scholarly journals On the cohomology of comodules over smash coproducts

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.

scholarly journals Counterexamples to Hochschild-Kostant-Rosenberg in characteristic p

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.

scholarly journals Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree

2016 ◽  
Vol 9 (2) ◽  
pp. 607-686
Author(s):  
Robert Lipshitz ◽  
Peter S. Ozsváth ◽  
Dylan P. Thurston
Keyword(s):  
Spectral Sequence ◽  
Floer Homology ◽  
Double Cover ◽  
Spectral Sequences

Motivic cohomology of quadrics and the coniveau spectral sequence

2010 ◽  
Vol 6 (3) ◽  
pp. 547-589 ◽  
Author(s):  
Nobuaki Yagita
Keyword(s):  
Spectral Sequence ◽  
Spectral Sequences ◽  
Motivic Cohomology

AbstractWe study the coniveau spectral sequence for quadrics defined by Pfister forms. In particular, we explicitly compute the motivic cohomology of anisotropic quadrics over ℝ, by showing that their coniveau spectral sequences collapse from the -term

Spectral Sequences

2020 ◽  
pp. 45-56
Author(s):  
Loring W. Tu
Keyword(s):  
Projective Plane ◽  
Spectral Sequence ◽  
Fiber Bundles ◽  
Spectral Sequences ◽  
Complex Projective Plane ◽  
Computational Tool ◽  
Short Introduction ◽  
Complex Projective

This chapter focuses on spectral sequences. The spectral sequence is a powerful computational tool in the theory of fiber bundles. First introduced by Jean Leray in the 1940s, it was further refined by Jean-Louis Koszul, Henri Cartan, Jean-Pierre Serre, and many others. The chapter provides a short introduction, without proofs, to spectral sequences. As an example, it computes the cohomology of the complex projective plane. The chapter then details Leray's theorem. A spectral sequence is like a book with many pages. Each time one turns a page, one obtains a new page that is the cohomology of the previous page.

scholarly journals Cohomology of Modules Over -categories and Co--categories

2019 ◽  
Vol 72 (5) ◽  
pp. 1352-1385
Author(s):  
Mamta Balodi ◽  
Abhishek Banerjee ◽  
Samarpita Ray
Keyword(s):  
Hopf Algebra ◽  
Module Category ◽  
Spectral Sequences ◽  
Locally Finite ◽  
Hopf Module ◽  
Comodule Algebra ◽  
Hopf Modules ◽  
Derived Functors

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.

scholarly journals THE GENERALIZED HOMOLOGY OF PRODUCTS

2007 ◽  
Vol 49 (1) ◽  
pp. 1-10 ◽  
Author(s):  
MARK HOVEY
Keyword(s):  
Spectral Sequence ◽  
Derived Functors ◽  
The Right

Abstract.We construct a spectral sequence that computes the generalized homology E*(∏ Xα) of a product of spectra. The E2-term of this spectral sequence consists of the right derived functors of product in the category of E*E-comodules, and the spectral sequence always converges when E is the Johnson-Wilson theory E(n) and the Xα are Ln-local. We are able to prove some results about the E2-term of this spectral sequence; in particular, we show that the E(n)-homology of a product of E(n)-module spectra Xα is just the comodule product of the E(n)*Xα. This spectral sequence is relevant to the chromatic splitting conjecture.

scholarly journals Model category structures and spectral sequences

2019 ◽  
Vol 150 (6) ◽  
pp. 2815-2848
Author(s):  
Joana Cirici ◽  
Daniela Egas Santander ◽  
Muriel Livernet ◽  
Sarah Whitehouse
Keyword(s):  
Spectral Sequence ◽  
Commutative Ring ◽  
Model Category ◽  
Spectral Sequences ◽  
Model Structures

AbstractLet R be a commutative ring with unit. We endow the categories of filtered complexes and of bicomplexes of R-modules, with cofibrantly generated model structures, where the class of weak equivalences is given by those morphisms inducing a quasi-isomorphism at a certain fixed stage of the associated spectral sequence. For filtered complexes, we relate the different model structures obtained, when we vary the stage of the spectral sequence, using the functors shift and décalage.

scholarly journals On Some Local Cohomology Spectral Sequences

2018 ◽  
Vol 2020 (19) ◽  
pp. 6197-6293 ◽  
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.

scholarly journals Koszul complexes and spectral sequences associated with Lie algebroids

2020 ◽  
Author(s):  
Ugo Bruzzo ◽  
Vladimir N. Rubtsov
Keyword(s):  
Spectral Sequence ◽  
Complex Manifold ◽  
Global Section ◽  
Koszul Complex ◽  
Inner Product ◽  
Closed Field ◽  
Spectral Sequences ◽  
Lie Algebroid ◽  
Atiyah Algebroid ◽  
Regular Scheme

AbstractWe study some spectral sequences associated with a locally free $${{\mathscr {O}}}_X$$ O X -module $${{\mathscr {A}}}$$ A which has a Lie algebroid structure. Here X is either a complex manifold or a regular scheme over an algebraically closed field k. One spectral sequence can be associated with $${{\mathscr {A}}}$$ A by choosing a global section V of $${{\mathscr {A}}}$$ A , and considering a Koszul complex with a differential given by inner product by V. This spectral sequence is shown to degenerate at the second page by using Deligne’s degeneracy criterion. Another spectral sequence we study arises when considering the Atiyah algebroid $${{{\mathscr {D}}}_{{{\mathscr {E}}}}}$$ D E of a holomolorphic vector bundle $${{\mathscr {E}}}$$ E on a complex manifold. If V is a differential operator on $${{\mathscr {E}}}$$ E with scalar symbol, i.e, a global section of $${{{\mathscr {D}}}_{{{\mathscr {E}}}}}$$ D E , we associate with the pair $$({{\mathscr {E}}},V)$$ ( E , V ) a twisted Koszul complex. The first spectral sequence associated with this complex is known to degenerate at the first page in the untwisted ($${{\mathscr {E}}}=0$$ E = 0 ) case.

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