Motive (mathematics)

In algebraic geometry , the theory of motifs is a presumably universal cohomology theory of schemes , from which the De-Rham cohomology , the l-adic cohomology and the crystalline cohomology of the algebraic varieties associated with the scheme over different fields can be derived.

The theory of motifs was developed by Alexander Grothendieck and first introduced in a letter to Jean-Pierre Serre in 1964. It should find its generalization in the theory of mixed motives , the derived category of which was constructed by Vladimir Voivodsky .

The name comes from Yuri Manin , who attended the Grothendieck seminar at IHES in May 1967 , where he learned the concept of Grothendieck's motifs himself, and who published the first work on the subject in 1968, in which he also calculated the motif of an inflation without using Grothendieck's standard conjectures.

Quote

Of all the things that I was allowed to discover and bring to light, this world of motifs still strikes me as the most fascinating, the most charged with mystery - the very core of the deep identity of "geometry" and "arithmetic". And the "yoga of motifs". . . It is perhaps the most powerful tool that I uncovered during this first period of my math life as a mathematician.

Contrary to what happened in the usual topology, one finds oneself there before an alarming plethora of different cohomological theories. One has the distinct impression (but in a way that remains vague) that each of these theories "is the same", that they "produce the same results". To express this relationship between these different cohomological theories, I formulated the term "motif", which is associated with an algebraic variety. By this term I mean to suggest that it is the "common motive" (or "common ground") behind this multitude of cohomological invariants associated with an algebraic variety, or in fact behind all a priori possible cohomological invariants.

(Alexandre Grothendieck: Récoltes et semailles: Réflexions et témoignages sur un passé de mathématicien. Université des Sciences et Technologies du Languedoc, Montpellier, et Center National de la Recherche Scientifique, 1986.)

Definition (Grothendieck)

The category of subjects

A motif is a triple of a smooth projective variety , an idempotent correspondence and an integer . ${\displaystyle (X,p,m)}$ ${\displaystyle X}$ ${\displaystyle p\colon X\vdash X}$ ${\displaystyle m}$

Morphisms between motifs and are the elements of , with the group of correspondences denoted by degree and the group of their equivalence classes modulo numerical equivalence . ${\displaystyle (X,p,m)}$ ${\displaystyle (Y,q,n)}$ ${\displaystyle Corr_{\sim }^{n-m}(X,Y)\otimes \mathbb {Q} }$ ${\displaystyle Corr^{n-m}(X,Y)}$ ${\displaystyle X\vdash Y}$ ${\displaystyle n-m}$ ${\displaystyle Corr_{\sim }^{n-m}(X,Y)}$

Universal property

For every smooth projective variety one has an associated motif , where the diagonal is in. ${\displaystyle X}$ ${\displaystyle {\mathcal {M}}(X)=(X,\Delta _{X},0)}$ ${\displaystyle \Delta _{X}}$ ${\displaystyle X\times X}$

There is a universal "realization" functor from the category of motives to the category of - graduated Abelian groups , so that for each smooth projective variety its Chow group is the realization of . ${\displaystyle \mathbb {Z} }$ ${\displaystyle X}$ ${\displaystyle CH^{*}(X)}$ ${\displaystyle {\mathcal {M}}(X)}$

The realization forms on the image of the homomorphism ${\displaystyle (X,p,m)}$

${\displaystyle CH^{*}(X)\to CH^{*}(X\times X)\to CH^{*}(X\times X)\to CH^{*}(X)}$

from, the first image from the inclusion in the first factor, the second image from the average product with , and the third figure is induced by the projection onto the second factor. ${\displaystyle p}$

System of realizations

To a motif belongs a system of realizations (some authors like Deligne also use this as a definition of a motif), that is

• ${\displaystyle \mathbb {Q} }$Modules and , ${\displaystyle M_{B}}$ ${\displaystyle M_{dR}}$
• a module and ${\displaystyle A^{f}}$ ${\displaystyle M_{A^{f}}}$
• a module for each prime number ${\displaystyle p}$ ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle M_{cris^{p}}}$
• with morphisms between the (basic change of) modules, ${\displaystyle comp_{dR,B},comp_{A^{f},B},comp_{cris^{p},dR}}$
• with filters and , ${\displaystyle W}$ ${\displaystyle H}$
• with an effect on and ${\displaystyle Gal({\overline {\mathbb {Q} }},\mathbb {Q} )}$ ${\displaystyle M_{A^{f}}}$
• with a "Frobenius" automorphism on each . ${\displaystyle M_{cris^{p}}}$

