# Fortnightly links (35)

Goncalo Tabuada, Finite generation of the numerical Grothendieck group is a really fun paper for me to see, because it is probably the first time that I really understood an argument using topological Hochschild homology, to prove the very down-to-earth statement that $\mathrm{K}_0^{\mathrm{num}}$ is finitely generated for a smooth and proper dg category over a finite field. I might come back to this result in a later blogpost, in a different setting.

What is interesting about this result is that it is related to the fact that the category of noncommutative numerical motives is abelian semisimple. The analogous result for numerical motives in the usual sense is a landmark result due to Jannsen, and it follows now from the noncommutative result and the comparison result between motives and numerical motives. Awesome! Another awesome piece of mathematics is a notion of Hasse–Weil zeta function for dg categories. Awesome!

Matthew Emerton, Toby Gee: Dimension theory and components of algebraic stacks are notes on (you never guess), dimension theory and the notion of irreducible components for algebraic stacks. What is interesting about these notes is that they now constitute their own sections in the Stacks project (Multiplicities of components of algebraic stacks and Dimension theory of algebraic stacks). The idea of the workshop is to have a similar outcome for the topics addressed in the workshop (I am really looking forward to it!)