Motive (algebraic geometry)

The objects of ${\displaystyle \operatorname {Corr} (k)}$ are simply smooth projective varieties over k. The morphisms are correspondences. They generalize morphisms of varieties ${\displaystyle X\to Y}$, which can be associated with their graphs in ${\displaystyle X\times Y}$, to fixed dimensional Chow cycles on ${\displaystyle X\times Y}$.

It will be useful to describe correspondences of arbitrary degree, although morphisms in ${\displaystyle \operatorname {Corr} (k)}$ are correspondences of degree 0. In detail, let X and Y be smooth projective varieties and consider a decomposition of X into connected components:

${\displaystyle X=\coprod _{i}X_{i},\qquad d_{i}:=\dim X_{i}.}$

If ${\displaystyle r\in \mathbb {Z} }$, then the correspondences of degree r from X to Y are

${\displaystyle \operatorname {Corr} ^{r}(k)(X,Y):=\bigoplus _{i}A^{d_{i}+r}(X_{i}\times Y),}$

where ${\displaystyle A^{k}(X)}$ denotes the Chow-cycles of codimension k. Correspondences are often denoted using the "⊢"-notation, e.g., ${\displaystyle \alpha :X\vdash Y}$. For any ${\displaystyle \alpha \in \operatorname {Corr} ^{r}(X,Y)}$ and ${\displaystyle \beta \in \operatorname {Corr} ^{s}(Y,Z),}$ their composition is defined by

${\displaystyle \beta \circ \alpha$ :=\pi _{XZ*}\left(\pi _{XY}^{*}(\alpha )\cdot \pi _{YZ}^{*}(\beta )\right)\in \operatorname {Corr} ^{r+s}(X,Z),}

where the dot denotes the product in the Chow ring (i.e., intersection).

Returning to constructing the category ${\displaystyle \operatorname {Corr} (k),}$ notice that the composition of degree 0 correspondences is degree 0. Hence we define morphisms of ${\displaystyle \operatorname {Corr} (k)}$ to be degree 0 correspondences.

The following association is a functor (here ${\displaystyle \Gamma _{f}\subseteq X\times Y}$ denotes the graph of ${\displaystyle f:X\to Y}$):

${\displaystyle F:{\begin{cases}\operatorname {SmProj} (k)\longrightarrow \operatorname {Corr} (k)\\X\longmapsto X\\f\longmapsto \Gamma _{f}\end{cases}}}$

Just like ${\displaystyle \operatorname {SmProj} (k),}$ the category ${\displaystyle \operatorname {Corr} (k)}$ has direct sums (X ⊕ Y := X ∐ Y) and tensor products (X ⊗ Y := X × Y). It is a preadditive category. The sum of morphisms is defined by

${\displaystyle \alpha +\beta$ :=(\alpha ,\beta )\in A^{*}(X\times X)\oplus A^{*}(Y\times Y)\hookrightarrow A^{*}\left(\left(X\coprod Y\right)\times \left(X\coprod Y\right)\right).}

The transition to motives is made by taking the pseudo-abelian envelope of ${\displaystyle \operatorname {Corr} (k)}$:

${\displaystyle \operatorname {Chow} ^{\operatorname {eff} }(k):=Split(\operatorname {Corr} (k))}$.

In other words, effective Chow motives are pairs of smooth projective varieties X and idempotent correspondences α: X ⊢ X, and morphisms are of a certain type of correspondence:

${\displaystyle \operatorname {Ob} \left(\operatorname {Chow} ^{\operatorname {eff} }(k)\right):=\{(X,\alpha )\mid (\alpha :X\vdash X)\in \operatorname {Corr} (k){\mbox{ such that }}\alpha \circ \alpha =\alpha \}.}$

${\displaystyle \operatorname {Mor} ((X,\alpha ),(Y,\beta )):=\{f:X\vdash Y|f\circ \alpha =f=\beta \circ f\}.}$

Composition is the above defined composition of correspondences, and the identity morphism of (X, α) is defined to be α : X ⊢ X.

The association,

${\displaystyle h:{\begin{cases}\operatorname {SmProj} (k)&\longrightarrow \operatorname {Chow^{eff}} (k)\\X&\longmapsto [X]:=(X,\Delta _{X})\\f&\longmapsto [f]:=\Gamma _{f}\subset X\times Y\end{cases}}}$,

where Δ_X := [id_X] denotes the diagonal of X × X, is a functor. The motive [X] is often called the motive associated to the variety X.

As intended, Chow^eff(k) is a pseudo-abelian category. The direct sum of effective motives is given by

${\displaystyle ([X],\alpha )\oplus ([Y],\beta ):=\left(\left[X\coprod Y\right],\alpha +\beta \right),}$

The tensor product of effective motives is defined by

${\displaystyle ([X],\alpha )\otimes ([Y],\beta ):=(X\times Y,\pi _{X}^{*}\alpha \cdot \pi _{Y}^{*}\beta ),}$

where

${\displaystyle \pi _{X}:(X\times Y)\times (X\times Y)\to X\times X,\quad {\text{and}}\quad \pi _{Y}:(X\times Y)\times (X\times Y)\to Y\times Y.}$

The tensor product of morphisms may also be defined. Let f₁ : (X₁, α₁) → (Y₁, β₁) and f₂ : (X₂, α₂) → (Y₂, β₂) be morphisms of motives. Then let γ₁ ∈ A^*(X₁ × Y₁) and γ₂ ∈ A^*(X₂ × Y₂) be representatives of f₁ and f₂. Then

${\displaystyle f_{1}\otimes f_{2}:(X_{1},\alpha _{1})\otimes (X_{2},\alpha _{2})\vdash (Y_{1},\beta _{1})\otimes (Y_{2},\beta _{2}),\qquad f_{1}\otimes f_{2}:=\pi _{1}^{*}\gamma _{1}\cdot \pi _{2}^{*}\gamma _{2}}$,

where π_i : X₁ × X₂ × Y₁ × Y₂ → X_i × Y_i are the projections.

