nLab
pure motive

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

Category of correspondences

Let Corr (k)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 (h(X),h(Y))= iA n i(X i×Y), Corr_\sim(h(X),h(Y)) = \bigoplus_i A^{n_i}_\sim(X_i \times Y) \,,

where (X i)(X_i) are the irreducible components of XX and n in_i are their respective dimensions. The composition of two morphisms αCorr(X,Y)\alpha \in Corr(X,Y) and βCorr(Y,Z)\beta \in Corr(Y,Z) is given by

p XZ,*(p XY *(α).p YZ *(β)) p_{XZ,*} (p_{XY}^*(\alpha) . p_{YZ}^*(\beta))

where p XYp_{XY} denotes the projection X×Y×ZX×YX \times Y \times Z \to X \times Y and so on, and .. denotes the intersection product in X×Y×ZX \times Y \times Z.

There is a canonical contravariant functor

h:SmProj(k)Corr (k) h \colon SmProj(k) \to Corr_\sim(k)

from the category of smooth projective varieties over kk given by mapping XXX \mapsto X and a morphism f:XYf : X \to Y to its graph, the image of its graph morphism Γ f:XX×Y\Gamma_f : X \to X \times Y.

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

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

Category of effective pure motives

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

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

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

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

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

is the the motive of XX.

Category of pure motives

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

h(P k 1)=h(Spec(k))Lh(\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 (k,A)Mot eff(k,A)[L 1], Mot_\sim(k, A) \coloneqq Mot^{eff}_\sim(k,A)[\mathbf{L}^{-1}] \,,

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

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

Category of pure Chow motives

When the relation \sim is rational equivalence then A *A^*_\sim are the Chow groups, and Mot (k)=Mot rat(k)Mot_\sim(k) = Mot_{rat}(k) 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 LL-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 February 7, 2014 05:48:48 by Urs Schreiber (89.204.153.37)