nLab
norm map

Context

Representation theory

Ingredients

  • , ,

Definitions

, ,

  • ,

  • , ,

  • ,

  • ,

  • ,

  • ,

  • ,

  • ,

  • , , ,

  • , ,

  • ,

  • ,

  • , , ,

Geometric representation theory

  • , ,

  • , , ,

  • , , ,

  • ,

  • , , ,

Theorems

Contents

Idea

In representation theory a norm map is a canonical morphism from coinvariants to invariants of a given action which, in suitably well behaved cases is given by group averaging.

(e.g Lurie, constructions 6.1.6.4, 6.1.6.8, 6.1.6.18)

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

X GX GX tG. X_G \longrightarrow X^G \longrightarrow X^{t G} \,.

(e.g Lurie, def. 6.1.6.24)

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

References

The general abstract construction is due to

Review with an eye towards discussion of topological cyclic homology is in section I.1 of

Last revised on July 23, 2017 at 16:52:55. See the history of this page for a list of all contributions to it.