Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

Posts Tagged ‘Grothendieck’

Grothendieck anagram

Posted by Andreas Holmstrom on January 6, 2013

Looking back on the mathematical year 2012, the most exciting thing happening was probably Mochizuki‘s work on the abc conjecture. Something I had hoped to see during 2012, was the writings of Grothendieck, restored to the Grothendieck circle website. Sadly, this has not happened, as Grothendieck himself apparently is opposed to it. However, it is not clear (to an outsider) if he is opposed to having a website dedicated to his memory and/or publishers making money out of his writings, or if he is actually opposed to his hardcore algebraic geometry texts being made available for free to interested mathematicians. Looking carefully at the name Alexander Grothendieck, one observes that permuting the letters yields the sentence “Hardcore EGA, extend link!”. Although not a decisive argument in the moral/legal debate over Grothendieck’s letter, perhaps it means something 😉

One a similar note, the sentence “hi’ risk abstract banana hack” is an anagram of “Banach-Tarski”, while “Plain Anarchy Got Us! Shriek! Ahhh!!” is an anagram of “Hartshorne playing shakuhachi”.

Finally, a little anagram puzzle for those of you who need a small recreational break (can also be used as homework for students you for some reason do not like). Let n be a positive integer, and let S be the set of integers between zero and n (inclusive). Let N be the number of anagrams expressing valid arithmetic equalities between elements in the set S. Example: Twelve plus one = eleven plus two. Try to compute N for small values of n. Do you see a pattern? How does N grow with n? What happens if you replace the word “integer” with the word “rational number with bounded height”?

Happy New Year 2013!

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Toen on homotopy types of algebraic varieties

Posted by Andreas Holmstrom on October 4, 2009

Two recent conversations both reminded me of a short note of Toen, with the title Homotopy types of algebraic varieties. This note explains in only eight pages several exciting ideas, which I find interesting especially because they point towards some possible future interactions between homotopy theory and arithmetic geometry.

He starts out by a conceptual discussion of classical Weil cohomology theories, which were discussed in this earlier post. The idea is that the cohomological invariants should be refined into some notion of “homotopy type”, the relation being somewhat analogous to the relation in algebraic topology, between the cohomology and the homotopy type of, say, a CW complex. He then goes on to sketch how this can be made precise, using the language of stacks and schematic homotopy types.

Towards the end of the paper, he speculates about a possible connection between the homotopy types of a variety and rational points on the variety. The study of rational points is one of the main themes of arithmetic geometry, as they correspond to integer or rational solutions of (systems of) polynomial equations. The famous section conjecture of Grothendieck, explained in these notes of Kim, is supposed to give a conceptual proof of Faltings’ theorem, aka the Mordell conjecture. Faltings’ theorem says that a curve of genus at least 2, defined over \mathbb{Q}, only has a finite number of rational points. Toen suggests a generalization of the section conjecture to higher-dimensional varieties, using his notion of homotopy types.

Another main theme of arithmetic geometry is L-functions of various kinds. To any variety over \mathbb{Q}, one can attach an L-function, which encodes lots of information about the arithmetic properties of the variety. Many outstanding conjectures in number theory are formulated in terms of these functions, for example the Riemann hypothesis and the Birch-Swinnerton-Dyer conjecture, as well as many other more accessible conjectures. The building blocks of an L-function are precisely the various Weil cohomology groups, and one could speculate about the significance of Toen’s conceptual approach to Weil cohomologies. Could it give us some new tools for approaching questions about L-functions? Or could it be that L-functions are not the right thing to consider, but that the notion of homotopy types could lead us to some better objects of study?

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Update on Harada’s proof – no error after all

Posted by Andreas Holmstrom on February 4, 2009

In a previous post I wrote that there is a mistake in Masana Harada’s proof of the standard conjectures. Now it seems that I was wrong about this. As James Milne kindly points out in a comment,  his paper is indeed misquoted, but the argument of Harada is still valid, because, and I quote, “the Tate conjecture (including num=hom) implies that the category of motives over finite fields is generated by abelian varieties, and so the standard conjectures for abelian varieties over finite fields then implies it for all varieties over finite fields”.

Also, Harada posted a revised version of his second preprint a few days ago, fixing a mistake in the proof of Theorem 6.1. 

Apparently the proof of Harada builds on an unsuccessful attempt by Thomason to prove the Tate conjecture. Is there anyone who knows where to find a copy of the original preprint of Thomason?

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Maltsinotis: Grothendieck and homotopical algebra

Posted by Andreas Holmstrom on January 26, 2009

More notes from the Grothendieck conference: Maltsiniotis spoke on Grothendieck and homotopical algebra – here is my scan. In my opinion this talk was one of the best, given that the subject is often regarded as quite inaccessible, and still he managed to make it very enjoyable. The talk covered basic notions and ideas from Grothendieck’s work in the 80s, when he was thinking about stacks, higher categories, and homotopical algebra (among many other things). The original manuscripts of Grothendieck are available online; Pursuing stacks is available on the Grothendieck circle under Unpublished mathematical texts, and here is Les Derivateurs.

