Some comments on motivic nilpotence

Abstract

We discuss some results and conjectures related to the existence of the non-nilpotent motivic maps η \eta and μ 9 \mu _9 . To this purpose, we establish a theory of power operations for motivic H ∞ H_{\infty } -spectra. Using this, we show that the naive motivic analogue of the unstable Kahn-Priddy theorem fails. Over the complex numbers, we show that the motivic T T -spectrum S [ η − 1 , μ 9 − 1 ] S[\eta ^{-1},\mu _9^{-1}] is closely related to higher Witt groups, where S S is the motivic sphere spectrum and η \eta and μ 9 \mu _9 are explicit elements in π ∗ ∗ ( S ) \pi _{**}(S) .