26
$\begingroup$

It has been almost 60 years since Grothendieck conceived the conjectural theory of motives in order to grasp the common behavior of the most important (Weil) cohomology theories.

But what is the status of the theory of motives today?

What is the closest approach (or alternative) to Grothendieck's original ideas as of today (and why is that)? What are the most promising approaches (or alternatives) which are being "prepared" as we speak?

$\endgroup$
1

4 Answers 4

19
$\begingroup$

Of course Grothendieck gave a construction of a category of what we would now call pure motives, but he couldn't prove that it was abelian etc. without assuming the standard conjectures. These conjectures are still wide open. There are currently several alternative constructions of motives, but arguably the one closest in spirit to the original is André's construction of pure motives using motivated cycles. These are basically the cycles that would be algebraic if one assumed the standard conjectures. By throwing these in, one gets all the desired properties for this category.

$\endgroup$
12
$\begingroup$

I think one recent conceptual advance is Scholze's proposal of a universal cohomology theory valued in "shtukas over $\operatorname{Spec}(\mathbb{Z})$" -- see his ICM address, Section 9. The proposal is that there is a sort of concrete linear algebra category encoding all known cohomology theories. It needn't be "universal" as for Grothendieck's category, but there should be natural algorithms for recovering known cohomology theories from this universal one.

So this suggests an "alternative" to the theory of motives, perhaps the way Hodge structures provide an "alternative" over $\mathbb{C}$ and torsion $\mathbb{Q}_{\ell}[F]$-modules provide an alternative over $\mathbb{F}_q$. (In both cases, one should include more adjectives about the categories, but let me not be so rigorous.) Of course, both those replacements have been quite successfully applied to a range of problems while dodging problems related to the standard conjectures. And one similarly expects that this shtuka cohomology theory would actually encode a great deal of arithmetic and geometric information that would not be accessible from the "bare" theory of motives.

There are many issues: shtukas over $\mathbb{Z}$ aren't defined (doing so would be itself an enormous advance), and one runs into standard conjecture problems in addressing some of the well-posed, concrete questions Scholze considers (see the end of his Section 9). But personally, I found his picture to be quite novel and exciting when I read it.

$\endgroup$
3
$\begingroup$

Here is an excellent recent overview article by Clement Dupont from Grothendieck Centre in Montpellier, covering the origins and most of the recent developments for the non-expert mathematician. This article is in French, however, it is written so nicely that it should provide very profitable reading even with very modest french skills La Gazette de la Société Mathématique de France 178 (octobre 2023)

$\endgroup$
3
  • 1
    $\begingroup$ Unfortunately I cannot open the document ... $\endgroup$
    – THC
    Jan 18 at 10:47
  • 3
    $\begingroup$ The article is titled "Motifs : un tour d’horizon", pp.32-47 of the newsletter. You can download the entire newsletter by clicking on the "Téléchargement" link under "Open access", and entering your email address to receive a download link. $\endgroup$ Jan 18 at 11:35
  • 1
    $\begingroup$ Using DeepL to translate this document is not a bad idea... $\endgroup$
    – Akira
    Jan 18 at 12:26
3
$\begingroup$

This is not very recent but it is worth mentioning. One approach to the existence of category of mixed motives $MM(k)$ ($k$ a field of characteristic zero) is via the existence of the motivic Galois group $\mathbf{G}_{mot}(k)$ since $MM(k)$ is expected be a neutral Tannakian category. One candidate of such a group is the Nori motivic Galois group, $\mathbf{G}_{Nori}(k)$. There is a different approach by Ayoub when $k \subset \mathbb{C}$, called the weak Tannakian formalism. Let me recall some general nonsense: let $(\mathcal{M},\mathbf{1},\otimes), (\mathcal{N},\mathbf{1},\otimes)$ be two symmetric monoidal category and $f \colon \mathcal{M} \to \mathcal{N}$ a symmetric monoidal functor having a right adjoint $g \colon \mathcal{N} \to \mathcal{M}$.

Theorem (Ayoub). Assume that there exists a monoidal functor $e \colon \mathcal{N} \to \mathcal{M}$ such that $f \circ e \simeq id_{\mathcal{N}}$ and $e$ has a right adjoint $u$ and furthermore the coprojection $g(A) \otimes M \simeq g(A \otimes f(M))$ is an isomorphism, then the object $fg\mathbf{1}$ is a Hopf algebra.

Let's denote by $\mathbf{DA}(k,\mathbb{Q})$ the category of étale motives with rational coefficients, and $$(\operatorname{Bti}^* \dashv \operatorname{Bti}_*) \colon \mathbf{DA}(k,\mathbb{Q}) \longrightarrow \mathbf{D}(\mathbb{Q})$$ the Betti realization, where $\mathbf{D}(\mathbb{Q})$ is the derived category of $\mathbb{Q}$-vector spaces. Since $\operatorname{Bti}^*$ has a section given by sending a $\mathbb{Q}$-vector space $V$ to the constant presheaf with values in $V$. The theorem above can be applied to this situation to endow $H=\operatorname{Bti}^*\operatorname{Bti}_*\mathbf{1}$ a Hopf algebra structure and Ayoub showed furthermore that $H$ has no homology in negative degree, so it is a $\mathbb{Q}$-Hopf algebra. The scheme $\operatorname{Spec}(H)$ is called the Ayoub motivic Galois group $\mathbf{G}_{Ayoub}(k)$. The magic thing is

Theorem (Choudhurry, Gallauer, Souza). $\mathbf{G}_{Ayoub}(k) \simeq \mathbf{G}_{Nori}(k)$.

So at least, theoretically, we have one more reason to believe that this is a candidate for the real motivic Galois group.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.