nLab higher chromatic Chern character

Idea

The traditional Chern character on complex K-theory happens to be a map from complex oriented cohomology theory of chromatic level 1 to one of chromatic level 0 (ordinary cohomology).

More generally, one may ask for higher chromatic Chern characters that map a generalized cohomology theory of some chromatic level to another of some other level.

This has been considered for

• maps from elliptic cohomology/tmf to K-theory, see at elliptic Chern character

• maps from Morava E-theory at level $(n+1)$ to Morava E-theory at level $n$, tensored with some ring, see at E-theoretic Chern character.

Related concepts

• transchromatic character

References

A survey is for instance in the introduction of

• Takeshi Torii, Milnor operations and the generalized Chern character, Geom. Topol. Monogr. 10 (2007) 383-421 (arXiv:0903.4708)

• Takeshi Torii, HKR characters, p-divisible groups and the generalized Chern character, (pdf)