Talk:Motive (algebraic geometry)

Untitled

Note about wikifying mathematicians. I firstly wikified those who I considered important according to my mathematical taste. It happened to agree with Fields medal. ilya 01:22, 15 Jan 2004 (UTC)

TeX Style

How do we want to denote categories? I would prefer $\operatorname {Corr}$ or $\operatorname {\underline {Corr}}$ to $Corr$, $SmProj$ etc. Just writing words as if they were the product of several variables seems to be bad style.

Spaetzle 16:05, 08 Apr 2011 (UTC)

References

This article needs specific references cited, not just external links. Is it original research?

Lmcelhiney 06:59, 24 December 2006 (UTC)

I've rewritten the article and added references. I did not add references for each theorem, definition etc., but the references do cover all of the material and I tried to specify which part they cover. Also, it is not original research. Jakob.scholbach 01:57, 31 March 2007 (UTC)

Algebraic cycles

I think at least some of the material on algebraic cycles should be moved to that article. The discussion of the 'standard conjectures' really belongs there (the implications for motives should stay here). Also, some of the discussion of equivalence relations for algebraic cycles should probably be moved out: its importance is not limited to the topic of motives at all. Charles Matthews 21:25, 3 April 2007 (UTC)

Yes. Let's do that. I guess the mathematical organization is: 1. algebraic cycles, 2. equivalence relations, 3. Weil cohomology theories, 4. standard conjectures (they are not only about cycles, but do involve the W.c.t.). 5. motives. Perhaps it's best to have separate articles for all of these points. I will also try to minimize-eliminate the current vague style in the article of algebraic cycles. On the other hand, at least for me personally, the motivic interpretation of the standard conjectures is the most neat one and so probably also some considerable part of the conjectures should remain in this article. Jakob.scholbach 03:46, 4 April 2007 (UTC)

The vagueness is related to my not having thought about these things since the early 1980s ... Charles Matthews 21:37, 5 April 2007 (UTC)

No problem; I didn't mean to criticize you. I provided a definition there and wrote also something about intersection products (see intersection theory). I'll also write a bit on Weil cohomology theories and take the material from the last section in algebraic cycles there. Jakob.scholbach 00:52, 8 April 2007 (UTC)

Direction of morphisms

In the definitions section, it is stated that correspondences generalize morphisms of algebraic varieties X → Y. It is also claimed a little below that there is a functor F:SmProj(k) → Corr(k) sending f:X → Y to the graph $\Gamma _{f}\subseteq X\times Y$. This is inconsistent with the definition given of Corr(X, Y) which is in terms of cycles whose codimension agrees with the dimension of X. Such cycles would be associated to morphisms f:Y → X. This is also the correct direction if morphisms on motives go in the same direction as morphisms in cohomology, which is contravariant.

I have seen similar apparent mixups in other accounts of motives. I assume authors are copying from each other and propagating a typo and/or basic confusion. The discrepancy should either be repaired or, if there is a good reason for it, explained. — Preceding unsigned comment added by Ajrmmm (talk • contribs) 07:21, 13 June 2015 (UTC) --Ajrmmm (talk) 07:24, 13 June 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Motive (algebraic geometry)/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

needs more on the relation to algebraic K-theory Jakob.scholbach 00:34, 22 April 2007 (UTC)

Last edited at 00:34, 22 April 2007 (UTC). Substituted at 02:21, 5 May 2016 (UTC)

Different Motivic Theories

There should be separate pages discussing the various (mixed) motivic theories:

• geometric mixed motives (https://pdfs.semanticscholar.org/2b04/2f81bc16df356e7efb35ac2504ef0aadd5ff.pdf)
• homotopy-invariant sheaves
• nori mixed-motives
• non-commutative mixed-motives (tabauda and Kontesvich)
• relative mixed motives (Cinski, Deglise) https://hal.inria.fr/hal-00440908v2/document — Preceding unsigned comment added by 67.165.219.170 (talk) 01:00, 13 August 2017 (UTC)

For writing articles on various motivic theory, you must have reliable secondary sources, not only primary sources. Before writing such articles, the relations and differences between these theories must be explained in this article. Wikipedia is an encyclopedia, not a repository, see WP:What Wikipedia is not. D.Lazard (talk) 22:42, 19 September 2017 (UTC)

I disagree about the usage of primary and secondary sources because typically secondary sources at this level end up being a re-working of the original content. For example, Hartshorne's algebraic geometry is a secondary source which is based upon EGA. Most of the definitions and theorems in his book are the same, except for a few simplifications. I agree that this page should discuss relations between the various categories of mixed motives, but this should not be necessary before writing the articles. This is partly because some of the relations are open problems! (see https://mathoverflow.net/questions/240474/references-voevodsky-motives-are-the-derived-category-of-nori-motives). Also, I do not see how including articles about various incarnations of mixed motives is treating wikipedia as a repository. They are not external links/internet directories, nor are they internal links. In addition, they are not media files nor photographs. What you're saying is that wikipedia should not be a repository for knowledge about a field of math. I whole heartedly disagree seeing as there is comprehensive coverage for many different fields, e.g. for functional analysis. — Preceding unsigned comment added by 75.166.193.229 (talk) 03:05, 20 September 2017 (UTC)

You do not understand my points. Secondary sources are required for justifying the importance of mathematical results. A result or a theory that is not cited nor used nor validated by others than its author(s) does not belong to Wikipedia, although it belongs to a repository. Moreover, in this specific case, the number of related but different motivic theories is a witness of a problem that is not well understood by the mathematicians community. In such a case nobody can predict which of these theories will remain to be notable, and which will be forget. As it is not the role of Wikipedia to make such a prediction, Wikipedia policy is to not emphasize on such theories that are not yet well established. This is a further reason of citing these theory in this article instead of creating an article for each of these theories. It is possible that I have not well understood the problem. If this is the case, this means that the first thing to do is to explain it clearly in the article. That would be true encyclopedic content. D.Lazard (talk) 09:09, 20 September 2017 (UTC)

Todo

Geometric Mixed Motives

• Realization of Voevodsky's motives, Annette Huber, https://pdfs.semanticscholar.org/2b04/2f81bc16df356e7efb35ac2504ef0aadd5ff.pdf — Preceding unsigned comment added by Wundzer (talk • contribs) 21:52, 15 March 2020 (UTC)
• List properties these motives have
• Give example computations with these properties with cohomological realizations
  • includes Gysin morphisms
  • includes blowups
• Discuss the relation with effective chow motives
• Embed geometric category into etale motives and nisnevich motives.

Nisnevich Sheaves with Transfer

• Discuss the construction of the mixed motives as Nisnevich sheaves with transfers
• Mention the importance of the Nisnevich topology by mentioning the application to proposition 3.1.3 of https://faculty.math.illinois.edu/K-theory/0074/tmotives.pdf and discuss some of the constructions in https://faculty.math.illinois.edu/K-theory/0074/tmotives.pdf

Discuss Motivic Homotopy Theory

A good place to start is https://faculty.math.illinois.edu/K-theory/0305/nowmovo.pdf

Artin Motives

There should be some more discussion about Artin motives, in particular their realizations in Betti and l-adic cohomology theories. For instance, check out https://mathoverflow.net/questions/282957/motives-associated-to-a-number-field?rq=1 for a discussion. — Preceding unsigned comment added by 65.153.185.86 (talk) 22:54, 13 December 2019 (UTC)