compact Lie group, it turns out that is not so hard to identify $D_R$: it is quasi-isomorphic over $R$ to the chains on $S^{\text{ad}} =$ the one point compactification of the adjoint representation.

When $G=S^1$ one gets in this case $D_R = C_\ast(S^1)$ with trivial action, and the norm map in this case is a map $$C_*(S^1) \otimes_{k} M_{hS^1} \to M^{hS^1}$$ which is the same as writing a map $$\Sigma M_{hS^1} \to M^{hS^1} .$$ Yet another approach appears in the paper:

Klein, John R. Axioms for generalized Farrell-Tate cohomology. J. Pure Appl. Algebra 172 (2002), no. 2-3, 225–238.

2 added 6 characters in body

I have a different way to approach to this sort of question which appears in the paper:

Klein, John R. The dualizing spectrum of a topological group. Math. Ann. 319 (2001), no. 3, 421–456

The above was written using the language of naive $G$-spectra, but one can get the construction in the differential graded setting by means of very simple idea.

The idea is this: if $R$ is an algebra augmented over a commutative ring $k$, then we have a composition pairing $$\hom_{R\text{-mod}}(k,R) \otimes_{R} \hom_{R\text{-mod}}(R,M) \to \hom_{R\text{-mod}}(k,M)$$ that can be defined for any left $R$-module $M$. We are using here the observation that $R$ is a bimodule over itself to give $\hom_{R\text{-mod}}(-,R)$ and $\hom_{R\text{-mod}}(R,-)$
the structure of $R$-modules (the first of these is a right module and the second one is a left module).

Notice that $M = \hom_{R\text{-mod}}(R,M)$ canonically, so we can rewrite the pairing as $$\hom_{R\text{-mod}}(k,R) \otimes_{R} M \to \hom_{R\text{-mod}}(k,M)$$

There is a derived version of this construction too: we can replace $k$ by a free resolution $\hat k$ over $R$, and $M$ by any chain complex over $R$ to get a pairing $$D_R \otimes_R M \to \hom_{R\text{-mod}}(\hat k,M)$$ where $D_R = \hom_{R\text{-mod}}(\hat k,R)$ and where $\otimes_R$ now means derived tensor product.

The target is the derived invariants of $R$ acting on $M$, and I tend to denote it by $M^R$. M^{hR}$. So with these changes, we have a map $$D_R \otimes_R M \to M^R M^{hR} .$$ I called this the norm map. Tate cohomology is defined to be homotopy cofiber of this map (algebraic mapping cone). When$R = C_*(G)$is the dga given by the chains on a$d$dimensional compact Lie group, it turns out that is not so hard to identify$D_R$: it is quasi-isomorphic over$R$to the chains on$S^{\text{ad}} =$the one point compactification of the adjoint representation. When$G=S^1$one gets in this case$D_R = C_\ast(S^1)$with trivial action, and the norm map in this case is a map $$C_*(S^1) \otimes_{k} M_{hS^1} \to M^{hS^1}$$ which is the same as writing a map $$\Sigma M_{hS^1} \to M^{hS^1} .$$ Yet another approach appears in the paper: Klein, John R. Axioms for generalized Farrell-Tate cohomology. J. Pure Appl. Algebra 172 (2002), no. 2-3, 225–238. 1 I have a different way to approach to this sort of question which appears in the paper: Klein, John R. The dualizing spectrum of a topological group. Math. Ann. 319 (2001), no. 3, 421–456 The above was written using the language of naive$G$-spectra, but one can get the construction in the differential graded setting by means of very simple idea. The idea is this: if$R$is an algebra augmented over a commutative ring$k$, then we have a composition pairing $$\hom_{R\text{-mod}}(k,R) \otimes_{R} \hom_{R\text{-mod}}(R,M) \to \hom_{R\text{-mod}}(k,M)$$ that can be defined for any left$R$-module$M$. We are using here the observation that$R$is a bimodule over itself to give$\hom_{R\text{-mod}}(-,R)$and$\hom_{R\text{-mod}}(R,-)$the structure of$R$-modules (the first of these is a right module and the second one is a left module). Notice that$M = \hom_{R\text{-mod}}(R,M)$canonically, so we can rewrite the pairing as $$\hom_{R\text{-mod}}(k,R) \otimes_{R} M \to \hom_{R\text{-mod}}(k,M)$$ There is a derived version of this construction too: we can replace$k$by a free resolution$\hat k$over$R$, and$M$by any chain complex over$R$to get a pairing $$D_R \otimes_R M \to \hom_{R\text{-mod}}(\hat k,M)$$ where$D_R = \hom_{R\text{-mod}}(\hat k,R)$and where$\otimes_R$now means derived tensor product. The target is the derived invariants of$R$acting on$M$, and I tend to denote it by$M^R$. So with these changes, we have a map $$D_R \otimes_R M \to M^R .$$ I called this the norm map. Tate cohomology is defined to be homotopy cofiber of this map (algebraic mapping cone). When$R = C_*(G)$is the dga given by the chains on a$d$dimensional compact Lie group, it turns out that is not so hard to identify$D_R$: it is quasi-isomorphic over$R$to the chains on$S^{\text{ad}} =$the one point compactification of the adjoint representation. When$G=S^1$one gets in this case$D_R = C_\ast(S^1)\$ with trivial action, and the norm map in this case is a map $$C_*(S^1) \otimes_{k} M_{hS^1} \to M^{hS^1}$$ which is the same as writing a map $$\Sigma M_{hS^1} \to M^{hS^1} .$$ Yet another approach appears in the paper:

Klein, John R. Axioms for generalized Farrell-Tate cohomology. J. Pure Appl. Algebra 172 (2002), no. 2-3, 225–238.