Definitions and examples of algeebraic Morava K-theories

TL;DR

The paper investigates algebraic Morava K-theories AK(n) by comparing their definitions as quotients of algebraic cobordism with definitions arising from tensoring motivic AP(n) with K(n)^*. Using the motivic AHSS, Milnor operations Q_i, and weight degree, it derives concrete descriptions for AK(n) on classifying spaces, simple Lie groups, and norm varieties, and connects these to CH^*(X) and étale motivic cohomology. It shows that AP(n)^{2*,*}(X) ≅ Ω^*(X)/I_n for smooth X and that AK(n)^{2*,*}(X) can be computed via Ω^*(X) and K(n)^*, yielding explicit results for BG, SO_m, and related groups, including non-nilpotence phenomena in the A^1-stable setting. The work further develops the role of Čech complexes and Rost varieties in computing AK(n) on norm varieties and Rost motives, and provides detailed mod 2 motivic cohomology computations for quadrics over ℝ, highlighting the interaction between motivic cohomology, Milnor operations, and the algebraic Morava K-theory framework. Overall, the paper clarifies the structure and computations of algebraic Morava K-theories across key algebraic-geometric and motivic contexts, with implications for classifying spaces, algebraic groups, and real quadrics.

Abstract

Algebraic Morava K-theories are defined by Sechin,Vishik and others as quotients of algebraic cobordisms. On the other hand, the author had defined them as some (two degrees) cohomology theories. In this paper, we compare these theories.

Theorems & Definitions (33)

• Lemma 1.1
• Theorem 1.2
• Theorem 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• Lemma 5.1
• Corollary 5.2
• Lemma 5.3