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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]

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]

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]

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.