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. Â In each such complex, the interesting map, R->R, is multiplication by a (homogeneous) form.
The tensor product differential is a bit beastly. Â The modules in the complex are just new free modules (because the tensor product of free modules is another free module). Â However, the differential is a Frankensteinian mix of differentials from the two complexes you tensor together. Â 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. Â 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. Â 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. Â Then we started tensoring complexes together like mad. Â We’ll pick up next week with an equivalent characterization of the differential of the Koszul complex. Â This alternative description is much more straightforward. Â For one, it’s easier to show that if you start with a regular sequence over a polynomial ring, the Koszul complex is acyclic. Â 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.