To proceed to motives, we adjoin to Chow^eff(k) a formal inverse (with respect to the tensor product) of a motive called the Lefschetz motive. The effect is that motives become triples instead of pairs. The Lefschetz motive L is

${\displaystyle L:=(\mathbb {P} ^{1},\lambda ),\qquad \lambda$ :=pt\times \mathbb {P} ^{1}\in A^{1}(\mathbb {P} ^{1}\times \mathbb {P} ^{1})} .

If we define the motive 1, called the trivial Tate motive, by 1 := h(Spec(k)), then the elegant equation

${\displaystyle [\mathbb {P} ^{1}]=\mathbf {1} \oplus L}$

holds, since

${\displaystyle \mathbf {1} \cong \left(\mathbb {P} ^{1},\mathbb {P} ^{1}\times \operatorname {pt} \right).}$

The tensor inverse of the Lefschetz motive is known as the Tate motive, T := L⁻¹. Then we define the category of pure Chow motives by

${\displaystyle \operatorname {Chow} (k):=\operatorname {Chow} ^{\operatorname {eff} }(k)[T]}$.

A motive is then a triple

${\displaystyle (X\in \operatorname {SmProj} (k),p:X\vdash X,n\in \mathbb {Z} )}$

such that morphisms are given by correspondences

${\displaystyle f:(X,p,m)\to (Y,q,n),\quad f\in \operatorname {Corr} ^{n-m}(X,Y){\mbox{ such that }}f\circ p=f=q\circ f,}$

and the composition of morphisms comes from composition of correspondences.

As intended, ${\displaystyle \operatorname {Chow} (k)}$ is a rigid pseudo-abelian category.

In order to define an intersection product, cycles must be "movable" so we can intersect them in general position. Choosing a suitable equivalence relation on cycles will guarantee that every pair of cycles has an equivalent pair in general position that we can intersect. The Chow groups are defined using rational equivalence, but other equivalences are possible, and each defines a different sort of motive. Examples of equivalences, from strongest to weakest, are

• Rational equivalence
• Algebraic equivalence
• Smash-nilpotence equivalence (sometimes called Voevodsky equivalence)
• Homological equivalence (in the sense of Weil cohomology)
• Numerical equivalence

The literature occasionally calls every type of pure motive a Chow motive, in which case a motive with respect to algebraic equivalence would be called a Chow motive modulo algebraic equivalence.

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

Given a smooth variety X and a variety Y call an integral closed subscheme ${\displaystyle W\subset X\times Y}$ which is finite over X and surjective over a component of Y a prime correspondence from X to Y. Then, we can take the set of prime correspondences from X to Y and construct a free A-module ${\displaystyle C_{A}(X,Y)}$. Its 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.

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

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 ${\displaystyle [X]}$. If we localize this category with respect to the smallest thick subcategory (meaning it is closed under extensions) containing morphisms

${\displaystyle [X\times \mathbb {A} ^{1}]\to [X]}$

and

${\displaystyle [U\cap V]{\xrightarrow {j_{U}'+j_{V}'}}[U]\oplus [V]{\xrightarrow {j_{U}-j_{V}}}[X]}$

then we can form the triangulated category of effective geometric motives ${\displaystyle {\mathcal {DM}}_{\text{gm}}^{\text{eff}}(k,A).}$ 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 Mayer–Vietoris sequence.

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

Using the triangulated structure we can construct a triangle

${\displaystyle \mathbb {L} \to [\mathbb {P} ^{1}]\to [\operatorname {Spec} (k)]{\xrightarrow {[+1]}}}$

from the canonical map ${\displaystyle \mathbb {P} ^{1}\to \operatorname {Spec} (k)}$. We will set ${\displaystyle A(1)=\mathbb {L} [-2]}$ and call it the Tate motive. Taking the iterative tensor product lets us construct ${\displaystyle A(k)}$. If we have an effective geometric motive 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 M an effective geometric mixed motive and n an integer representing the twist by the Tate motive. The hom-groups are then the colimit

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

There are several elementary examples of motives which are readily accessible. One of them being the Tate motives, denoted ${\displaystyle \mathbb {Q} (n)}$, ${\displaystyle \mathbb {Z} (n)}$, or ${\displaystyle A(n)}$, depending on the coefficients used in the construction of the category of Motives. These are fundamental building blocks in the category of motives because they form the "other part" besides Abelian varieties.

The motive of a curve can be explicitly understood with relative ease: their Chow ring is just

$${\displaystyle \mathbb {Z} \oplus {\text{Pic}}(C)}$$

for any smooth projective curve ${\displaystyle C}$, hence Jacobians embed into the category of motives.

Motivic cohomology itself had been invented before the creation of mixed motives by means of algebraic K-theory. The above category provides a neat way to (re)define it by

${\displaystyle H^{n}(X,m):=H^{n}(X,\mathbb {Z} (m)):=\operatorname {Hom} _{DM}(X,\mathbb {Z} (m)[n]),}$

where n and m are integers and ${\displaystyle \mathbb {Z} (m)}$ is the m-th tensor power of the Tate object ${\displaystyle \mathbb {Z} (1),}$ which in Voevodsky's setting is the complex ${\displaystyle \mathbb {P} ^{1}\to \operatorname {pt} }$ shifted by –2, and [n] means the usual shift in the triangulated category.

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