Jan1 answered gluing of DG-categories Jan1 comment gluing of DG-categories You need to be able to identify the dg-categories with their respective opposite categories. For example it would make sense for dg-categories of sheaves on a scheme, where you have duals ($E \mapsto \mathbf{R}\mathrm{Hom}(E, O_X)$). Jan1 comment gluing of DG-categories How exactly are you defining $\varphi^\mathrm{op}$? Dec30 comment V.I. Arnold's high school problem This is a very easy olympiad problem. Take a look at "The USSR olympiad problem book", you'll be surprised at what else Russian twelve-year olds can do. Dec30 answered Maps to projective space determined by a line bundle Dec26 comment K-theory of complete intersection Perhaps the title should be changed to "Grothendieck group of..." instead of "K-theory of...". Dec23 comment A statement for a triangulated category generated by a subset As soon as the inclusion admits a right adjoint, there is a canonical equivalence of categories $\left^\perp \stackrel{\sim}{\to} D/\left$. Also $\left$ is automatically thick under this assumption. See section 1.2 of Beilinson-Vologodsky for a reference. Dec21 comment Reconstructing the Chow ring from the derived category @SébastienPalcoux, this is noncommutative geometry in the sense of Kontsevich, when one replaces a variety by the triangulated category of perfect complexes on it. Dec19 comment Is Euclid dead? @MonroeEskew, sadly, I doubt you will see a proof of these things in a high school geometry course (in the US, at least; I think the Russian curriculum is another story). Dec19 comment Reconstructing the Chow ring from the derived category @Sasha, thanks, but I consider $\mathbf{D}(X)$ without the tensor structure. Dec19 comment Reconstructing the Chow ring from the derived category @ya-tayr, good point, you only get the ring structure on $\mathrm{CH}_*(X)$ when you consider the monoidal structure on $\mathbf{D}(X)$. Dec19 revised Reconstructing the Chow ring from the derived category added 22 characters in body Dec19 revised Reconstructing the Chow ring from the derived category added 130 characters in body Dec19 asked Reconstructing the Chow ring from the derived category Dec14 awarded Necromancer Dec14 awarded Yearling Dec14 revised Stable infinity categories vs dg-categories added 786 characters in body Dec12 comment Derived Category. It may be helpful to think first about the corresponding questions in the "baby" case of homological algebra, i.e. modules over a commutative ring. Nov15 answered Stable infinity categories vs dg-categories Oct21 comment About the category of chain complexes and Grothendieck categories. See also the nlab page: ncatlab.org/nlab/show/…