Truncated derived functors and spectral sequences

AbstractLet $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category ${\mathcal{C}}$ as modules over the smash extension ${\mathcal{C}}\#H$. We construct Grothendieck spectral sequences for the cohomologies as well as the $H$-locally finite cohomologies of these objects. We also introduce relative $({\mathcal{D}},H)$-Hopf modules over a Hopf comodule category ${\mathcal{D}}$. These generalize relative $(A,H)$-Hopf modules over an $H$-comodule algebra $A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational $\text{Hom}$ objects and higher derived functors of coinvariants.

Download Full-text

On Some Local Cohomology Spectral Sequences

International Mathematics Research Notices ◽

10.1093/imrn/rny186 ◽

2018 ◽

Vol 2020 (19) ◽

pp. 6197-6293 ◽

Cited By ~ 2

Author(s):

Josep Àlvarez Montaner ◽

Alberto F Boix ◽

Santiago Zarzuela

Keyword(s):

Local Cohomology ◽

Sufficient Conditions ◽

Inverse System ◽

Spectral Sequences ◽

Local Cohomology Modules ◽

Decomposition Formula ◽

Derived Functors ◽

Finite Partially Ordered Set ◽

The Right ◽

Decomposition Theorems

Abstract We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The 1st type of spectral sequences involves the left derived functors of the colimit of the direct system that we obtain by applying a family of functors to a single module. For the 2nd type we follow a completely different strategy as we start with the inverse system that we obtain by applying a covariant functor to an inverse system. The spectral sequences involve the right derived functors of the corresponding limit. We also have a version for contravariant functors. In all the introduced spectral sequences we provide sufficient conditions to ensure their degeneration at their 2nd page. As a consequence we obtain some decomposition theorems that greatly generalize the well-known decomposition formula for local cohomology modules of Stanley–Reisner rings given by Hochster.

Download Full-text

Derived Categories: A Quick Tour

10.1093/acprof:oso/9780199296866.003.0002 ◽

2007 ◽

Author(s):

D. Huybrechts

Keyword(s):

Abelian Category ◽

Derived Categories ◽

Derived Category ◽

Homotopy Category ◽

Intermediate Step ◽

Spectral Sequences ◽

Separate Section ◽

Derived Functors ◽

Left And Right

This chapter briefly outlines the main steps in the construction of the derived category of an arbitrary abelian category. The homotopy category of complexes is considered as an intermediate step, which is then localized with respect to quasi-isomorphisms. Left and right derived functors are explained in general, and particular examples are studied in more detail. Spectral sequences are treated in a separate section.

Download Full-text

On the cohomology of comodules over smash coproducts

Journal of Algebra and Its Applications ◽

10.1142/s0219498819501895 ◽

2019 ◽

Vol 18 (10) ◽

pp. 1950189

Author(s):

S. Caenepeel ◽

T. Guédénon

Keyword(s):

Spectral Sequence ◽

Spectral Sequences ◽

Derived Functors

We consider the category of comodules over a smash coproduct coalgebra [Formula: see text]. We show that there is a Grothendieck spectral sequence connecting the derived functors of the Hom functors coming from [Formula: see text]-colinear, [Formula: see text]-colinear and rational [Formula: see text]-colinear morphisms. We give several applications and connect our results to existing spectral sequences in the literature.

Download Full-text

Classes of extensions and resolutions

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1961.0014 ◽

1961 ◽

Vol 254 (1039) ◽

pp. 155-222 ◽

Cited By ~ 46

Keyword(s):

Vector Bundle ◽

General Theory ◽

Abelian Category ◽

Homology Theory ◽

Spectral Sequences ◽

Adjoint Functors ◽

Dedekind Domains ◽

Derived Functors ◽

Natural Transformations ◽

Natural Classes

A class of resolutions of objects of an abelian category determines a theory of derived functors if each morphism between objects extends to a morphism, unique to within homotopies, between their resolutions. This paper is primarily concerned with resolutions canonically associated with certain natural classes of extensions (E-functors), and the known examples are constructed by using pairs of adjoint functors. An inclusion between two E-functors on the same category induces natural transformations between functors derived from their associated resolutions, and other relations exist in the form of invariant exact couples. The relations simplify for the special and frequently occurring class of ‘central’ inclusions of E-functors; in particular the operations of forming satellites of a functor on the two resolutions commute. Amongst various applications the general theory provides generalizations of: results on groups of extensions of modules over Dedekind domains; the Hochschild—Serre spectral sequences in the homology theory of groups; the spectral sequences for coherent algebraic sheaves that determine Ext by means of vector bundle resolutions and affine coverings.

Download Full-text

Counterexamples to Hochschild-Kostant-Rosenberg in characteristic p

Forum of Mathematics Sigma ◽

10.1017/fms.2021.20 ◽

2021 ◽

Vol 9 ◽

Author(s):

Benjamin Antieau ◽

Bhargav Bhatt ◽

Akhil Mathew

Keyword(s):

Spectral Sequence ◽

Spectral Sequences ◽

Characteristic P ◽

De Rham

Abstract We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP and crystalline-TP spectral sequences need not degenerate.

Download Full-text