Motivic decomposition of anisotropic varieties of type F₄into generalized Rost motives

Abstract

We prove that the Chow motive of an anisotropic projective homogeneous variety of type F 4 is isomorphic to the direct sum of twisted copies of a generalized Rost motive. In particular, we provide an explicit construction of a generalized Rost motive for a generically splitting variety for a symbol in K M 3 (k)/3. We also establish a motivic isomorphism between two anisotropic non-isomorphic projective homogeneous varieties of type F 4 . All our results hold for Chow motives with integral coefficients.

Motivic decomposition of a compactification of a Merkurjev-Suslin variety

“…Namely, Rost obtained the celebrated decomposition of a norm quadric (see [18]) and later Voevodsky found some direct summand, called a generalized Rost motive, in the motive of any norm variety (see [19]). Note that the F 4 -varieties from [16] can be considered as a mod-3 analog of a Pfister quadric (more precisely, of a maximal Pfister neighbour). In its turn, our variety can be considered as a mod-3 analog of a norm quadric.…”

Rost motives, affine varieties, and classifying spaces

“…Assume now that comes from the first Tits construction, consider its Serre–Rost invariant (see [, § 40]) and the respective Rost motive given in [, Theorem 1.1]. (We remark that formally the coefficients in the Galois cohomology here are instead of , but this does not make any substantial difference in the present situation.)…”

Integral motives, relative Krull–Schmidt principle, and Maranda-type theorems

“…Assume that our group G does not have splitting fields of degree 2 and 3. Then the motivic decompositions of X modulo 2 and modulo 3 are known (see [NSZ09] and [PSZ08, Section 7]). Namely, we have over F M(X) ⊗ Z/3Z = 7 i=0 R 3 (i) and M(X) ⊗ Z/2Z = ⊕ i∈{0,1,2,4,5,7,8,10,11,12} R 2 (i)…”