On the tensor product of Grothendieck categories (original) (raw)
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Sheaves on Grothendieck constructions
Cornell University - arXiv, 2022
In this paper we introduce a generalisation of a covariant Grothendieck construction to the setting of sites. We study the basic properties of defined site structures on Grothendieck constructions as well as we treat the cohomological aspects of corresponding toposes of sheaves. Despite the fact that the toposes of G-equivariant sheaves Sh G (X) have been introduced in literature, their cohomological aspects have not been treated properly in a desired fashion. So in the end of the paper we study some of the acyclic families, introduce new type of acyclic resolutions which we call the G-equivariant Godement resolutions, the degree of actions, and some other basic cohomological concepts arising in Sh G (X).
Local and stable homological algebra in Grothendieck abelian categories
Homology Homotopy and Applications, 2009
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Categories parametrized by schemes and representation theory in complex rank
Journal of Algebra, 2013
Many key invariants in the representation theory of classical groups (symmetric groups Sn, matrix groups GLn, On, Sp 2n ) are polynomials in n (e.g., dimensions of irreducible representations). This allowed Deligne to extend the representation theory of these groups to complex values of the rank n. Namely, Deligne defined generically semisimple families of tensor categories parametrized by n ∈ C, which at positive integer n specialize to the classical representation categories. Using Deligne's work, Etingof proposed a similar extrapolation for many non-semisimple representation categories built on representation categories of classical groups, e.g., degenerate affine Hecke algebras (dAHA). It is expected that for generic n ∈ C such extrapolations behave as they do for large integer n ("stabilization").
A functorial formalism for quasi-coherent sheaves on a geometric stack
Expositiones Mathematicae, 2015
A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to a presentation of a geometric stack are equivalent categories. As a consequence, we show that the category of quasi-coherent sheaves on a geometric stack is a Grothendieck category. We also associate, in a 2-functorial way, to a 1-morphism of geometric stacks f : X → Y, an adjunction f * ⊣ f * for the corresponding categories of quasi-coherent sheaves that agrees with the classical one defined for schemes. This construction is described both geometrically in terms of the small flat site and algebraically in terms of comodules over the Hopf algebroid. CONTENTS 25 6. Properties and functoriality of quasi-coherent sheaves 33 7. Describing functoriality via comodules 40 8. Deligne-Mumford stacks and functoriality for the étale topology 45 References 49
Locally finitely presented categories with no flat objects
Forum Mathematicum, 2000
If X is a quasi-compact and quasi-separated scheme, the category Qcoh(X) of quasi-coherent sheaves on X is locally finitely presented. Therefore categorical flat quasi-coherent sheaves in the sense of [30] naturally arise. But there is also the standard definition of flatness in Qcoh(X) from the stalks. So it makes sense to wonder the relationship (if any) between these two notions. In this paper we show that there are plenty of locally finitely presented categories having no other categorical flats than the zero object. As particular instance, we show that Qcoh(P n (R))) has no other categorical flat objects than zero, where R is any commutative ring.