The title pretty much sums it up for the day. We’re working toward a strong foundational understanding of the Koszul complex and how to build it from copies of 0 -> R -> R -> 0. In each such complex, the interesting map, R->R, is multiplication by a (homogeneous) form.
The tensor product differential is a bit beastly. The modules in the complex are just new free modules (because the tensor product of free modules is another free module). However, the differential is a Frankensteinian mix of differentials from the two complexes you tensor together. Let’s extend the gruesome metaphor for a minute: if you want to stitch together a bunch of human parts to get another human, you have to be careful about how you stitch them together. So too with the differentials; here, there are a bunch of (-1)^e, where e depends on… stuff. (I have to remind myself that I’m writing a blog, not a textbook!)
TL;DR: These signs that occur so that the resulting module homomorphisms square to zero.
The morning started with a crash course on tensor products of modules. I assigned a few standard homework exercises that focus on using the universal property to exhibit the basic isomorphisms that we need for the tensor product of complexes. Then we started tensoring complexes together like mad. We’ll pick up next week with an equivalent characterization of the differential of the Koszul complex. This alternative description is much more straightforward. For one, it’s easier to show that if you start with a regular sequence over a polynomial ring, the Koszul complex is acyclic. That is, it’s a minimal free resolution of the quotient of R by the ideal generated by the regular sequence.
We’re making some progress on the research project, too.
I’m optimistic for a fun weekend in Salem and looking forward to next week’s math.