Milnor K-theory modulo 2 embeds into the Picard group of invertible Morava K-theory motives, with quadrics providing the construction and with Chow motives recoverable from Morava motives in the low-dimensional case.
Artin shapes
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abstract
We introduce and study on examples a notion of the Artin shape for a motive related to a projective homogenous variety. We apply it to the problem of finding the complete motivic decomposition of the variety. Our examples cover unitary involution varieties as well as some varieties given by a quadratic Weil transfer. Some of the decompositions obtained dispel prior expectations on how motivic decompositions of projective homogeneous varieties can look like.
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Invertible Morava motives in quadrics
Milnor K-theory modulo 2 embeds into the Picard group of invertible Morava K-theory motives, with quadrics providing the construction and with Chow motives recoverable from Morava motives in the low-dimensional case.