- Some limits in motivic spectra I (Jan. 2021).
This is the first of a pair of posts where we will examine how limits in the category of motivic spectra can be poorly behaved. In this post we examine elliptic curves in order to conclude that over an algebraically closed field of characteristic zero the p-completion of the unit is not cellular.
- Spectral sequence calculations are NP-complete (Dec. 2020).
Many problems in algebraic topology require making computations in spectral sequences which have the structure of a module over an algebra. We will show that the problem of determining whether a given differential can occur as part of a self-consistent collection of differentials is NP-complete. This verifies the folk theorem that "spectral sequence computations can be hard".
- Fp localization vs. Fp nilpotent completion (Nov. 2020).
There are two natural notions of Fp-localization on the category of spectra.
The first is the Bousfield localization at the Eilenberg--MacLane spectrum Fp, the second is the Fp nilpotent completion (to which the Fp-Adams spectral sequence converges). In good situations, such as for bounded below spectra, Bousfield showed these two localization procedures agree. In this post we give an example where they differ.
|