A new instalment of fortnightly links.
Bertrand Toën, Problèmes de modules formels, which is his Bourbaki lecture from last week (available on YouTube). I haven't watched the video yet, but reading the pdf turned out to really clear up some confusions I had regarding formal moduli problems in the context of derived algebraic geometry.
The lecture by Gaitsgory from the same day also seems very interesting, I haven't found the notes for that one though. At some point you should be able to download all the texts from the Bourbaki website, or you can go to the Institut Henri Poincaré, on the first floor outside the library door you can find physical copies.
Greg Stevenson, A tour of support theory for triangulated categories through tensor triangular geometry is another very interesting introductory read. As usual, Greg's writing is very lucid.
Daniel Bergh, Valery Lunts, Olaf Schnürer, Geometricity for derived categories of algebraic stacks is another very interesting preprint. Orlov introduced the notion of geometricity for smooth and proper dg categories in a preprint from a little while back. Smooth and proper dg categories are the noncommutative analogues of smooth and proper (or projective) varieties, and he isolates a subclass of dg categories that can be embedded in the bounded derived category of an actual smooth and proper variety. He then goes on to prove that this class is closed under gluing, which tells us that finite-dimensional algebras are also geometric.
In their preprint Bergh--Lunts--Schnürer prove that smooth and proper Deligne--Mumford stacks can be suitably destackified, which allows one to apply the closure under gluing result, showing that these stacks are also geometric. The stacky techniques used in proving this are very cool! At some point I might write a blogpost about the analogue of Orlov's blow-up formula for root stacks, which is one of the main ingredients of the proof.
SGA4 has seen some more digitizing action. When Grothendieck requested that these efforts were halted (see also Yves Laszlo's page about this) the efforts for SGA4 went underground, but now they are again publicly available. The layout of the result is pretty awesome right now, and you can even take a look at the Git repository.
That's it for this fortnight.