Invertible Morava motives in quadrics

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 $\alpha$ in $\mathrm{K}^{\mathrm{M}}_{n+1}(k)/2$ we construct an invertible $\mathrm{K}(n)$-motive $L_\alpha$ in a way that is natural in the base field and additive in $\alpha$. This can be seen as categorification of $\mathrm{K}^{\mathrm{M}}_{n+1}(k)/2$ in motives. The motives $L_\alpha$ 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 $L_\alpha$ as a direct summand of the $\mathrm{K}(n)$-motive of $X$ can be seen as evidence that $\alpha$ is a cohomological invariant of $X$. We study this occurrence for quadrics and relate it to Kahn's Descent conjecture.