Holmstrom Pure motives

Pure motives

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Pure motives

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Pure motives

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Pure motives

See also Motives, Weil cohomology, Algebraic cycles, and most importantly, the “Pure” chapter in the book project.

(Certain forms of) pure motives are sometimes referred to by the names Chow motives, Classical motives, Grothendieck motives.

See also Finite-dimensional motives

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Pure motives

Milne and Ramachandran: Integral motives and special values of zeta functions (no separate page for integral motives, or the also-mentioned Weil motives)

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Pure motives

Some notes from André:

Fix a commutative ring of coeffs, and an adequate equiv relation. Start with the cat of smooth projective $k$-schemes, and then perform three steps: Replace morphisms by correspondences mod $\sim$ with coeffs in $F$ (maybe of degree zero). Take the pseudo-abelian envelope. Invert the Lefschetz motive.

The first step gives an $F$-linear cat, with a tensor structure.

The second step gives a pseudo-abelian cat, the cat of pure effective motives.

Def of reduced motive of a variety with a rational point.

Third step gives us Tate twists (?)

Tensor structure. Duality, so the cat of pure motives is a rigid tensor cat.

Some exercises!

Def: “The Tate motive”. The phrase “Tate motives” also refers to all finite sums of twists of the unit motive.

Subtensor cats: Given a subcat $V$ of the cat smooth projectives, stable under product and disjoint sum, we can consider the smallest full subcat of the cat of pure motives, stable sum, tensor, direct factor, and duals, and containing all motives coming from $V$. This will also be a rigid $F$-linear pseuod-abelian tensor cat. Example: Finite étale $k$-schemes give Artin motives, a cat which is independent of the choice of equivalence relation; if $char(F)=0$, the cat is tensor equivalent to the cat of Galois reps. Example: The cat of motives generated by a variety.

4.2 Functorialities and first properties.

Change of adequate equiv relation. The obvious surjective homomorphisms of cycle groups give rise to a canonical full tensor functor between the cats of pure motives. It seems to be a conjecture that this is essentially surjective.

Change of coefficients. In most cases (I think if () holds) this the same as tensoring each Hom group with the new coeffs.

Base change. Get an extension of scalars functor, which has a right adjoint (induced by the “underlying $k$-scheme functor) if the extension is finite separable. The base change functor on the level of smooth projective schemes has both a left and a right adjoint (underlying $k$-scheme and Weil restriction, respectively), possibly under the same condition on the extension. Weil restriction induces a functor on Chow motives (Karpenko).

The universal property of Chow motives, for “Weil-cohomology-like” functors on the cat of smooth projective varieties. Details omitted here.

As a consequence, we have:

Prop: Giving the data of a Weil cohomology with coeffs in a field $K$ containing $F$ is equivalent to giving a tensor functor

satisfying $H^i(\mathbf{1}(-1)) = 0$ for $i \neq 2$.

Functors like this are called realizations. The classical realizations send the Tate twist in Chow motives to the Tate twist described earlier for the various cohomologies.

Prop: The dimension of the $K$-vector space $H^i(M)$ is independent of the choice of classical Weil cohomology. Question: What about non-classical ones???

Prop: Suppose that $F$ is an algebra over the rationals. Let $f:X \to Y$ be a morphism of smooth projective $k$-schemes. Then $f^*: h(Y) \to h(X)$ admits a left inverse in $M^{\sim}(k)_F$, so $h(Y)$ is a direct factor of $h(X)$.

Proof of last prop uses alterations.

4.3 Examples

Manin’s identity principle. A consequence of Yoneda, allows us to determine when morphisms of motives is an iso, or when two such morphisms coincide, or when two maps fits together in a split short exact seq. Method: a motive represents a contravariant functor, and we can check the above statements “pointwise” on all varieties (so I think that the point here is that we do not need to check pointwise on all motives).

Application to computation of some motives:

Projective bundles, in particular projective $n$-space, gives the formula $h(P) = \oplus_0^n h(X)(-r)$ where $X$ is the base.

