# Fortnightly links (64)

Theo Raedschelders and Michel Van den Bergh: The Tannaka–Krein formalism and (re)presentations of universal quantum groups is an appendix intended for the republication of Manin's book Quantum groups and noncommutative geometry, surveying some recent work by Theo and Michel. Theo has explained many of these ideas to me over the years, and it's nice to read them all in 1 place.

Amnon Neeman, Grothendieck duality made simple is a wonderful preprint to read. It gives a historical overview of what is understood by Grothendieck duality, and its different approaches. There are many interesting remarks and observations, and it ends with the following invitation:

The computations will involve Hochschild homology and cohomology—terms like $S\otimes_{S^{\mathrm{e}}}\mathbf{R}\mathrm{Hom}_R(S,S\otimes_RN)$ are bound to appear. Fortunately the world is full of experts in Hochschild homology and cohomology, and once they take an interest they will undoubtedly be able to move these computations much further than the handful of us, the few people who have been working on Grothendieck duality. Let's face it: in our tiny group none is adept at handling the Hochschild machinery. The Hochschild experts should feel invited to move right in.

As I started my PhD thinking about Grothendieck duality, the invitation is very appealing to me, but maybe it also applies to some of the readers.

Amnon Neeman, The category $[\mathcal{T}^{\mathrm{c}}]^{\mathrm{op}}$ as functors on $\mathcal{T}_{\mathrm{c}}^{\mathrm{b}}$ is another preprint by Amnon, describing locally finite cohomological functors as functors represented by perfect complexes in far more general settings than just over a field. It is amazing how this interplay between abstract triangulated category theory and algebraic geometry can be made to work.