Homology, Homotopy and Applications

Volume 16 (2014)

Number 2

Motives and oriented cohomology of generically cellular varieties

Pages: 275 – 288

DOI: https://dx.doi.org/10.4310/HHA.2014.v16.n2.a15

Author

Alexander Neshitov (Department of Mathematics and Statistics, University of Ottawa, Ontario, Canada; and St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia)

Abstract

For a cellular variety $X$ over a field $k$ of characteristic 0 and an algebraic oriented cohomology theory $\mathtt{h}$ of Levine-Morel we construct a filtration on the cohomology ring $\mathtt{h}(X)$ such that the associated graded ring is isomorphic to the Chow ring of $X$. Using this filtration we establish the following comparison result between Chow motives and $\mathtt{h}$-motives of generically cellular varieties: any irreducible Chow-motivic decomposition of a generically cellular variety $Y$ gives rise to an $\mathtt{h}$-motivic decomposition of $Y$ with the same generating function. Moreover, under some conditions on the coefficient ring of $\mathtt{h}$ the obtained $\mathtt{h}$-motivic decomposition will be irreducible. We also prove that if the Chow motives of two twisted forms of $Y$ coincide, then their $\mathtt{h}$-motives coincide as well.

Keywords

flag variety, oriented cohomology theory, algebraic group, algebraic cobordism

2010 Mathematics Subject Classification

14F43, 20G10

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Published 30 November 2014