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We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context - or equivalently for a strongly stratifying ideal - we show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. One of the main consequences is that Han's conjecture holds for an algebra admitting a strongly (co-)stratifying chain whose steps verify Han's conjecture. If Han's conjecture is true for local algebras and an algebra Λ admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for Λ. [ABSTRACT FROM AUTHOR]

Let B ⊂ A be a left or right bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long nearly exact sequence to show that B satisfies Han's conjecture if and only if A does, regardless if the extension splits or not. We provide conditions ensuring that an extension by arrows and relations is left or right bounded. Finally we give a structure result for extensions of an algebra given by a quiver and admissible relations, and examples of non split left or right bounded extensions. [ABSTRACT FROM AUTHOR]

Let G be a group acting on a small category C over a field k , that is C is a G - k -category. We first obtain an unexpected result: C is resolvable by a category which is G - k -equivalent to it, on which G acts freely on objects. This resolving category enables to show that if the coinvariants and the invariants functors are exact, then the coinvariants and invariants of the Hochschild-Mitchell (co)homology of C are isomorphic to the trivial component of the Hochschild-Mitchell (co)homology of the skew category C [ G ]. If the action of G is free on objects, then there is a canonical decomposition of the Hochschild-Mitchell (co)homology of the quotient category C / G along the conjugacy classes of G. This way we provide a general frame for monomorphisms which have been described previously in low degrees. [ABSTRACT FROM AUTHOR]

For an extension of associative algebras B⊂A over a field and an A‐bimodule X, we obtain a Jacobi–Zariski long nearly exact sequence relating the Hochschild homologies of A and B, and the relative Hochschild homology, all of them with coefficients in X. This long sequence is exact twice in three. There is a spectral sequence which converges to the gap of exactness. [ABSTRACT FROM AUTHOR]

We describe how the Hochschild (co)homology of a bound quiver algebra changes when deleting or adding arrows to the quiver. The main tools are relative Hochschild (co)homology, the Jacobi-Zariski long exact sequence obtained by A. Kaygun, and a length one relative projective resolution of tensor algebras. [ABSTRACT FROM AUTHOR]

We provide a formula for the change of the dimension of the first Hochschild cohomology vector space of bound quiver algebras when adding new arrows. For this purpose we show that there exists a short exact sequence which relates the first cohomology vector spaces of the algebras to the first relative cohomology. Moreover, we show that the first Hochschild homologies are isomorphic when adding new arrows. [ABSTRACT FROM AUTHOR]

We define the fundamental group of a Hopf algebra over a field. For this purpose we first consider gradings of Hopf algebras and Galois coverings. The latter are given by linear categories with new additional structure which we call Hopf linear categories over a finite group. We compare this invariant to the fundamental group of the underlying linear category, and we compute those groups for families of examples. [ABSTRACT FROM AUTHOR]

We consider a homogeneous action of a finite group on a free linear category over a field in order to prove that the subcategory of invariants is still free. Moreover we show that the representation type is preserved when considering invariants. [ABSTRACT FROM AUTHOR]

Assume that HT ht is a bounded HT ht -bimodule, with index of nilpotency HT ht and projective dimension HT ht . The Theorem and its proof read as follows: 6.5 Theorem Let HT ht be an extension of HT ht -algebras and let HT ht be an HT ht -bimodule. [Extracted from the article]

To a regular projection of a knot we associate a finite dimensional non-commutative associative algebra which is self-injective and special biserial. [ABSTRACT FROM AUTHOR]

Consider the intrinsic fundamental group à la Grothendieck of a linear category, introduced in our earlier papers using connected gradings. In this article we prove that any full convex subcategory is incompressible, in the sense that the group map between the corresponding fundamental groups is injective. We start by proving the functoriality of the intrinsic fundamental group with respect to full subcategories, based on the study of the restriction of connected gradings. [ABSTRACT FROM AUTHOR]

Let k be a commutative ring. We study the behavior of coverings of k-categories through fiber products and find a criterion for a covering to be Galois or universal. [ABSTRACT FROM AUTHOR]

We consider $${\mathsf{N}}$$ -complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology (called generalized cohomology by M. Dubois-Violette) only vanishes on injective functors providing a well defined functor on the stable category. For left truncated $${\mathsf{N}}$$ -complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive $${\mathsf{N}}$$ -complexes is proved to be isomorphic to an Ext functor of an indecomposable $${\mathsf{N}}$$ -complex inside the abelian functor category. Finally we show that for the monoidal structure of $${\mathsf{N}}$$ -complexes a Clebsch-Gordan formula holds, in other words the fusion rules for $${\mathsf{N}}$$ -complexes can be determined. [ABSTRACT FROM AUTHOR]

We classify the twisted tensor products of a finite set algebra with a two elements set algebra using colored quivers obtained through considerations analogous to Ore extensions. This provides also a classification of entwining structures between a finite set algebra and the grouplike coalgebra on two elements. The resulting 2-nilpotent algebras have particular features with respect to Hochschild (co)homology and cyclic homology. [ABSTRACT FROM AUTHOR]

Given a finite-dimensional monomial algebra A, we consider the trivial extension TA and provide formulae, depending on the characteristic of the field, for the dimensions of the summands HH1(A) and Alt(DA) of the first Hochschild cohomology group HH1(TA). From these a formula for the dimension of HH1(TA) can be derived. [ABSTRACT FROM AUTHOR]

We classify graded Hopf algebras structures over path coalgebras, that is over free pointed coalgebras, using Hopf quivers which are analogous to Cayley graphs. [Copyright &y& Elsevier]

The article examines finite dimensional Hopf algebras that are quotients of the upper-triangular subalgebra of the quantum groups that were introduced by V. G. Drinfel'd and M. Jimbo. It offers information on finite dimensional algebras which are either tame, finite, or wild representation types. It adds that the algebraic structures give rise to solutions of the Yang-Baxter equation by the universal R-matrix that is acting on it and the modular category of representations.