Type Spectral Sequence Research Articles

A Hochschild-Kostant-Rosenberg theorem and residue sequences for logarithmic Hochschild homology

This paper incorporates the theory of Hochschild homology into our program on log motives. We discuss a geometric definition of logarithmic Hochschild homology of animated pre-log rings and construct an André-Quillen type spectral sequence. The latter degenerates for derived log smooth maps between discrete pre-log rings. We employ this to show a logarithmic version of the Hochschild-Kostant-Rosenberg theorem and that logarithmic Hochschild homology is representable in the category of log motives. Among the applications, we deduce a generalized residue sequence involving blow-ups of log schemes.

The standard cohomology of regular Courant algebroids

For any regular Courant algebroid E over a smooth manifold M with characteristic distribution F and ample Lie algebroid AE, we prove that there exists a canonical homological vector field on the graded manifold AE[1]⊕(TM/F)⁎[2] such that the resulting dg manifold ME, which we call the minimal model of the Courant algebroid E, encodes all cohomological information of E. Indeed, the standard cohomology of E can be identified with the cohomology of the function space on ME, which can be computed by a Hodge-to-de Rham type spectral sequence. We apply this result to generalized exact Courant algebroids and those arising from regular Lie algebroids.

On the cohomology of pro-fusion systems

We prove the Cartan–Eilenberg stable elements theorem and construct a Lyndon–Hochschild–Serre type spectral sequence for pro-fusion systems. As an application, we determine the continuous mod-[Formula: see text] cohomology ring of [Formula: see text] for any odd prime [Formula: see text].

Morse–Floer theory for superquadratic Dirac-geodesics

In this paper we present the full details of the construction of a Morse–Floer type homology related to the superquadratic perturbation of the Dirac-geodesic model. This homology is computed explicitly using a Leray–Serre type spectral sequence and this computation leads us to several existence results of Dirac-geodesics.

Hecke operators in Bredon (co)homology, K-(co)homology and Bianchi groups

In this paper, we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum–Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the [Formula: see text]-theory of the reduced [Formula: see text]-algebra of the group. We show the power of this method giving explicit computations for the group [Formula: see text]. In order to carry out these computations we use an Atiyah–Segal type spectral sequence together with the Bredon homology of the classifying space for proper actions.

Gabriel–Zisman Cohomology and Spectral Sequences

Extending constructions by Gabriel and Zisman, we develop a functorial framework for the cohomology and homology of simplicial sets with very general coefficient systems given by functors on simplex categories into abelian categories. Furthermore we construct Leray type spectral sequences for any map of simplicial sets. We also show that these constructions generalise and unify the various existing versions of cohomology and homology of small categories and as a bonus provide new insight into their functoriality.

Lie algebroid cohomology as a derived functor

We show that the hypercohomology of the Chevalley–Eilenberg–de Rham complex of a Lie algebroid L over a scheme with coefficients in an L-module can be expressed as a derived functor. We use this fact to study a Hochschild–Serre type spectral sequence attached to an extension of Lie algebroids.

Stratified fiber bundles, Quinn homology and brane stability of hyperbolic orbifolds

We revisit the problem of stability of string vacua involving hyperbolic orbifolds using methods from homotopy theory and K-homology. We propose a definition of Type II string theory on such backgrounds that further carry stratified systems of fiber bundles, which generalize the more conventional orbifold and symmetric string backgrounds, together with a classification of wrapped branes by a suitable generalized homology theory. For spaces stratified fibered over hyperbolic orbifolds we use the algebraic K-theory of their fundamental groups and Quinn homology to derive criteria for brane stability in terms of an Atiyah–Hirzebruch type spectral sequence with its lift to K-homology. Stable D-branes in this setting carry stratified charges which induce new additive structures on the corresponding K-homology groups. We extend these considerations to backgrounds which support H-flux, where we use K-groups of twisted group algebras of the fundamental groups to analyze stability of locally symmetric spaces with K-amenable isometry groups, and derive stability conditions for branes wrapping the fibers of an Eilenberg–MacLane spectrum functor.

Semitopologization in motivic homotopy theory and applications

We study the semi-topologization functor of Friedlander-Walker from the perspective of motivic homotopy theory. We construct a triangulated endo-functor on the stable motivic homotopy category $\mathcal{SH}(\C)$, which we call \emph{homotopy semi-topologization}. As applications, we discuss the representability of several semi-topological cohomology theories in $\mathcal{SH}(\C)$, a construction of a semi-topological analogue of algebraic cobordism, and a construction of Atiyah-Hirzebruch type spectral sequences for this theory.

Spectral sequences for the cohomology rings of a smash product

Stefan and Guichardet have provided Lyndon–Hochschild–Serre type spectral sequences which converge to the Hochschild cohomology and Ext groups of a smash product. We show that these spectral sequences carry natural multiplicative structures, and that these multiplicative structures can be used to calculate the cup product on Hochschild cohomology and the Yoneda product on an Ext algebra.

