Ascent of Finiteness of Flat Dimension

The main focus of this talk is to prove ascent of finiteness of flat dimension through local homomorphisms. Descent is known classically, e.g., it is in Cartan–Eilenberg’s book on homological algebra, but the corresponding ascent property is surprising because of the need to use derived categories. We present a short proof of ascent due to Dwyer–Greenlees–Iyengar, and discuss the main ingredient of the proof, which is the classification due to Hopkins and Neeman of thick subcategories of the category of perfect complexes over a commutative noetherian ring R. At the end, we discuss other directions these ideas of Hopkins and Neeman lead, in particular Balmer’s result that says one can reconstruct a scheme from its derived category. The main sources for this talk are [Iye06] and [DGI06]. For basic results on unbounded complexes of modules at a (hopefully) accessible level, and for more thorough proofs for many of the preliminary results we do not prove here, we point the reader to [BN93], and a previous note by the author [Mur15].