# Contents

## Idea

Grothendieck conjectured that every Weil cohomology theory factors uniquely through some category, which he called the category of motives. For smooth projective varieties (over some field $k$) such a category was given by Grothendieck himself, called the category of pure Chow motives. For general smooth varieties the category is still conjectural, see at mixed motives.

## Construction

Fix some adequate equivalence relation $\sim$ (e.g. rational equivalence). Let ${Z}^{i}\left(X\right)$ denote the group of $i$-codimensional algebraic cycles and let ${A}_{\sim }^{i}\left(X\right)$ denote the quotient ${Z}^{i}\left(X\right)/\sim$.

### Category of correspondences

Let ${\mathrm{Corr}}_{\sim }\left(k\right)$, the category of correspondences, be the category whose objects are smooth projective varieties and whose hom-sets are the direct sum

${\mathrm{Corr}}_{\sim }\left(h\left(X\right),h\left(Y\right)\right)=\underset{i}{⨁}{A}_{\sim }^{{n}_{i}}\left({X}_{i}×Y\right)\phantom{\rule{thinmathspace}{0ex}},$Corr_\sim(h(X),h(Y)) = \bigoplus_i A^{n_i}_\sim(X_i \times Y) \,,

where $\left({X}_{i}\right)$ are the irreducible components of $X$ and ${n}_{i}$ are their respective dimensions. The composition of two morphisms $\alpha \in \mathrm{Corr}\left(X,Y\right)$ and $\beta \in \mathrm{Corr}\left(Y,Z\right)$ is given by

${p}_{\mathrm{XZ},*}\left({p}_{\mathrm{XY}}^{*}\left(\alpha \right).{p}_{\mathrm{YZ}}^{*}\left(\beta \right)\right)$p_{XZ,*} (p_{XY}^*(\alpha) . p_{YZ}^*(\beta))

where ${p}_{\mathrm{XY}}$ denotes the projection $X×Y×Z\to X×Y$ and so on, and $.$ denotes the intersection product in $X×Y×Z$.

There is a canonical contravariant functor

$h:\mathrm{SmProj}\left(k\right)\to {\mathrm{Corr}}_{\sim }\left(k\right)$h \colon SmProj(k) \to Corr_\sim(k)

from the category of smooth projective varieties over $k$ given by mapping $X↦X$ and a morphism $f:X\to Y$ to its graph, the image of its graph morphism ${\Gamma }_{f}:X\to X×Y$.

The category of correspondences is symmetric monoidal with $h\left(X\right)\otimes h\left(Y\right)≔h\left(X×Y\right)$.

We also define a category ${\mathrm{Corr}}_{\sim }\left(k,A\right)$ of correspondences with coefficients in some commutative ring $A$, by tensoring the morphisms with $A$; this is an $A$-linear category additive symmetric monoidal category.

### Category of effective pure motives

The Karoubi envelope (pseudo-abelianisation) of ${\mathrm{Corr}}_{\sim }\left(k,A\right)$ is called the category of effective pure motives (with coefficients in $A$ and with respect to the equivalence relation $\sim$), denoted ${\mathrm{Mot}}_{\sim }^{\mathrm{eff}}\left(k,A\right)$.

Explicitly its objects are pairs $\left(h\left(X\right),p\right)$ with $X$ a smooth projective variety and $p\in \mathrm{Corr}\left(h\left(X\right),h\left(X\right)\right)$ an idempotent, and morphisms from $\left(h\left(X\right),p\right)$ to $\left(h\left(Y\right),q\right)$ are morphisms $h\left(X\right)\to h\left(Y\right)$ in ${\mathrm{Corr}}_{\sim }$ of the form $q\circ \alpha \circ p$ with $\alpha \in {\mathrm{Corr}}_{\sim }\left(h\left(X\right),h\left(Y\right)\right)$.

This is still a symmetric monoidal category with $\left(h\left(X\right),p\right)\otimes \left(h\left(Y\right),q\right)=\left(h\left(X×Y\right),p×q\right)$. Further it is Karoubian, $A$-linear and additive.

The image of $X\in \mathrm{SmProj}\left(k\right)$ under the above functor

$h:\mathrm{SmProj}\left(k\right)\to {\mathrm{Corr}}_{\sim }\left(k,A\right)\to {\mathrm{Mot}}_{\sim }^{\mathrm{eff}}\left(k,A\right)$h \colon SmProj(k) \to Corr_\sim(k,A) \to Mot^{eff}_\sim(k,A)

is the the motive of $X$.

### Category of pure motives

There exists a motive $L$, called the Lefschetz motive, such that the motive of the projective line decomposes as

$h\left({P}_{k}^{1}\right)=h\left(Spec\left(k\right)\right)\oplus L$h(\mathbf{P}^1_k) = h(\Spec(k)) \oplus \mathbf{L}

To get a rigid category we formally invert the Lefschetz motive and get a category

${\mathrm{Mot}}_{\sim }\left(k,A\right)≔{\mathrm{Mot}}_{\sim }^{\mathrm{eff}}\left(k,A\right)\left[{L}^{-1}\right]\phantom{\rule{thinmathspace}{0ex}},$Mot_\sim(k, A) \coloneqq Mot^{eff}_\sim(k,A)[\mathbf{L}^{-1}] \,,

the category of pure motives (with coefficients in $A$ and with respect to $\sim$).

This is a rigid, Karoubian, symmetric monoidal category. Its objects are triples $\left(h\left(X\right),p,n\right)$ with $n\in Z$.

### Category of pure Chow motives

When the relation $\sim$ is rational equivalence then ${A}_{\sim }^{*}$ are the Chow groups, and ${\mathrm{Mot}}_{\sim }\left(k\right)={\mathrm{Mot}}_{\mathrm{rat}}\left(k\right)$ is called the category of pure Chow motives.

### Category of pure numerical motives

When the relation $\sim$ is numerical equivalence, then one obtains numerical motives.

## References

• Yuri Manin, Correspondences, motifs and monoidal transformations , Math. USSR Sb. 6 439, 1968(pdf, web)

• Tony Scholl?, Classical motives, in Motives, Seattle 1991. Proc Symp. Pure Math 55 (1994), part 1, 163-187 (pdf)

• James Milne, Motives – Grothendieck’s Dream (pdf)

• Minhyong Kim, Classical Motives: Motivic $L$-functions (pdf)

• Bruno Kahn, pdf slides on pure motives

• R. Sujatha, Motives from a categorical point of view, Lecture notes (2008) (pdf)

Section 8.2 of

Revised on November 7, 2013 23:40:03 by Urs Schreiber (82.169.114.243)