# Fortnightly links (48)

Bertrand Toën, Gabriele Vezzosi: Trace formula for dg-categories and Bloch's conductor conjecture I is the preprint proving Bloch's conductor formula using techniques from derived and noncommutative algebraic geometry. I've blogged about this result on several occasions, as I think it's one of the more exciting applications of abstract machinery to very down-to-earth questions.

Akhil Mathew, Kaledin's degeneration theorem and topological Hochschild homology is another amazing preprint in derived and noncommutative algebraic geometry. It reproves Kaledin's degeneration in a very clean way using the machinery of topological Hochschild and cyclic homology. Cool!

Benjamin Antieau, Gabriele Vezzosi: A remark on the Hochschild–Kostant–Rosenberg theorem in characteristic $p$ is a fun paper about one of my favourite results: the Hochschild–Kostant–Rosenberg decomposition of Hochschild (co)homology. In this preprint, the authors apply a cute duality result to improve $\operatorname{dim}X<p$ to $\operatorname{dim}X\leq p$. What is a funny coincidence is that this method was (first?) applied in the Deligne–Illusie paper proving degeneration of the Hodge-to-de Rham spectral sequence, and it's its noncommutative analogue (the Hochschild-to-cyclic spectral sequence) which is the subject of Kaledin's degeneration result.