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pure motive

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

Category of effective pure motives

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

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

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

the category of pure motives (with coefficients in RR 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. 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 LL-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)

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