Zhao Yu Ma on homological vanishing and optimal ranges

Really nice new preprint up by Zhao Yu Ma. Earlier this year Mark Shusterman and I posted a paper which I somehow haven’t gotten around to blogging about yet. But the point (it got long, there are actually a few points, I should say a point) is that if V is a braided vector space over a field k, the sequence of homology groups fit together into a ring — let’s call it R — and is an R-module, which you’re gonna hope to show is more or less finitely generated by doing some homological algebra in the category of R-modules. This really goes all the way back to my original paper with Venkatesh and Westerland, which treats a special case of this. Anyway, a key point in this paper with Mark is that life is really good if R is actually finite-dimensional over k, which doesn’t happen in the Hurwitz space cases we used to exclusively think about. In cases like this, it turns out that vanishes in some range i < cn + d. In later work with Tran and Westerland, we went past this point of view and…