Witt equivalence of global fields

A Simplicial derivation procedure for crossed module valued functors is established. As a consequence, a theory of higher Baer invariants and relative homology is settle on crossed modules. Then we illustrative how this method is efficient, as it extends classical formulations and results, like various known five term sequences on crossed modules by Conduché and Ellis (1989), the author (1993) and Ladra and R-Grandjeán (1994).     

Central Extensions of Precrossed Modules

We classify the precrossed module central extensions using the second cohomology group of precrossed modules. We relate these central extensions to the relative central group extensions of Loday, and to other notions of centrality defined in general contexts. Finally we establish a Universal Coefficient Theorem for the (co)homology of precrossed modules, which we use to describe the precrossed module central extensions in terms of the generalized Galois theory developed by Janelidze.     

Homological Invariants for Connected DG Algebras

In this article, we study various homological invariants of differential graded (DG for short) modules over a connected DG algebra following Frankild–Jørgensen. Two different versions of homological dimensions (resolutional and functorial) are defined. In some cases, they are proved to be simply the bound of the cohomology of the DG module. Some homological identities, such as Auslander–Buchsbaum formula and Bass formula, are proved for compact DG modules over a connected DG algebra.     

Ganea Term for the Homology of Precrossed Modules

We construct a Ganea term for the homology of precrossed modules, which generalizes the classical Ganea term for the integral homology of groups. We also introduce a central precrossed submodule which relates the Ganea term with capable, unicentral and perfect precrossed modules. Finally, we apply these constructions to the resolution of some open questions in the theory of universal central extensions of precrossed and crossed modules.     

Actor of a Precrossed Module

Applying the equivalence of the category of precrossed modules with the category of groups with two additional unary operations satisfying the corresponding conditions, the construction of an actor is given in terms of Whitehead group of generalized regular derivations, defined in the article, and the automorphism group of a precrossed module. The analogous approach to this problem in the case of crossed modules leads to the well-known construction given in the works of Lue and Norrie.     

Semi-infinite homological algebra

Summary The paper provides a homological algebraic foundation for semi-infinite cohomology. It is proved that semi-infinite cohomology of infinite dimensional Lie algebras is a two-sided derived functor of a functor that is intermediate between the functors of invariants and coinvariants. The theory of two-sided derived functors is developed. A family of modules including a module generalizing the universal enveloping algebra appropriate to the setting of two sided derived functors is introduced. A vanishing theorem for such modules is proved.Oblatum 28-IX-1992 & 11-I-1993Research supported in part by NSF grant DMS-8505550     

Actions in the Category of Precrossed Modules in Lie Algebras

For any object L in the category of precrossed modules in Lie algebras PXLie, we construct the object Act(L), which we call the actor of this object. From this construction, we derive the notions of action, center, semidirect product, derivation, commutator, and abelian precrossed module in PXLie. We show that the notion of action is equivalent to the one given in semi-abelian categories, and Act(L) is the split extension classifier for L. In the case of a crossed module in Lie algebras we show how to recover its actor in the category of crossed modules from its actor in the category of precrossed modules.     

Cohomology theories for complexes

We introduce and study a complete cohomology theory for complexes, which provides an extended version of Tate–Vogel cohomology in the setting of (arbitrary) complexes over associative rings. Moreover, for complexes of finite Gorenstein projective dimension a notion of relative Ext is introduced. On the basis of these cohomology groups, some homological invariants of modules over commutative noetherian local rings, such as Martsinkovsky’s ξ-invariants and relative and Tate versions of Betti numbers, are extended to the framework of complexes with finite homology. The relation of these invariants with their prototypes is explored.     

A non-abelian tensor product of precrossed modules in lie algebras

Abstract

In this paper, we introduce the non-abelian tensor square of precrossed modules in Lie algebras and investigate some of its properties. In particular, for an arbitrary Lie algebra L, we study the relation of the second homology of a precrossed L-module and the non-abelian exterior square. Also, we show how this non-abelian tensor product is related to the universal central extensions (with respect to the subcategory of crossed modules) of a precrossed module.     

