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Any abelian group M with an action of a finite group $G$ has a Tate cohomology object $\hat H(G;M)$ in the derived category of chain complexes. There are several ways to define this. One is as the tensor product $W \otimes_G M$ with a particular (typically unbounded) complex of projective $\Bbb Z[G]$-modules which happens to have trivial homology. One is to instead use $Hom^G(W',M)$ for $W'$ a similar complex. One is as the mapping cone of a norm map $$ M_{hG} \to M^{hG} $$ from derived coinvariants to derived invariants, where we view $M$ as a complex concentrated in degree zero.

We can attempt to generalize this construction to the derived category of $\Bbb Z[G]$-modules, or of $R[G]$-modules for $R$ a commutative ring. If $M_*$ is a chain complex of $\Bbb Z[G]$-modules, these give us three definitions of the Tate construction on $M_*$: $$ \begin{align*} Tate^\oplus(M)_n &= \bigoplus_{p+q=n} W_p \otimes_G M_q\\ Tate^\prod(M)_n &= \prod_{p+q=n} W_p \otimes_G M_q\\ Tate(M)_n &= \bigcup_N \prod_{p+q = n, p \leq N} W_p \otimes_G M_q \end{align*} $$ (There are natural maps $Tate^\oplus \to Tate \to Tate^{\prod}$ which are isomorphisms on bounded complexes.)

The first construction is nice because it commutes with filtered colimits, and the second with filtered limits. The third is neither, but it does have a norm-cofiber sequence and a Tate cohomology spectral sequence $$ \hat H^p(G; H^q(M)) \Rightarrow H^{p+q}(Tate(M)). $$ The third construction is also the only one that preserves quasi-isomorphisms, and thus the only one that (naturally) extends to the unbounded derived category.

Are there derived-category interpretations for the constructions $Tate^\oplus$ or $Tate^\prod$?

Such an interpretation probably needs to take in more information than an object of the derived category of $\Bbb Z[G]$-modules. The main motivation for asking is that I'd like to see what is necessary in more general contexts (e.g. modules over a differential graded algebra).

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  • $\begingroup$ Dave Benson or John Greenlees would be my go-to people for this. Maybe you've already asked them... $\endgroup$ Mar 30, 2016 at 11:32
  • $\begingroup$ @Jesper Both of those are good suggestions. I haven't asked either yet, and it looks like I'll probably be doing so soon. $\endgroup$ Mar 31, 2016 at 18:45
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    $\begingroup$ Another characterisation of Tate cohomology $\hat H^n(G, M)$ is as $Hom(\mathbf Z, M[n])$, where the Hom-set is in the category of singularities, defined as the quotient of the usual derived category $D^b(Mod_{\mathbf Z G})$ modulo its subcategory of perfect complexes. A natural extension thus seems to be (I remember this is also mentioned somewhere in Gaitsgory's work, but don't seem to find it right now) to consider the Hom-set or -space in the quotient of Ind-Coherent sheaves modulo Quasi-coherent sheaves (in the $\infty$-sense). $\endgroup$
    – Jakob
    Jun 3, 2019 at 13:10

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