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.

scholarly journals Nonabelian holomorphic Lie algebroid extensions

2015 ◽  
Vol 26 (05) ◽  
pp. 1550040 ◽  
Author(s):  
Ugo Bruzzo ◽  
Igor Mencattini ◽  
Vladimir N. Rubtsov ◽  
Pietro Tortella
Keyword(s):  
Line Bundle ◽  
Spectral Sequence ◽  
Lie Algebras ◽  
Hodge Structure ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Serre Spectral Sequence ◽  
Atiyah Algebroid

We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the Hochschild–Serre spectral sequence for Lie algebras to the holomorphic Lie algebroid setting. As an application, we show that the hypercohomology of the Atiyah algebroid of a line bundle has a natural Hodge structure.

scholarly journals Coniveau spectral sequences of classifying spaces for exceptional and Spin groups

2011 ◽  
Vol 152 (2) ◽  
pp. 251-278
Author(s):  
M. KAMEKO ◽  
M. TEZUKA ◽  
N. YAGITA
Keyword(s):  
Spectral Sequence ◽  
Algebraic Group ◽  
Closed Field ◽  
Spectral Sequences ◽  
Classifying Spaces ◽  
Spin Groups ◽  
Algebraically Closed Field ◽  
Cohomological Invariants ◽  
Simply Connected

AbstractLet k be an algebraically closed field of ch(k) = 0 and G be a simple simply connected algebraic group G over k. By using results about cohomological invariants, we compute the coniveau spectral sequence for classifying spaces BG.

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.

Self-adjoint operators on inner product spaces over fields of power series

2008 ◽  
Vol 58 (4) ◽  
Author(s):  
Hans Keller ◽  
Ochsenius Herminia
Keyword(s):  
Power Series ◽  
Adjoint Operator ◽  
Inner Product ◽  
Closed Field ◽  
Product Spaces ◽  
Binary Forms ◽  
Inner Product Spaces ◽  
Finite Dimensional ◽  
Orthogonal Decompositions ◽  
Adjoint Operators

AbstractTheorems on orthogonal decompositions are a cornerstone in the classical theory of real (or complex) matrices and operators on ℝn. In the paper we consider finite dimensional inner product spaces (E, ϕ) over a field K = F((χ 1, ..., x m)) of generalized power series in m variables and with coefficients in a real closed field F. It turns out that for most of these spaces (E, ϕ) every self-adjoint operator gives rise to an orthogonal decomposition of E into invariant subspaces, but there are some salient exceptions. Our main theorem states that every self-adjoint operator T: (E, ϕ) → (E, ϕ) is decomposable except when dim E is a power of 2 with exponent at most m, and ϕ is a tensor product of pairwise inequivalent binary forms. In the exceptional cases we provide an explicit description of indecomposable operators.

scholarly journals Motivic and $$\ell $$-adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

2022 ◽  
Vol 28 (2) ◽  
Author(s):  
Massimo Pippi
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Local Ring ◽  
Global Section ◽  
Regular Local Ring ◽  
Regular Scheme ◽  
Zero Locus

AbstractWe study the motivic and $$\ell $$ ℓ -adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n.

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 Spectral sequences for commutative Lie algebras

2020 ◽  
Vol 28 (2) ◽  
pp. 123-137
Author(s):  
Friedrich Wagemann
Keyword(s):  
Spectral Sequence ◽  
Lie Algebras ◽  
Spectral Sequences ◽  
Characteristic 2 ◽  
Leibniz Cohomology

AbstractWe construct some spectral sequences as tools for computing commutative cohomology of commutative Lie algebras in characteristic 2. In a first part, we focus on a Hochschild-Serre-type spectral sequence, while in a second part we obtain spectral sequences which compare Chevalley--Eilenberg-, commutative- and Leibniz cohomology. These methods are illustrated by a few computations.

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