Homology with Models and Tor

We consider the extension of the notion of a projective module to that of a projective functor relative to a model set (as in Dold, MacLane, Oberst, 1967). Then taking projective resolutions of functors, we consider the usual associated homology.We show that in some cases, including the classical simplicial homology of topological spaces, the model set can be replaced by a model set having only one element. We show that when the model set consists of a single element the homology modules can be interpreted as values of the Torsion functor. In the case of simplicial homology of topological spaces these Tors will be shown to be analogous to the Tors which occur in group homology.     

Relations Between Derived Hochschild Functors via Twisting

Let be a regular ring, and let A, B be essentially finite type -algebras. For any functor F: D(ModA) × ? × D(ModA) → D(ModB) between their derived categories, we define its twist F^!: D(ModA) × ? × D(ModA) → D(ModB) with respect to dualizing complexes, generalizing Grothendieck's construction of f^!. We show that relations between functors are preserved between their twists, and deduce that various relations hold between derived Hochschild (co)-homology and the f^! functor. We also deduce that the set of isomorphism classes of dualizing complexes over a ring (or a scheme) form a group with respect to derived Hochschild cohomology, and that the twisted inverse image functor is a group homomorphism.     

Hochschild cohomology of ring objects in monoidal categories

We define the Hochschild complex and cohomology of a ring object in a monoidal category enriched over abelian groups. We interpret the cohomology groups and prove that the cohomology ring is graded-commutative.     

Non-abelian Tensor Product of Leibniz Algebras and an Exact Sequence in Leibniz Homology

Abstract

Let 𝔪 and 𝔫 be two-sided ideals of a Leibniz algebra 𝔤 such that 𝔤 = 𝔪 + 𝔫. The goal of the paper is to achieve the exact sequence Ker(𝔪  𝔫 + 𝔫  𝔪 → 𝔤) → HL ₂(𝔤) → HL ₂(𝔤/𝔪) ⊕ HL ₂(𝔤/𝔫) → 𝔪 ∩ 𝔫/ [𝔪,𝔫] → HL ₁(𝔤) → HL ₁(𝔤/𝔪) ⊕ HL ₁(𝔤/𝔫) → 0, where HL denotes the Leibniz homology with trivial coefficients of a Leibniz algebra and denotes a non-abelian tensor product of Leibniz algebras.     

Cyclic Homology of Hopf Comodule Algebras and Hopf Module Coalgebras

Abstract

In this paper we construct a cylindrical module A ? ? for an ?-comodule algebra A, where the antipode of the Hopf algebra ? is bijective. We show that the cyclic module associated to the diagonal of A ? ? is isomorphic with the cyclic module of the crossed product algebra A ? ?. This enables us to derive a spectral sequence for the cyclic homology of the crossed product algebra. We also construct a cocylindrical module for Hopf module coalgebras and establish a similar spectral sequence to compute the cyclic cohomology of crossed product coalgebras.     

Homology with Trivial Coefficients of Leibniz n-Algebras

Abstract

We introduce homology K-vector spaces with trivial coefficients for Leibniz n-algebras and we obtain exact sequences relating them. As a consequence we get a Hopf formula and several results on central extensions of Leibniz n-algebras.     

Hochschild Cohomology Ring of an Order of a Simple Component of the Rational Group Ring of the Generalized Quaternion Group

We will give an efficient bimodule projective resolution of an order Γ, where Γ is an order of a simple component of the rational group ring ? Q _{2 ^r} of the generalized quaternion 2-group Q _{2 ^r} of order 2 ^r+2. Moreover, we will determine the ring structure of the Hochschild cohomology HH*(Γ) by calculating the Yoneda products using this bimodule projective resolution.     

Hochschild Cohomology of an Algebra Associated with a Circular Quiver

In this article we show that an algebra A = K Γ/(f(X ^s )) has a periodic projective bimodule resolution of period 2, where KΓ is the path algebra of the circular quiver Γ with s vertices and s arrows over a commutative ring K, f(x) is a monic polynomial over K and X is the sum of all arrows in KΓ. Moreover, by means of this projective bimodule resolution, we compute the Hochschild cohomology group of A, and we give a presentation of the Hochschild cohomology ring HH?(A) by the generators and the relations in the case K is a field.     

Hochschild Cohomology Ring of the Mod-2 Group Ring of the Generalized Quaternion Group

We will determine the ring structure of the Hochschild cohomology HH?( ₂ Q _t ) of the mod-2 group ring  ₂ Q _t for arbitrary generalized quaternion groups Q _t of order 4t by calculating the ordinary cup product in H?(Q _t , _ψ  ₂ Q _t ).