nLab pure motive

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.

Related concepts

• sheaf with transfer

• motive

• mixed motive

• Voevodsky motive

• noncommutative motive

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

• Alain Connes, Matilde Marcolli, Noncommutative Geometry, Quantum Fields and Motives (pdf)

category: algebraic geometry