In the case of the motif associated with a scheme , is ${\displaystyle X}$ ${\displaystyle {\mathcal {M}}(X)}$

• ${\displaystyle M_{B}}$the Betti cohomology of , ${\displaystyle X(\mathbb {C} )}$
• ${\displaystyle M_{dR}}$the de Rham cohomology of , ${\displaystyle X(\mathbb {C} )}$
• ${\displaystyle M_{A^{f}}}$the L-adic cohomology over any field of the characteristic with its effect, ${\displaystyle p\not =l}$ ${\displaystyle Gal({\overline {\mathbb {Q} }},\mathbb {Q} )}$
• ${\displaystyle M_{cris^{p}}}$the crystalline cohomology of with its Frobenius homomorphism , ${\displaystyle X(\mathbb {Q} _{p})}$
• ${\displaystyle W}$the weight filtration of the cohomology of , ${\displaystyle X(\mathbb {C} )}$
• ${\displaystyle H}$the Hodge filter of the cohomology of . ${\displaystyle X(\mathbb {C} )}$

Motives as a universal cohomology theory

Motives form a universal cohomology theory if (in any cohomology theory) the cohomology class of each numerically zero-equivalent algebraic cycle vanishes. This conjecture is a weak form of the Lefschetz standard conjecture , from which, together with the Hodge standard conjecture, a proof of the Weil conjectures (proven with other methods by Deligne) would result. It has been proven in characteristic 0 for Abelian varieties and would generally follow from the Hodge conjecture .

properties

• The morphisms form finite-dimensional - vector spaces . ${\displaystyle Mor((X,p,m),(Y,q,n))}$ ${\displaystyle \mathbb {Q} }$
• The motifs form an additive category , i.e. H. one can form direct sums of motives.
• Every idempotent endomorphism of a motif breaks it down as the direct sum of its core and image .
• The category of motifs is abelian and semi-simple .
• A tensor product is defined on the category of motifs , so that the Künneth formula applies.
• Each motif has a dual motif and an evaluation image with a universal property . ${\displaystyle M}$ ${\displaystyle M^{\vee }}$ ${\displaystyle M^{\vee }\otimes M\to 1}$

L-functions of motifs

For pure motifs above , one defines their zeta function as a characteristic polynomial of the Frobenius homomorphism (if the motif is of odd weight) or its inverse (if the motif is of even weight). For direct sums of pure motifs , the zeta function is defined as the product of the zeta functions of the pure summands. ${\displaystyle M_{p}}$ ${\displaystyle F_{p}}$ ${\displaystyle Z(M_{p},t)}$ ${\displaystyle F_{p}}$

For motifs over one can reduce the motif to a motif over for almost all ("good") prime numbers and then define it ${\displaystyle \mathbb {Q} }$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle M_{p}}$ ${\displaystyle F_{p}}$

${\displaystyle \zeta (M,s):=\Pi _{p\ gut}Z(M_{p},p^{-s})}$.

This function is called the motivic L-function.

The modularity assumption from the Langlands program states that every motivic L-function is an alternating product of automorphic L-functions .

literature

• Uwe Jannsen , Steven Kleiman , Jean-Pierre Serre : Motives , Proceedings of Symposia in Pure Mathematics 55, Providence, RI: American Mathematical Society, 1994. ISBN 978-0-8218-1636-3
• Yves André : Une introduction aux motifs (motifs purs, motifs mixtes, périodes) , Panoramas et Synthèses 17, Paris: Société Mathématique de France, 2004. ISBN 978-2-85629-164-1
• Jacob Murre , Jan Nagel, Chris Peters : Lectures on the theory of pure motives , American Mathematical Society 2010
• Barry Mazur : What is a Motive? , Notices AMS, Volume 51, November 2004, pp. 1214-1216, pdf

Web links

• James Milne : Motives - Grothendieck's Dream , Milne website about it
• Milne, What is a Motive? , pdf
• Charles Weibel : Review by Vladimir Voevodsky, Andrei Suslin, Eric Friedlander: Cycles, transfers and motivic homology theories, Princeton UP 2000 , Bulletin AMS, Volume 39, 2002, pp. 127–143 (with an introduction to motifs)
• Pierre Deligne : What is a Motive? (Video of a lecture at the Institute for Advanced Study)

Individual evidence

1. ↑ Manin, Correspondences, Mofifs and monoidal transformations, Math. USSR-Sb., Volume 6, 1968, pp. 439-470, Russian version at mathnet.ru
2. ↑ Manin, Forgotten motives: the varieties of scientific experience, in: Leila Schneps, Alexandre Grothendieck, a mathematical portrait, International Press, Boston 2014, Arxiv