There is much one could say about the homotopical ideas of Grothendieck, and I am not really competent enough to say much. There is one small thing that I would like to comment on, which I have found strange since I first saw it, and which even experts, including Grothendieck, refer to as a mystery. It is the appearance of the \Delta category in the definition of simplicial sets. A simplicial set is by definition a contravariant functor from \Delta to Sets (see nLab or Wikipedia). In general, a simplicial object in a category C is a contravariant functor from \Delta to C.

Somehow an amazing miracle happens when we pass from the category Sets (extremely boring category, no geometry at all), to the category of such functors into Sets, in which the objects model all homotopy types, so in some sense encompass all the complexity of algebraic topology. A similar thing happens when we pass from the category of abelian groups, which is reasonably boring, to the category of simplicial abelian groups, which correspond, via the Dold-Kan correspondence, to complexes of abelian groups. These form the key tool for extracting cohomological data from geometric objects, and they are the building blocks for the rich world of derived categories. In a similar way one can pass from algebras to simplicial algebras, and get something which is in some sense behaves like the category of dg-algebras – not sure about the details of this though. Yet another example is the shift from schemes to simplicial schemes, which was one of the fundamental ideas in Deligne’s groundbreaking work on mixed Hodge theory in the 70s. In each of these processes, it seems that one passes to a category that is much richer, and which is a much more natural setting for the study of many problems. Why is this?

To make a long story short,  Grothendieck asks where on earth the category \Delta came from. He commences a study of the class of categories which behave like the \Delta category, i.e. which have the amazing property that presheaves on the category model homotopy types, and he calls such a category a test category. More about this in the scanned notes above. 

After Grothendieck, and especially in recent years, quite a lot of work has been done to develop his vision on homotopical algebra and higher categories. We refer to the webpages of Ronnie Brown, Denis-Charles Cisinski, Bertrand Toen, George Maltsinotis and of course to the n-category cafe. Something to look forward to, is that Pursuing stacks is being edited and hopefully soon published, together with much of Grothendieck’s correspondence with various people at the time.

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Bloch: Motivic cohomology

Posted by Andreas Holmstrom on January 23, 2009

Spencer Bloch gave a nice overview of motivic cohomology. I could not take complete notes, as he was using slides, but fortunately he posted the slides online here.

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Toen: Nonabelian Hodge structures

Posted by Andreas Holmstrom on January 23, 2009

More notes from the Grothendieck conference: Bertrand Toen talked about how one can use derived algebraic geometry to define nonabelian Hodge structures. Here are the scanned notes.

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Ofer Gabber and Carlos Simpson

Posted by Andreas Holmstrom on January 19, 2009

Also on Monday, Carlos Simpson talked about descent, and Ofer Gabber talked about finiteness theorems in etale cohomology. I don’t have very coherent notes from these talks, so I will only point to the abstracts (Gabber and Simpson), both of which are very informative (2 pages each) and much more coherent than anything I could post myself. For Gabber’s rather technical results on finiteness theorems, see also the notes by Joel Riou from a study group in Paris.

I hope to come back to the notion of descent in some future posts, when I have understood it better.

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Nick Katz: The l-adic revolution in number theory

Posted by Andreas Holmstrom on January 19, 2009

On Monday, Nick Katz talked about l-adic representations, in particular the Sato-Tate conjecture. Here are my scanned notes. More notes to come.

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Articles on Grothendieck’s life and work

Posted by Andreas Holmstrom on January 19, 2009

Having returned from a very inspiring week at IHES, I will start going through my notes and post here whatever I think might be useful. To begin with, there are a number of articles on Grothendieck’s life and work, collected on this page. The first one is written by Winfried Scharlau, who is also writing a three-volume biography of Grothendieck (in German), see his web page. Of course, there are many other interesting things on the Grothendieck circle page.

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Error in the article of Harada

Posted by Andreas Holmstrom on January 10, 2009

I wrote earlier about the recent preprints of Harada, in which he claims to prove the standard conjectures of Grothendieck. Having asked some experts what they think about this, most seem sceptical, and my supervisor pointed out to me that there is at least one serious mistake in The Tate-Thomason conjecture. After Theorem 6.3 on page 19, he states that the Hodge conjecture for CM abelian varieties implies the standard conjecture of Hodge type for varieties of positive characteristics, and he gives a reference to Milne: Polarizations and Grothendieck’s Standard Conjectures. However, Milne only proves that the Hodge conjecture for CM abelian varieties implies the standard conjecture of Hodge type for abelian varieties, so the argument of Harada is not valid. Of course, if this is the only mistake in the article, his results would still be extremely interesting, but in any case it seems like the standard conjectures are certainly not proven in full.

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