# Fortnightly links (73)

Alexey Elagin, Valery Lunts, Olaf Schnürer: Smoothness for derived categories of algebras proves a really interesting result. It is known that $\mathbf{D}^{\mathrm{b}}(X)$ is smooth as a dg category, regardless of $X$ being smooth (the correct category to look at for this is $\operatorname{Perf}X$). Iyama asked whether $\mathbf{D}^{\mathrm{b}}(A)$ for a finite-dimensional algebra $A$ (not necessarily of finite global dimension) is smooth. The answer turns out to be yes!

They also discuss derived categories of sheaves of coherent algebras, and show that in the affine case with $A$ finite over its center the same smoothness holds. Fun challenge: prove it in the non-affine case!

Ryo Kanda: Non-exactness of direct products of quasi-coherent sheaves gives the details for the statement that on non-affine schemes there are not enough projective quasicoherent sheaves, or equivalently that direct products are not exact. It is remarkable that it took until now to see this, as this is a completely natural question to be asked in the development of algebraic geometry and homological algebra in the beginning of the 1960s. Thank you Ryo for finally settling this.

Lyalya Guseva: On the derived category of $\mathrm{IGr}(3,8)$ is another step in the complete (well, is it every complete?) understanding of derived categories of partial flag varieties. She constructs several full Lefschetz collections on the Grassmannian of isotropic 3-dimensional subspaces of an 8-dimensional symplectic vector space, i.e. the partial flag variety of Dynkin type $\mathrm{C}_4$ associated to the maximal parabolic $P_3$.

In case you were wondering, this variety is 12-dimensional, has Grothendieck group of rank 32, and its index is 6. And if I'm not mistaken, the likely next case to consider would have to be Dynkin type $\mathrm{D}_4$ with maximal parabolic $P_2$, where the associated partial flag variety is 9-dimensional, a Grothendieck group of rank 24 and index 5.