According to the activities page, I should be in Strasbourg right now. In fact, I am in Strasbourg! Here is a typical city picture:
Now, my intention is to write a series of blog posts briefly summarising the conference. This post is just a summary of summaries, and the upcoming posts will be the actual summary of the talks. The first one should appear today or tomorrow at the latest.
Now before you think I should be a better tourist and hence I should be outside sampling the delicious noisette ice cream that costs three euros, I actually need to summarise what has happened so far so that I understand the upcoming lectures. Otherwise I won’t get much out of the lectures, some of which contain much new material, and that would be a shame indeed. However I have a few extra days here during which I promise to be the most involved tourist imaginable, perhaps even visiting every single shop in the downtown area.
Conventions and definitions: Rings are unital and not necessarily commutative. Modules over rings are left modules. A local ring is a ring in which the set of nonunits form an ideal. A module is called projective if it is a direct summand of a free module.
Today I shall share with you the wonderful result that any projective module over a local ring is free. We shall follow Kaplansky (reference given below), who first proved this result.
Now modules are in fact my favourite mathematical objects. They are like vector spaces, except that they are interesting. Of course, this “interesting” can be irksome if one has to solve a problem and these interesting properties throw a wrench in the works. However, by themselves modules are certainly curious creatures worthy of intense and gruelling analysis!
Of course, when the idea of a module was first conceived, mathematicians attempted to port all kinds of ideas from vector spaces into the world of modules. Some, like the direct sum construction, worked flawlessly. Other concepts such as rank, fortunately or unfortunately depending on your perspective, did not turn out so well (think about it: if everything worked well with modules then there’d be much less interesting math).