Blow-ups: Relations between the motives of a space, its blow-up, the subscheme, and the exceptional divisor.

Curves: Under some conditions (geom conn, and either a rational point or $F$ containing the rationals), one has $h(X) = \mathbf{1} \oplus h^1(X) \oplus \mathbf{1}(-1)$.

Abelian varieties: Deep work, involving the Fourier-Mukai transform. Result: a unique decomposition $h(A) = \oplus_0^{2d} h^i(A)$ such that the correspondence defined by $[m]_A$ acts as multiplication by $m^i$ on $h^i(A)$. Also, an iso between $h^i(A)$ and $S^i h^1(A)$. Expression for the dual, and iso induced by a polarisation. Homs between two $h^1$‘s of abelian varieties equals the $Hom(A,B) \otimes_{\mathbb{Z}} F$ and is hence indep of the equiv relation.

Consideration of the cat of motives “generated by” Artin motives and $h^1$‘s of abelian varieties. Any motive of a variety dominated by a product of smooth projective curves lies in this cat.

Remark: A motive of a Fermat hypersurface is cut out form the motive of some abelian variety.

Exercises.

One conjectures that decompositions as above exist for any smooth projective variety. Descriptions of $h^0, h^1, h^{2d-1}, h^{2d}$. In particular, construction of an idempotent cutting out $h^1$, with relations to the Picard and Albanese varieties. In the case $F=\mathbb{Q}$, we get a description of the cat of motives of the form $h^1(X)$; it is equivalent to the cat of abelian varieties up to isogeny, so in particular it is abelian semisimple.

Remark: Get decomposition for surfaces.

Motives attached to modular forms. Deligne attached $\ell$-adic parabolic cohomology spaces to modular forms of weight at least two, for some congruence subgroup, using Kuga-Sato varieties. Scholl proved that these cohomology spaces come from a Chow motive. If one wants a decomposition respecting the Hecke action, it seems like one has to pass to motives wrt homological equivalence. I think the following is true: Given a normalised newform of weight at least 2, level $N$, and character $\chi$, with coeffs in a number field $F$, then Scholle constructs a motive in $M_{hom}(\mathbb{Q})_F$ with the right L-factor at good primes.

4.4 Tensor ideals and adequate equiv relations

Def of (twosided) ideal in an $F$-linear cat, and of quotienting out a cat by such and ideal. If the ideal is a tensor ideal (= monoidal ideal) in a tensor cat, then the quotient inherits a tensor structure.

Correspondence between adequate equivalence relations and tensor ideals in $CHM(k)_F$. Given an adequate relation, the corresponding cat of motives is obtained by quotienting out by the corresponding ideal, and then taking the pseudo-abelian envelope.

Remark: This correspondence does not work on the level of effective motives.

Remark: The correspondence allows us to define the product of two equivalence relations. Explicitly, this works out as

for $\alpha \in CH(X)_F$.

Example: The ideal corresponding to smash-nilpotence

Example: For $F$ a field, description of the ideal corr to numerical equiv.

4.5 Semisimplicity of numerical motives

Thm (Jannsen): If $F$ is a field, the cat $NM(k)_F = M_{num}(k)_F$ is abelian semisimple, and num is the only adequate equivalence relation for which this happens.

Proof: Purely categorical, except for the input of the existence of a Weil cohomology theory.

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Pure motives

A course on pure motives

http://ncatlab.org/nlab/show/pure+motive

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Pure motives

Manin: Correspondences, motifs and monoidal transformations (in Russian, English translation) (1968)

Kleiman: Motives, Algebraic Geometry, Oslo, 1970 (F. Oort, ed), Walters-Noordhoff, Groningen, 1972, pp53-82

André: Une introduction aux motifs. Chapter 4.

Scholl: Classical motives (in Motives volume I)

Murre: Lectures on motives?

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Pure motives

http://mathoverflow.net/questions/53220/could-the-kunneth-decomposition-of-a-motif-depend-on-the-choice-of-l

http://mathoverflow.net/questions/14587/understanding-the-definition-of-the-lefschetz-pure-effective-motive

nLab page on Pure motives