We define and study the $2$-category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using this in the context of two-dimensional local fields and two-dimensional adèle theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface.

We consider a finite group acting on a vector space and the corresponding skew group algebra generated by the group and the symmetric algebra of the space. This skew group algebra illuminates the resulting orbifold and serves as a replacement for the ring of invariant polynomials, especially in the eyes of cohomology. One analyzes the Hochschild cohomology of the skew group algebra using isomorphisms which convert between resolutions. We present an explicit chain map from the bar resolution to the Koszul resolution of the symmetric algebra which induces various isomorphisms on Hochschild homology and cohomology, some of which have appeared in the literature before. This approach unifies previous results on homology and cohomology of both the symmetric algebra and skew group algebra. We determine induced combinatorial cochain maps which invoke quantum differentiation (expressed by Demazure–BGG operators).

Let $G$ be a connected linear algebraic group over a field $k$. We say that $G$ is toric-friendly if for any field extension $K∕k$ and any maximal $K$-torus $T$ in $G$ the group $G\left(K\right)$ acts transitively on $\left(G∕T\right)\left(K\right)$. Our main result is a classification of semisimple (and under certain assumptions on $k$, of connected) toric-friendly groups.

We prove basic facts about reflexivity in derived categories over noetherian schemes, and about related notions such as semidualizing complexes, invertible complexes, and Gorenstein-perfect maps. Also, we study a notion of rigidity with respect to semidualizing complexes, in particular, relative dualizing complexes for Gorenstein-perfect maps. Our results include theorems of Yekutieli and Zhang concerning rigid dualizing complexes on schemes. This work is a continuation of part I (Algebra and Number Theory 4:1 (2010), 47–86), which dealt with commutative rings.