We show that various derived categories of torsion modules and contramodules over the adic completion of a commutative ring by a weakly proregular ideal are full subcategories of the related derived categories of modules. By the work of Dwyer–Greenlees and Porta–Shaul–Yekutieli, this implies an equivalence between the (bounded or unbounded) conventional derived categories of the abelian categories of torsion modules and contramodules. Over the adic completion of a commutative ring by an arbitrary finitely generated ideal, we obtain an equivalence between the derived categories of complexes of modules with torsion and contramodule cohomology modules. We also define two versions of the notion of a dedualizing complex over the adic completion of a commutative ring, one for an ideal with an Artinian quotient ring and the other one for a weakly proregular ideal, and use these to construct equivalences between the conventional as well as certain exotic derived categories of the abelian categories of torsion modules and contramodules. The philosophy of derived co–contra correspondence is discussed in the introduction.
We consider actions of reductive groups on a variety with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox ring all maximal open subsets such that the quotient is quasiprojective or embeddable into a toric variety. As applications, we obtain an explicit description of the chamber structure of the linearized ample cone and several Gelfand–MacPherson type correspondences relating quotients by reductive groups to quotients by torus actions. Moreover, our approach provides a general access to the geometry of many of the resulting quotient spaces
This paper deals with the problem of the classification of the local graded Artinian quotients K[x,y]/I where K is an algebraically closed field of characteristic 0. They have a natural invariant called Hilbert–Samuel sequence. We say that a Hilbert–Samuel sequence is of homogeneous finite type, if it is the Hilbert–Samuel sequence of a finite number of isomorphism classes of graded local algebras. We give the list of all the Hilbert–Samuel sequences of homogeneous finite type in the case of algebras generated by 2 elements of degree 1.
A group G acts infinitely transitively on a set Y if for every positive integer m, its action is m-transitive on Y. Given a real affine algebraic variety Y of dimension greater than or equal to 2, we show that, under a mild restriction, if the special automorphism group of Y (the group generated by one-parameter unipotent subgroups) is infinitely transitive on each connected component of the smooth locus Yreg , then for any real affine suspension X over Y, the special automorphism group of X is infinitely transitive on each connected component of Xreg . This generalizes a recent result given by Arzhantsev, Kuyumzhiyan, and Zaidenberg over the field of real numbers.
Recently,Moosa and Scanlon introduced(iterative)Hasse–Schmidt systems D and,given such a Hasse–Schmidt system, they defined (iterative)D-rings, generalizing rings with higher derivation as introduced by Hasse and Schmidt in 1937. We show that there is a bijection between Hasse–Schmidt systems D and cocommutative coalgebras D. For a given Hasse–Schmidt system D with associated coalgebra D we show that D-ring are in bijection to algebras with a D-measuring on them.Under these correspondences iterative Hasse–Schmidt systems D correspond to cocommutative bialgebras D and iterative В-rings correspond to D-module algabras.