Contents

# Contents

## Idea

Grothendieckconjectured 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 $Z^i(X)$ denote the group of $i$-codimensional algebraic cycles and let $A^i_\sim(X)$ denote the quotient $Z^i(X)/\sim$.

### Category of correspondences

Let $Corr_\sim(k)$, the category of correspondences, be the category whose objects are smooth projective varieties and whose hom-sets are the direct sum

$Corr_\sim(h(X),h(Y)) = \bigoplus_i A^{n_i}_\sim(X_i \times Y) \,,$

where $(X_i)$ are the irreducible components of $X$ and $n_i$ are their respective dimensions. The composition of two morphisms $\alpha \in Corr(X,Y)$ and $\beta \in Corr(Y,Z)$ is given by

$p_{XZ,*} (p_{XY}^*(\alpha) . p_{YZ}^*(\beta))$

where $p_{XY}$ denotes the projection $X \times Y \times Z \to X \times Y$ and so on, and $.$ denotes the intersection product in $X \times Y \times Z$.

There is a canonical contravariant functor

$h \colon SmProj(k) \to Corr_\sim(k)$

from the category of smooth projective varieties over $k$ given by mapping $X \mapsto X$ and a morphism $f : X \to Y$ to its graph, the image of its graph morphism $\Gamma_f : X \to X \times Y$.

The category of correspondences is symmetric monoidal with $h(X) \otimes h(Y) \coloneqq h(X \times Y)$.

We also define a category $Corr_\sim(k, R)$ of correspondences with coefficients in some commutative ring $R$, by tensoring the morphisms with $R$; this is a $R$-linear category additive symmetric monoidal category.

### Category of effective pure motives

The Karoubi envelope (pseudo-abelianisation) of $Corr_\sim(k, R)$ is called the category of effective pure motives (with coefficients in $R$ and with respect to the equivalence relation $\sim$), denoted $Mot^eff_\sim(k, R)$.

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

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

The image of $X \in SmProj(k)$ under the above functor

$h \colon SmProj(k) \to Corr_\sim(k,A) \to Mot^{eff}_\sim(k,R)$

is the the motive of $X$.

### Category of pure motives

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

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

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

$Mot_\sim(k, R) \coloneqq Mot^{eff}_\sim(k,R)[\mathbf{L}^{-1}] \,,$

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

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

### Category of pure Chow motives

When the relation $\sim$ is rational equivalence then $A^*_\sim$ are the Chow groups, and $Mot_\sim(k) = Mot_{rat}(k)$ is called the category of pure Chow motives. This category has the advantage that it is universal for Weil cohomology theories: that is, every Weil cohomology factors uniquely through it.

### Category of pure numerical motives

When the relation $\sim$ is numerical equivalence, then one obtains numerical motives. This category has the advantage of being a semisimple abelian category. In fact, Uwe Jannsen proved that numerical equivalence is the only adequate equivalence relation that gives a semisimple abelian category of pure motives.

## References

• Daniel Dugger, Navigating the Motivic World. (pdf) A draft of a user-friendly introduction to motives, especially good if you’re coming to this topic through algebraic topology.

• Uwe Jannsen, Motives, numerical equivalence, and semi-simplicity, Invent. Math. 107.3 (1992): 447-452. (pdf)

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

• Bruno Kahn, pdf slides on pure motives

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

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

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

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

Section 8.2 of

Last revised on May 18, 2022 at 02:06:44. See the history of this page for a list of all contributions to it.