Motives and oriented cohomology of a linear algebraic group

TL;DR

This paper constructs a filtration on the cohomology of cellular varieties over characteristic zero fields, relating it to Chow rings, and applies it to compute algebraic cobordism rings of specific algebraic groups, also establishing motives comparison results.

Contribution

It introduces a new filtration on the cohomology of cellular varieties that links oriented cohomology to Chow rings and applies this to compute cobordism rings of certain algebraic groups.

Findings

Computed algebraic cobordism rings for groups of type G2, SO_n, Spin_m, and PGL_k.

Established a correspondence between Chow motives and oriented cohomology motives for generically cellular varieties.

Proved that isomorphic Chow motives imply isomorphic oriented cohomology motives under certain conditions.

Abstract

For a cellular variety $X$ over a field $k$ of characteristic 0 and an algebraic oriented cohomology theory $\hh$ of Levine-Morel we construct a filtration on the cohomology ring $\hh(X)$ such that the associated graded ring is isomorphic to the Chow ring of $X$. Taking $X$ to be the variety of Borel subgroups of a split semisimple linear algebraic group $G$ over $k$ we apply this filtration to relate the oriented cohomology of $G$ to its Chow ring. As an immediate application we compute the algebraic cobordism ring of a group of type $G_2$, of groups $SO_n$ and $Spin_m$ for $n=3,4$ and $m=3,4,5,6$ and $PGL_k$ for $k\geqslant 2$. Using this filtration we also establish the following comparison result between Chow motives and $\hh$-motives of generically cellular varieties: any irreducible Chow-motivic decomposition of a generically split variety $Y$ gives rise to a $\hh$-motivic decomposition of $Y$ with the same generating function. Moreover, under some conditions on the coefficient ring of $\hh$ the obtained $\hh$-motivic decomposition will be irreducible. We also prove that if Chow motives of two twisted forms of $Y$ coincide, then their $\hh$-motives coincide as well.