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Geometric mixed-motives were originally introduced by Vladimir Voevodsky as a formulation for the conjectural category of mixed-motives over a field of characteristic 0. Their flavor is much of the same as Grothendieck's original category of pure motives. His process for constructing this category is by starting with an additive category, forming the homotopy category and then localizing by some desired properties satisfied by a cohomology theory.

Construction

Notation

Here we will fix a field ${\displaystyle k}$ of characteristic ${\displaystyle 0}$ and let ${\displaystyle A=\mathbb {Q} ,\mathbb {Z} }$ be our coefficient ring. Set ${\displaystyle {\mathcal {Var}}/k}$ as the category of of quasi-projective varieties over ${\displaystyle k}$ are separated schemes of finite type. We will also let ${\displaystyle {\mathcal {Sm}}/k}$ be the subcategory of smooth varieties.

Smooth Varieties with Correspondences

Given a smooth variety ${\displaystyle X}$ and a variety ${\displaystyle Y}$ call an integral closed subscheme ${\displaystyle W\subset X\times Y}$ which is finite over ${\displaystyle X}$ and surjective over a component of ${\displaystyle Y}$ a prime correspondence from ${\displaystyle X}$ to ${\displaystyle Y}$. Then, we can take the set of prime correspondences from ${\displaystyle X}$ to ${\displaystyle Y}$ and construct a free ${\displaystyle A}$-module ${\displaystyle C_{A}(X,Y)}$. It's elements are called finite correspondences. Then, we can form an additive category ${\displaystyle {\mathcal {SmCor}}}$ whose objects are smooth varieties and morphisms are given by smooth correspondences. The only non-trivial part of this "definition" is the fact that we need to describe compositions. These are given by a push-pull formula from the theory of Chow rings.

Examples

Typical examples of prime correspondences come from the graph ${\displaystyle \Gamma _{f}\subset X\times Y}$ of a morphism of varieties ${\displaystyle f:X\to Y}$.

Localizing the Homotopy Category

From here we can form the homotopy category ${\displaystyle K^{b}({\mathcal {SmCor}})}$ of bounded complexes of smooth correspondences. Here smooth varieties will be denoted ${\displaystyle }$. If we localize this category with respect to the smallest thick subcategory (meaning it is closed under extensions) containing morphisms

${\displaystyle \to }$

and

${\displaystyle {\xrightarrow {j_{U}'+j_{V}'}}\oplus {\xrightarrow {j_{U}-j_{V}}}}$

then we can form the triangulated category of effective geometric motives ${\displaystyle {\mathcal {DM}}_{gm}^{eff}(k,A)}$. Note that the first class of morphisms are localizing ${\displaystyle \mathbb {A} ^{1}}$-homotopies of varieties while the second will give the category of geometric mixed motives the Meyer-Vietoris sequence.

Also, note that this category has a tensor structure given by the product of varieties, so ${\displaystyle \otimes =}$.

Inverting the Tate-Motive

Using the triangulated structure we can construct a triangle

${\displaystyle \mathbb {L} \to \to {\xrightarrow {}}}$

from the canonical map ${\displaystyle \mathbb {P} ^{1}\to {\text{Spec}}(k)}$. We will set ${\displaystyle A(1)=\mathbb {L} }$ and call it the tate motive. Taking the iterative tensor product let's us construct ${\displaystyle A(k)}$. If we have an effective geometric motive ${\displaystyle M}$ we let ${\displaystyle M(k)}$ denote ${\displaystyle M\otimes A(k)}$. Moreover, this behaves functorially and forms a triangulated functor. Finally, we can define the category of geometric mixed motives ${\displaystyle {\mathcal {DM}}_{gm}}$ as the category of pairs ${\displaystyle (M,n)}$ for ${\displaystyle M}$ an effective geometric mixed motive and ${\displaystyle n}$ an integer representing the twist by the Tate motive. The hom-groups are then the colimit

${\displaystyle {\text{Hom}}_{\mathcal {DM}}((A,n),(B,m))\lim _{k\geq -n,-m}{\text{Hom}}_{{\mathcal {DM}}_{gm}^{eff}}(A(k+n),B(k+m))}$

References

• Voevedsky, Vladimir, Triangulated categories of motives over a field, url=https://faculty.math.illinois.edu/K-theory/0074/ {{citation}}: Missing pipe in: |title= (help)

• Draft AfC submissions