# Passing motivic decompositions from rational to algebraic equivalence

It is well known that there are several adequate equivalence relations for algebraic cycles (see https://en.wikipedia.org/wiki/Adequate_equivalence_relation for a list including definitions).

The category of motives $\mathcal{M}_k$ over a field $k$, known as Grothendieck-Chow-Motives is based on choosing rational equivalence for $\sim$. But one can also choose other equivalence relations for $\sim$, and thus get a motivic category $\mathcal{M}^\sim_k$ , which might have different properties (being Tannakian for example).

Assume we have a motivic decomposition of a smooth, projective variety $X$ over $k$,

$M(X) = \bigoplus_{i\in I} M_i$,

in $\mathcal{M}_k$.

1.Is it known how the decomposition of $X$ in $\mathcal{M}^\sim_k$ will differ from the above?

1.1.Is it true, that one will always have less or at most the same number of summands?

2.Can you give a specific example (aside from projective spaces)?

I am interested in the case for $\sim$ to be $alg$, but every example is welcome, considering how little is probably known.

To give more context. I am trying to find out more about criteria for the motive of a variety to be not decomposable. So lifting and descent properties are what i am really interested in.

• Firstly, rational equivalence corresponds to Chow motives, whereas Grothendieck motives are the ones corresponding to numerical equivalence (in the usual terminology). Secondly, which decompositions would you like to consider? Note that it is not known whether Chow motives can be decomposed into indecomposable summands, and there exist non-isomorphic indecomposable decompositions of certain Chow motives with integral coefficients. May 6, 2016 at 19:33
• 1.I was under the assumption that Grothendieck defined what is known today as Grothendieck-Chow-Motives (rational equicalence). I will change that. 2.I know that integer coefficients might get us two non isomorphic decompositions (by results of Merkurjev, Chernousov and some Semenov,Petrov paper). I just wanted to have at least some example, no matter what restrictions are necessary.You are right. To ensure uniqueness we should only consider $\mathbb{F}_p$ coefficients.
– nxir
May 6, 2016 at 20:39
• So, you are interested in finite coefficients? This is an interesting "choice"; yet a rather "non-classical" one; so I doubt that many references are available. Did you try to look at motives of curves (and of Abelian varieties)? May 7, 2016 at 8:29
• And certainly, a relevant reference for your question is "Nilpotence theorem for cycles algebraically equivalent to zero", by Vladimir Voevodsky. Yet it does not say that much on motives with finite coefficients. May 7, 2016 at 8:37
• For a concrete calculation i would be interested in motives of projective, homogeneous varieties of algebraic groups.
– nxir
May 9, 2016 at 9:30