# Fortnightly links (32)

Amnon Neeman, Strong generators in $\mathbf{D}^{\mathrm{perf}}(X)$ and $\mathbf{D}_{\mathrm{coh}}^{\mathrm{b}}(X)$ is an amazing paper. Amnon generalises Bondal--Van den Bergh's existence of a strong generator in $\mathbf{D}^{\mathrm{perf}}(X)$ all the way up to quasicompact separated schemes admitting an affine open cover where each piece is of finite global dimension, which was originally only known for smooth varieties. And then for $\mathbf{D}_{\mathrm{coh}}^{\mathrm{b}}(X)$ he uses de Jong's alterations to construct a strong generator, building on the result for $\mathbf{D}^{\mathrm{perf}}(X)$ in the case where closed subschemes of $X$ admit regular alterations, which was previously known for separated schemes of finite type over a field. The generality in which regular alterations exist includes mixed characteristic, and it is a question people are interested in for totally different reasons. It is awesome to see them being used for such foundational questions regarding triangulated categories.

Karsten Naert, Mixing and twisting is a paper Karsten lectured about in Antwerp 2 weeks ago. In various classifications of (algebraic) groups, there are certain weird classes. Depending on your situation, these are the Suzuki–Ree groups, mixed groups, and pseudo-reductive groups. These only exist in small characteristic (depending on the context either $2$, $3$ or both), where the presence of extra automorphisms of the Dynkin diagrams used in the classification wreaks havoc in the classification, making reductive groups not suffice. Karsten shows that by extending the category of schemes to either twisted or mixed schemes (where you take into account roots of the Frobenius endomorphism) you can make the classification again uniform. Jokingly he likes to say that we need a field of $\mathbb{F}_{\sqrt{p}}$ elements.

When I saw the announcement for Dennis Gaitsgory's lecture series on Hirzebruch–Riemann–Roch as a categorical trace at the Max Planck Institute I felt sad for not being able to attend. But luckily the recordings are now online.