On non-Abelian Leibniz cohomology

In this note, by using a generalized notion of the Leibniz algebra of derivations, we present the constructions of the zero, first, and second non-Abelian Leibniz cohomologies with coefficients in crossed modules, which generalize the classical zero, first, and second Leibniz cohomology. For Lie algebras we compare the non-Abelian Leibniz and Lie cohomologies. We describe the second non-Abelian Leibniz cohomology via extensions of Leibniz algebras by crossed modules.     

Directions for the Long Exact Cohomology Sequence in Moore Categories

A new method for realizing the first and second order cohomology groups of an internal abelian group in a Barr-exact category was introduced by Bourn (Cahiers Topologie Géom Différentielle Catég XL:297–316, 1999; J Pure Appl Algebra 168:133–146, 2002). The main role, in each level, is played by a direction functor. This approach can be generalized to any level n and produces a long exact cohomology sequence. By applying this method to Moore categories we show that they represent a good context for non-abelian cohomology, in particular for the Baer Extension Theory.      

On universal central extensions of precrossed and crossed modules

We study the connection between universal central extensions in the categories of precrossed and crossed modules. They are compared with several kinds of universal central extensions in the categories of groups, epimorphisms of groups, groups with operators and modules over a group. We study the relationship between the homologies defined in these categories. Applications to relative algebraic K-theory are also obtained.     

On Gorenstein injective discrete modules over profinite groups

Gorenstein homological algebra was introduced in categories of modules. But it has proved to be a fruitful way to study various other categories such as categories of complexes and of sheaves. In this paper, the research of relative homological algebra in categories of discrete modules over profinite groups is initiated. This seems appropriate since (in some sense) the subject of Gorenstein homological algebra had its beginning with Tate homology and cohomology over finite groups. We prove that if the profinite group has virtually finite cohomological dimension then every discrete module has a Gorenstein injective envelope, a Gorenstein injective cover and we study various cohomological dimensions relative to Gorenstein injective discrete modules. We also study the connection between relative and Tate cohomology theories.     

Generalized local duality,canonical modules,and prescribed bound on projective dimension

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on generalized local cohomology, and in addition we introduce and study a generalization of the notion of canonical module.     

Ideal bicombings for hyperbolic groups and applications

For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established in Monod and Shalom (Orbit equivalence rigidity and bounded cohomology, preprint, to appear) hold for all non-elementary hyperbolic groups and their non-elementary subgroups. We also derive superrigidity results for actions of general irreducible lattices on a large class of hyperbolic metric spaces.     

Hopf cyclic cohomology and Hodge theory for proper actions on complex manifolds

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.     

Prebasic submodules over valuation rings

Summary Basic concepts in the theory of modules over valuation rings are introduced. The notion of height is used to define indicators of elements, whose irregularities are investigated. The indicator leads to the new notion of smoothness, a property which does not originate from abelian groups. Invariants generalizing the finite Ulm-Kaplansky invariants of abelian p-groups, as well as the Baer invariants for completely decomposable torsion-free abelian groups, are defined, and several results relating these invariants of a module to those of submodules are proved. All these concepts lead to the notion of prebasic submodules, which seems to be the right analogue of the basic subgroups in abelian groups.The authors gratefully acknowledge the support of the National Science Foundation and the Italian C.N.R.     

Cohomological reduction by split pairs

A new method is developed to compare cohomology in module categories of different rings. This method does in general not produce isomorphisms, but surjective (or injective) maps between extension groups of modules over the two rings involved. Applications of this method are given to abstract problems—we recover and extend results on the strong no loops conjecture—and to algebras naturally coming up in invariant theory—we relate the cohomology of Brauer algebras with that of various symmetric groups.     

Link homology theories from symplectic geometry

For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conjecture them to be equal to those of Khovanov and Rozansky after a collapse of the bigrading. Our work is a generalization of that of Seidel and Smith, who treated the case n=2.