标题： 可逆的马罗瓦动机在二次曲面中 显示英文标题

标题： Invertible Morava motives in quadrics

摘要： 我们将域$k$的 Milnor K-理论中任意元素模 2 关联到一个在$k$上的可逆 Morava K-理论动机。 具体来说，对于$\alpha$在$\mathrm{K}^{\mathrm{M}}_{n+1}(k)/2$中，我们以一种在基域上自然且在$\alpha$上可加的方式构造一个可逆$\mathrm{K}(n)$-动机$L_\alpha$。 这可以看作是动机中$\mathrm{K}^{\mathrm{M}}_{n+1}(k)/2$的范畴化。 动机$L_\alpha$被构造为二次曲线的$\mathrm{K}(n)$动机的直和项，我们开发了研究后者所需的框架。 我们证明，将场扩展到维度大于等于$2^{n+1}-1$的二次曲线的函数域不会丢失关于$\mathrm{K}(n)$动机结构的任何信息。 这是基于对二次曲线的 Morava K 理论中“对角分解”的研究。 对于维度小于$2^{n+1}-1$的二次曲线，我们证明它们的 Chow 动机可以从它们的$\mathrm{K}(n)$动机“重构”，尽管后者在结构上更简单。 我们对该结果的证明依赖于 Vishik 在代数 bordism 上使用的不稳定对称操作。 动机$L_\alpha$作为$\mathrm{K}(n)$-动机的直接子动机出现在$X$中，这可以看作是$\alpha$是$X$的上同调不变量的证据。我们研究了二次曲线中的这种情况，并将其与Kahn的下降猜想联系起来。 显示英文摘要

摘要： 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.