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We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.
We develop a version of Hodge theory for a large class of smooth cohomologically proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivari
We introduce and describe the $2$-category $mathsf{Grt}_{flat}$ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories $boxtimes$ restricts nicely to $mathsf{Grt}_{f
This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analyt
The title refers to the nilcommutative or $NC$-schemes introduced by M. Kapranov in math.AG/9802041. The latter are noncommutative nilpotent thickenings of commutative schemes. We consider also the parallel theory of nil-Poisson or $NP$-schemes, whic
This is an expanded version of the two papers Interpolation of Varieties of Minimal Degree and Interpolation Problems: Del Pezzo Surfaces. It is well known that one can find a rational normal curve in $mathbb P^n$ through $n+3$ general points. More r