nLab Tate spectrum

Idea

For ∞-actions of finite groups $G$ on objects $E$ in stable (∞,1)-categories, then the homotopy cofiber $X^{t G}$ of the norm map is called the Tate construction, sitting in a homotopy fiber sequence

$$X_G \stackrel{norm}{\longrightarrow} X^G \longrightarrow X^{t G} \,.$$

(e.g Lurie, def. 6.1.6.24)

Depending on which edition you have, chapter 6 may be chapter 7.

For the stable (∞,1)-category of spectra this is accordingly called the Tate spectrum.

Examples

• Tate spectra for cyclic groups play a role in the characterization of cyclotomic spectra.

• For $N$ a $G$-representation and $H N$ its Eilenberg-MacLane spectrum, then the Tate spectrum $H N^{t G}$ represents what is classically called Tate cohomology

  $$\hat H^{-n}(G,N) \;\simeq\; \pi_n( H N^{t G} )$$

  (Nikolaus-Scholze 17, p. 13) This may be generalized to possibly infinite discrete groups, see at Farrell-Tate cohomology.

References

The general abstract discussion is due to

• Jacob Lurie, section 6.1 of Higher Algebra

Review with an eye towards topological cyclic homology is in

• Thomas Nikolaus, Peter Scholze, section I of On topological cyclic homology (arXiv:1707.01799)