Invertible Morava motives in quadrics
Andrei Lavrenov, Pavel Sechin
TL;DR
The paper develops Morava K-theory motives for quadrics as a framework to categorify Milnor K-theory mod 2, pairing α ∈ K^{M}_{n+1}(k)/2 with invertible Morava motives L_α that split precisely when α vanishes over field extensions. It introduces universal splitting fields k(α) and shows that L_α acts as a canonical splitting object for Morava motives M_{K(n)}(X), yielding a robust mechanism to detect cohomological invariants of varieties, especially quadrics, and to relate Morava motives to Rost and Chow motives via base-change and descent. The work establishes base-change reflection principles (Rost Nilpotence, A-universal surjectivity/bijectivity) and proves a Morava MDT framework that parallels Chow MDT for quadrics, enabling recovery of Chow motivic decompositions from Morava data in the pre-stable regime. It also proves that Milnor K-theory modulo 2 is realized inside invertible Morava motives, with L_α providing a canonical, invertible summand whose presence encodes the nontrivial cohomology class α; L_α can be detected by comparing Tate summands over k and k(α) and, numerically, in K(n)_num. Overall, the paper links motivic decompositions, cohomological invariants, and splitting fields in a principled Morava-theoretic setting, offering a pathway to assign cohomological invariants to Chow motives and to extend Rost-type descent phenomena within Morava motives.
Abstract
We associate to any element in the Milnor K-theory of a field $k$ modulo 2 an invertible Morava K-theory motive over $k$. Specifically, for $α$ in $\mathrm{K}^{\mathrm{M}}_{n+1}(k)/2$ we construct an invertible $\mathrm{K}(n)$-motive $L_α$ in a way that is natural in the base field and additive in $α$. This can be seen as categorification of $\mathrm{K}^{\mathrm{M}}_{n+1}(k)/2$ in motives. The motives $L_α$ are constructed as direct summands of the $\mathrm{K}(n)$-motives of quadrics, and we develop the necessary framework for the study of the latter. We show that passing to the field of functions of quadrics of dimension greater than or equal to $2^{n+1}-1$ does not lose any information about the structure of $\mathrm{K}(n)$-motives. This is based on the study of "decomposition of the diagonal" in Morava K-theory of quadrics. For quadrics of dimension less than $2^{n+1}-1$, we show that their Chow motives can be "reconstructed" from their $\mathrm{K}(n)$-motives, although the latter appear structurally simpler. Our proof of this result relies on the use of the unstable symmetric operations of Vishik on algebraic cobordism. The occurrence of the motive $L_α$ as a direct summand of the $\mathrm{K}(n)$-motive of $X$ can be seen as evidence that $α$ is a cohomological invariant of $X$. We study this occurrence for quadrics and relate it to Kahn's Descent conjecture.