Michi’s blog

ComplexZeta asked me about the origins of my intuitions for homological algebra in my recent post. The answer got a bit lengthy, so I’ll put it in a post of its own.

I find Weibel very readable – once the interest is there. It’s a good reference, and not as opaque as, for instance, the MacLane + Hilton-Stammbach couplet can be at points.

The interest, however, is something I blame my alma mater for. Once upon a time, Jan-Erik Roos went to Paris and studied with Grothendieck. When he got back, he got a professorship at Stockholm University without having finished his PhD. He promptly made sure that nowadays (when he’s an Emeritus stalking the halls) there is not a single algebraist at Stockholm University without some sort of intuition for homological algebra.

So, my MSc advisor, Jörgen Backelin, gave me a subject building on from things that he touched in his PhD thesis, since I was obviously interested in combinatorics. And as such, nothing fits me better than looking at Ext and Tor over monomial rings (corresponding to coordinate hyperplane varieties…)

The other Very Interesting teacher, Jan-Erik Björk, at that university held a course in homological algebra that I attended. It was very handwavy, but with enough of deep understanding underneath that some things just clicked into place.

The story goes on. All in all, out of my 5 years at Stockholm University, at least 3 was spent doing homological algebra in addition to whatever else I was doing, and they were spent in a tight clique of undergrads and early grad students that all shared a high interest in the subject matter. In my transcript, I have an imposing distribution of courses:

where one credit corresponds to one week of fulltime study, roughly. The +20 for homological stuff is for my thesis project, which was on homological algebra, but wasn’t a lecture course. I also have some 5 or 10 points of homological stuff I never got exams done for. So all in all, I spent about a year fulltime with only homological algebra (slightly more distributed in time), and a year fulltime with only algebra of sorts that were not explicitly homological in nature.

And as they say, practice does make perfect. I have, from the various lecture courses I took, an intuition for the category of chain complexes, and for the derived category. I have an understanding for the basics of model categories. I have studied Operads and PROPs with Sergei Merkulov, and seen what happens when you take the basic stance that “We want to equate a structure with its free resolution”, and then run for the hills with it.

The other week I was discussing my graduation plans with my advisor, and he asked what my Rigorosum was going to be about. I told him I expected to do it in homological algebra, and he answered that he wasn’t certain that there was anyone available who’d be capable to accurately test my knowledge of the field. He is a group cohomologist, and he outblazes me when it comes to intuition for that – and especially when it comes to the topological notions in the field.

I, however, am comfortable thinking about differential graded modules and dg-algebras, and doing homological algebra with these. And this seems to place me, possibly, as the single person in my state with a good working knowledge of modern homological algebra.

The things I talked about in the post this follows up on are current knowledge. None of it is particularily original, but the presentation is a result of my personal history. You might very well get similar presentations if you ask my course mates from Stockholm University – but then, that school has a very special athmosphere when it comes to homological algebra.