Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

Archive for December, 2008

Resources on motivic homotopy theory

Posted by Andreas Holmstrom on December 14, 2008

I found a really nice page today, maintained by Aravind Asok, with notes and resources on motivic homotopy theory. Worth checking out.

Posted in Random things found on the web | Tagged: , | 2 Comments »

What is cohomology?

Posted by Andreas Holmstrom on December 13, 2008

Cohomology (or homology) means different things to different people. The common theme of all notions of cohomology, is the idea of using algebraic invariants to study geometric objects. More precisely, a cohomology theory is a functor from a geometric category (for example CW complexes or schemes) to an algebraic category (for example abelian groups, vector spaces, or modules). This is an extremely powerful idea, as the algebraic objects are often easier to work with, so a problem in geometry can be solved by transferring it to algebra.

We will be a bit sloppy in that we won’t really distinguish between cohomology and homology in the discussion below. Homology usually refers to functors which are covariant, while cohomology refers to functors which are contravariant.

Most commonly, the word cohomology is used to refer to singular cohomology, one of the fundamental notions of algebraic topology. More generally, algebraic topology studies and makes use of generalized cohomology and homology theories, such as K-theory, complex cobordism, and stable homotopy groups. Good online references for these things include this book of May (pdf) and the books of Hatcher.

In mathematics as a whole, there are over 400 different notions of cohomology. The reason for this multitude of cohomologies seems to be that almost any interesting functor from geometry to algebra is referred to as a cohomology theory, regardless of its properties. One of the very few things that all cohomology theories seem to have in common, is the appearance of long exact sequences, which is one of the most important tools for doing actual computations. More generally, the power of cohomology comes from the use of homological algebra, see for example these lecture notes (pdf) of Schapira.

Most of the cohomology theories in mathematics seem to appear in algebraic and arithmetic geometry. Many of these have helped solve some of the major mathematical problems of the past century. I will come back with more posts discussing these in more detail.

Posted in Cohomology theories in algebraic geometry | Tagged: , , | 6 Comments »

A proof of Grothendieck’s standard conjectures?!?

Posted by Andreas Holmstrom on December 11, 2008

Amazing news: Seven days ago, Masana Harada of Kyoto University posted a preprint on the K-theory archive claiming to prove Grothendieck’s standard conjectures on algebraic cycles. If this is true, it will no doubt be one of the major mathematical achievements of our time.

Who is Masana Harada? Well, he is assistant professor at Kyoto University. He was a student of Robert Thomason, getting his PhD from John Hopkins University in 1987. His list of reviewed publications on MathSciNet is very short (only four items), but presumable this list does not include articles written in Japanese?

The proof presented is contained in “The Tate-Thomason conjecture”, a 21-page preprint (number 919 on the K-theory archive), using results from another recent preprint by the same author (“Higher K-theory of algebraic curves”; number 915). The list of proof ingredients includes Deligne’s result on the Riemann hypothesis over finite fields, the results of Voevodsky-Rost (-Weibel-Suslin-Friedlander???) on the Milnor-Bloch-Kato conjecture, and some work of Kahn and Levine on Azumaya algebras. The whole article is full of homotopical machinery invented by people like Bousfield, Quillen and Thomason.

Although it is of course far too early to celebrate, I find it amazing that anyone dares to approach the standard conjectures! It will be exciting to see what comes out of all this.

Posted in Harada's proof of the standard conjectures | Tagged: , , | 2 Comments »

Starting a new blog

Posted by Andreas Holmstrom on December 11, 2008

Welcome! Cohomology theories in algebraic geometry has been a rich and fascinating topic for more than 50 years: The birth of sheaf cohomology in the 50s; the Grothendieck school in the 60s developing etale cohomology, leading to Deligne’s proof of the Weil conjectures; the work of Quillen in the 70s on algebraic K-theory; the ideas of Beilinson, Deligne and others in the 80s on L-functions, mixed motives, and filtrations on Chow groups; the work of Voevodsky in the 90s on motivic cohomology and the Milnor conjecture; and much more.

However, many of the techniques underlying these ideas can seem difficult. Hopefully this blog will be a place for all of us to improve our understanding of these matters…

Hope to be back soon with more motivic stuff.

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