The module category weight of compact exceptional Lie groups

We give a lower bound for the Lusternik-Schnirelmann category of compact exceptional Lie groups by computing the module category weight through analyzing several Eilenberg-Moore type spectral sequences.

D-branes on spaces stratified fibered over hyperbolic orbifolds

We apply the methods of homology and K-theory for branes wrapping spaces stratified fibered over hyperbolic orbifolds. In addition, we discuss the algebraic K-theory of any discrete co-compact Lie group in terms of appropriate homology and Atiyah–Hirzebruch type spectral sequence with its nontrivial lift to K-homology. We emphasize the fact that the physical D-branes properties are completely transparent within the mathematical framework of K-theory. We derive criteria for D-brane stability in the case of strongly virtually negatively curved groups. We show that branes wrapping spaces stratified fibered over hyperbolic orbifolds carry charge structure and change the additive structural properties in K-homology.

Behavior of the Eilenberg–Moore spectral sequence in derived string topology

The purpose of this paper is to give applications of the Eilenberg–Moore type spectral sequence converging to the relative loop homology algebra of a Gorenstein space, which is introduced in the previous paper due to the authors. Moreover, it is proved that the spectral sequence is functorial on the category of simply-connected Poincaré duality spaces over a space.

Equivariant semi-topological invariants, Atiyah’s $KR$-theory, and real algebraic cycles

We establish an Atiyah-Hirzebruch type spectral sequence relating real morphic cohomology and real semi-topological K-theory and prove it to be compatible with the Atiyah-Hirzebruch spectral sequence relating Bredon cohomology and Atiyah's KR-theory constructed by Dugger. An equivariant and a real version of Suslin's conjecture on morphic cohomology are formulated, proved to come from the complex version of Suslin conjecture and verified for certain real varieties. In conjunction with the spectral sequences constructed here this allows the computation of the real semi-topological K-theory of some real varieties. As another application of this spectral sequence we give an alternate proof of the Lichtenbaum-Quillen conjecture over $\R$, extending an earlier proof of Karoubi and Weibel.

Graph homology and graph configuration spaces

If R is a commutative ring, M a compact R-oriented manifold and G a finite graph without loops or multiple edges, we consider the graph configuration space MG and a Bendersky–Gitler type spectral sequence converging to the homology H*(MG, R). We show that its E1 term is given by the graph cohomology complex CA(G) of the graded commutative algebra A = H*(M, R) and its higher differentials are obtained from the Massey products of A, as conjectured by Bendersky and Gitler for the case of a complete graph G. Similar results apply to the spectral sequence constructed from an arbitrary finite graph G and a graded commutative DG algebra \({\mathcal{A}}\).

Bundles of coloured posets and a Leray-Serre spectral sequence for Khovanov homology

The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a Leray-Serre type spectral sequence. We then show how this theory finds application in Khovanov homology by producing a new spectral sequence converging to the Khovanov homology of a given link.

A few splitting criteria for vector bundles

We prove a few splitting criteria for vector bundles on a quadric hypersurface and Grassmannians. We give also some cohomological splitting conditions for rank 2 bundles on multiprojective spaces. The tools are monads and a Beilinson’s type spectral sequence generalized by Costa and Miro-Roig.

M-Blocks Collections and Castelnuovo-mumford Regularity in multiprojective spaces

AbstractThe main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on n-dimensional smooth projective varieties X with an n-block collection B which generates the bounded derived category To this end, we use the theory of n-blocks and Beilinson type spectral sequence to define the notion of regularity of a coherent sheaf F on X with respect to the n-block collection B. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we compare our definition of regularity with previous ones. In particular, we show that in case of coherent sheaves on ℙn and for the n-block collection Castelnuovo-Mumford regularity and our new definition of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a multiprojective space ℙn1x…x ℙnr with respect to a suitable n1 +…+ nr-block collection and we compare it with the multigraded variant of the Castelnuovo-Mumford regularity given by Hoffman and Wang in [14].

Cohomological characterization of vector bundles on multiprojective spaces

We show that Horrocks criterion for the splitting of vector bundles on P n can be extended to vector bundles on multiprojective spaces and to smooth projective varieties with the weak CM property (see Definition 3.11). As a main tool we use the theory of n-blocks and Beilinson type spectral sequences. Cohomological characterizations of vector bundles are also showed.

Cartan–Leray spectral sequence for Galois coverings of linear categories

We provide a Cartan–Leray type spectral sequence for the Hochschild–Mitchell (co)homology of a Galois covering of linear categories. We infer results relating the Galois group and Hochschild cohomology in degree one.