• Piotr Achinger and Maciej Zdanowicz, Some elementary examples of non-liftable schemes constructs examples in characteristic $p$ of schemes that do not admit a lifting to characteristic 0 or the ring of second Witt vectors. I always find it funny how you can abuse the Frobenius morphism in ever more twisted ways, or decide to blow up all $\mathbb{F}_p$-rational points, and get something interesting.

• Jack Hall and David Rydh: The telescope conjecture for algebraic stacks is a preprint by two nice gentlemen that I met in Banff two weeks ago. The telescope conjecture is a relationship between small and big versions of triangulated categories, and should nowadays rather be called the telescope conjecture as there are counterexamples (the first one due to Keller if I'm not mistaken). The paper shows that in the cases where one expects the hypothesis to hold (one needs noetherianity) it does indeed hold.

• Jack Hall and David Rydh: Mayer–Vietoris squares in algebraic geometry is a second preprint they produced recently. It also does what the title promises: discuss Mayer–Vietoris squares. Usually such a square is given by two open subschemes and their intersection, so that you are gluing objects in a topologically intuitive sense. This paper studies many other settings, unifying some of the already existing descent situations.