Redshift and multiplication for truncated Brown-Peterson spectra

Abstract

We equip BPxny with an E 3 -BP-algebra structure, for each prime p and height n. The algebraic K-theory of this E 3 -ring is of chromatic height exactly n `1.

Examples of chromatic redshift in algebraic $K$-theory

“…We remark that the question of height shifting has recently been independently addressed by Hahn and Wilson [HW20] in the case of truncated Brown-Peterson spectra BP n ; their approach is more computational in nature, and further proves Lichtenbaum-Quillen-type statements in this case. The nonvanishing of L T (n+1) K(BP n ) is closely related to Theorem A, though we do not know if either result implies the other.…”

“…The nonvanishing of L T (n+1) K(BP n ) is closely related to Theorem A, though we do not know if either result implies the other. Theorem A was previously known in the case n = 1, p ≥ 5 by [Aus10, AR02, BM08], and the cases p = 2, 3 also follow from [HW20].…”

Syntomic complexes and $p$-adic étale Tate twists

“…We also thank Benjamin Antieau, Kęstutis Česnavičius, Jeremy Hahn, Lars Hesselholt, Jacob Lurie, Wiesława Nizioł, Matthew Morrow, Peter Scholze, and Dylan Wilson for helpful discussions. In particular, we learned the idea that there should be a mixed characteristic analog of Cartier smoothness from Morrow, and some of the arguments used by Hahn-Wilson in [HW21] inspired some of ours used here.…”

Adams spectral sequences and Franke's algebraicity conjecture

“…Example 8.6. Fix a prime p and let BP xny be an E 1 -form of the truncated Brown-Peterson spectrum in the sense of [46], so that in particular we have for k " 2p ´2 ´n.…”