Comments for Motivic stuff

Comment on Intersection theory at Rigorous Trivialities by Charles Siegel

Charles Siegel — Tue, 03 Nov 2009 02:53:00 +0000

Well, just pointing out that there’s nothing new coming out of this, and it’s really more a set of notes than a book, being as I’m (mostly) following Fulton’s book, though I do have a couple of detours into other things planned (if I can understand it well enough, I might even try to talk about the Goresky-MacPherson Intersection Homology theory, to get a handle on singular varieties properly, though I’m not sure that that’s going to happen.)

Comment on Events by More conferences « Motivic stuff

More conferences « Motivic stuff — Sun, 18 Oct 2009 22:55:25 +0000

[...] Events [...]

Comment on Notes on p-adic Hodge theory by David Brown

David Brown — Fri, 16 Oct 2009 01:56:08 +0000

One more piece of advice from someone who recently learned p-adic hodge theory: from my experience I think a good first task is to understand how p-divisible groups fit into the picture — I explained a little of this at mathoverflow, but B&C’s notes have a lot of very clear information about this and citations to the literature.

Comment on What is cohomology? by Andreas Holmstrom

Andreas Holmstrom — Fri, 16 Oct 2009 01:42:59 +0000

I come from algebraic geometry, so when talking about sheaf cohomology I think of the scheme (or the manifold) as the first variable and the sheaf as the second variable. But any Hom or Ext bifunctor (when taking two sheaves or other objects in an abelian category as input) would be contravariant in the first variable and covariant in the second.

Comment on What is cohomology? by Akhil Mathew

Akhil Mathew — Fri, 16 Oct 2009 01:37:59 +0000

Ok, so it’s basically restriction in the first variable (or corestriction, I forget which). Thanks!

By the way, for sheaf cohomology are you referring to the functors of two sheaf variables, or is the underlying topological space one of the variables?

Comment on What is cohomology? by Andreas Holmstrom

Andreas Holmstrom — Fri, 16 Oct 2009 01:12:00 +0000

Aha, I now saw that you just wrote a nice post on group cohomology. And I forgot to say that for sheaf cohomology it’s similar, it behaves contravariantly in the first variable and covariantly in the second.

Comment on What is cohomology? by Andreas Holmstrom

Andreas Holmstrom — Fri, 16 Oct 2009 01:05:39 +0000

I think the behaviour of group cohomology is contravariant in the first variable and covariant in the second. You have to be a bit careful though in order to state precisely on which category you have your functor, see for example the brief and clear discussion in Brown: Cohomology of groups, starting at the bottom of p 78. Here is a direct link to the page at Google Books, hope it works. A different answer is that the group cohomology of a group is related to the singular cohomology of the corresponding Eilenberg-MacLane space, and similarly for group homology and singular homology. This is probably the historical reason for the terminology. You might enjoy Mac Lane's survey on the origin of group cohomology, which should be freely available here.

Comment on What is cohomology? by Akhil Mathew

Akhil Mathew — Fri, 16 Oct 2009 00:28:23 +0000

Just curious- why is the word “group cohomology” applied to a covariant functor? Is it just because group homology already exists? What about sheaf cohomology then?

Comment on Places by Mathematical mailing lists « Motivic stuff

Mathematical mailing lists « Motivic stuff — Fri, 09 Oct 2009 14:00:30 +0000

[...] their own mailing lists, for example the Fields Institute, MSRI, and the Newton Institute. See this list of research institutes for [...]

Comment on What is geometry? by Novice

Novice — Thu, 08 Oct 2009 10:36:29 +0000

This was useful. Thanks for recommending it.