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Is the functor from the category of projective varieties over a field $k$ to the category of pure motives over $k$, faithful? (Perhaps it is not full).


Is the functor from the category of affine varieties over a field $k$ to the category of mixed motives over $k$, faithful? (Perhaps it is not full).

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First of all, I wouldn't equate "captures everything" with "faithful functor". The functor from groups to sets which forgets the multiplication structure is faithful, but I wouldn't say that knowing a group has 512 elements tells you everything about it.

An example of nonisomorphic varieties with isomorphic motives: Consider $\mathbb{P}^2$ blown up at $5$ points. Blowing up four points (in general position) rigidifies the $\mathbb{P}^2$. There is then a two dimensional space of moduli for where you choose the fifth point. All of these nonisomorphic surfaces have isomorphic motives.

An example of the functor to motives not being faithful: The automorphism group of $\mathbb{P}^1$ is $PGL_2$. All of these nontrivial automorphisms act trivially on the motive.

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What equivalence relation are you using to define motives?

The finest that people normally consider is rational equivalence, in which case two morphisms whose graphs are rationally equivalent (which is to say, roughly speaking, two morphisms which can be deformed one into the other via a family parameterized by $\mathbb P^1$) will give the same morphism.

A trivial, but concrete, example: all constant morphisms from any variety to $\mathbb P^1$ will be identified in the category of motives.

If you use a coarser relation (homological equivalence or numerical equivalence, which are conjecturally the same, and lead to what are called Grothendieck motives) than even more morphisms of varieties will be identified in the category of motives.

Incidentally, these functors are certainly not full, since one takes a Karoubian closure as part of the construction of the category of motives, and then furthermore inverts the Lefschetz motive.

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Different varieties with the same zeta function would likely (and maybe provably in the case of varieties over a finite field) provide examples of non-faithfulness on objects. The zetas are defined without direct reference to motives or "motivic structures" on cohomology.

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If you work with Grothendieck motives modulo rational equivalence, you even have simpler examples : the motive of any split (projective) quadric of odd dimension $n$ is the same as the motive of $\mathbb{P}^n$.

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