# Fortnightly links (63)

Henning Krause: Completing perfect complexes (with appendices by Tobias Barthel and Bernhard Keller) explains how one can obtain $\mathbf{D}^{\mathrm{b}}(\operatorname{mod}R)$

*intrinsically*from $\operatorname{Perf}R$, using only its structure as a triangulated category, without any reference to other triangulated categories. Likewise for the bounded derived category of coherent sheaves on separated noetherian schemes. The construction uses the notion of Cauchy sequences of objects in a triangulated category, and in order to obtain the correct triangulated category one considers bounded Cauchy sequences, which in this setting means that the Hom's in the completed category vanish when computing Hom's from the Cauchy sequence to sufficiently high shifts in both directions. Bernhard explained this construction on a nice walk in the hills around Oberwolfach (kudos to him), it's a really fun idea.Shizhang Li: An example of liftings with different Hodge numbers constructs smooth projective 3-folds over the discrete valuation ring $\operatorname{Spec}\mathbb{Z}_p[\zeta_p]$ for some $p\geq 5$ such that at the special fibres (i.e. in characteristic $p$) they are isomorphic, but at the generic fibre (i.e. in characteristic 0) they have different Hodge numbers.

Alexander Polishchuk: $\mathrm{A}_\infty$-structures associated with pairs of 1-spherical objects and noncommutative orders over curves is a long preprint (I have to admit I haven't read much yet), combining different notions of noncommutative algebraic geometry. There are sheaves of orders on (stacky) curves, qgr categories in Artin–Zhang style projective geometry

*and*$\mathrm{A}_\infty$-categories. In particular, he shows that an $\mathrm{A}_\infty$-category generated by a pair of 1-spherical objects is Morita equivalent to the category of perfect complexes for some sheaf of orders on a stacky curve.