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 Mk over a field k, known as Grothendieck-Chow-Motives is based on choosing rational equivalence for . But one can also choose other equivalence relations for , and thus get a motivic category Mk , 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)=iIMi,

in Mk.

1.Is it known how the decomposition of X in Mk 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 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.

Answer

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Source : Link , Question Author : nxir , Answer Author : Community

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