arXiv subjects
Publications and source records attributed to Eduardo N. Marcos.
We introduce the $τ$-Hochschild (co)homology of a finite dimensional associative algebra $Λ$ by means of the higher Auslander-Reiten translate of O. Iyama. We show that the global dimension of $Λ$, Happel's question and Han's conjecture are related to the $τ$-Hochschild (co)homology.
In this article, we introduce the concept of partial actions of a group $G$ on quivers and demonstrate that for any given partial action of G on a quiver $Γ$, there exists another quiver, $Γ'$ with a full $G$-action. This is an enveloping action of the partial action of $G$ on $Γ$. We also introduce partial actions of groups on algebras by subalgebras instead of ideals and we define enveloping actions in this case. We show that any partial action of a group on a path algebra that is induced by a partial action on a quiver has an enveloping action.
In this paper we introduce, according to one of the main ideas of $τ$-tilting theory, the $τ$-Hochschild cohomology in degree one of a finite dimensional $k$-algebra $Λ$, where $k$ is a field. We define the excess of $Λ$ as the difference between the dimensions of the $τ$-Hochschild cohomology in degree one and the dimension of the usual Hochschild cohomology in degree one. One of the main results is that for a zero excess bound quiver algebra $Λ=kQ/I$, the Hochschild cohomology in degree two $\mathsf{HH}^2(Λ) $ is isomorphic to the space of morphisms $\mathsf{Hom}_{kQ-kQ}(I/I^2, Λ).$ This is useful to determine when $\mathsf{HH}^2(Λ)=0$ for these algebras. We compute the excess for hereditary, radical square zero and monomial triangular algebras. For a bound quiver algebra $Λ$, a formula for the excess of $Λ$ is obtained. We also give a criterion for $Λ$ to be $τ$-rigid.
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 admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for it.
Let $B\subset A$ 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.
Let $G$ be a group acting on a small category $\mathcal C$ over a field $k$, that is $\mathcal C$ is a $G$-$k$-category. We first obtain that $\mathcal C$ is resolvable by a category which is $G$-$k$-equivalent to it, on which $G$ acts freely on objects. This resolvent 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 $\mathcal C$ are isomorphic to the trivial component of the Hochschild-Mitchell (co)ho\-mo\-logy of the skew category $\mathcal C[G]$. Otherwise the corresponding spectral sequence can be settled. If the action of $G$ is free on objects, there is a canonical decomposition of the Hochschild-Mitchell (co)homology of the quotient category $\mathcal C/G$ along the conjugacy classes of $G$. This way we provide a general frame for monomorphisms which have been described previously in low degrees.
For an extension of associative algebras $B\subset A$ 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.
A main purpose of this paper is to prove that the class of finite dimensional algebras which verify Han's conjecture is closed under split bounded extensions.
We give a new and intrinsic proof of the transitivity of the braid group action on the set of full exceptional sequences of coherent sheaves on a weighted projective line. We do not use here the corresponding result of Crawley-Boevey for modules over hereditary algebras. As an application we prove that the strongest global dimension of the category of coherent sheaves on a weighted projective line $\XX$ does not depend on the parameters of $\XX$. Finally we prove that the determinant of the matrix obtained by taking the values of $n$ $\ZZ$-linear functions defined on the Grothendieck group $K_0(\XX) \simeq \ZZ^n $ of the elements of a full exceptional sequence is an invariant, up to sign.
Let $\mathcal C$ be category over a commutative ring $k$, its Hochschild-Mitchell homology and cohomology are denoted respectively $HH_*(\mathcal C)$ and $HH^*(\mathcal C).$ Let $G$ be a group acting on $\mathcal C$, and $\mathcal C[G]$ be the skew category. We provide decompositions of the (co)homology of $\mathcal C[G]$ along the conjugacy classes of $G$. For Hochschild homology of a $k$-algebra, this corresponds to the decomposition obtained by M. Lorenz. If the coinvariants and invariants functors are exact, we obtain isomorphisms $\left(HH_*(C)\right)_G\simeq HH^{\{1\}}_* (\mathcal C[G])$ and $\left(HH^*(\mathcal C)\right)^G\simeq HH^*_{\{1\}} (\mathcal C[G]), $ where $\{ {1}\}$ is the trivial conjugacy class of $G$. We first obtain these isomorphisms in case the action of $G$ is free on the objects of $\mathcal C$. Then we introduce an auxiliary category $M_G(\mathcal C)$ with an action of $G$ which is free on its objects, related to the infinite matrix algebra considered by J. Cornick. This category enables us to show that the isomorphisms hold in general, and in particular for the Hochschild (co)homology of a $k$-algebra with an action of $G$ by automorphisms. We infer that $\left(HH^*(\mathcal C)\right)^G$ is a canonical direct summand of $HH^*(\mathcal C[G])$. This provides a frame for monomorphisms obtained previously, and which have been described in low degrees.
We describe how the Hochschild (co)homology of a bound quiver algebra changes when adding or deleting arrows to the quiver. The main tools are relative Hochschild (co)homology, the Jacobi-Zariski long exact sequence obtained by A. Kaygun and a one step relative projective resolution of a tensor algebra.
We provide a formula for the change of the dimension of the first Hoch\-schild 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.
We present a study on the description of incidence algebras that are piecewise hereditary, which we denominate Phia algebras. We describe the quiver with relations of the Phia algebras of Dynkin type and introduce a new family of Phia algebras of extended Dynkin type, which we call ANS family, in reference to Assem, Nehring, and Skowroński. In this description, the important method was the one of cutting sets on trivial extensions, inspired by this we made of a computer program which shows exactly the cutting sets on the given trivial extension that result on incidence algebras.
Let $KΔ$ be the incidence algebra associated with a finite poset $(Δ,\preceq)$ over the algebraically closed field $K$. We present a study of incidence algebras $KΔ$ that are piecewise hereditary, which we denominate PHI algebras. We investigate the strong global dimension, the simply conectedeness and the one-point extension algebras over a PHI algebras. We also give a positive answer to the so-called Skowroński problem for $KΔ$ a PHI algebra which is not of wild quiver type. That is for this kind of algebra we show that $HH^1(KΔ)$ is trivial if, and only if, $KΔ$ is a simply connected algebra. We determine an upper bound for the strong global dimension of PHI algebras; furthermore, we extend this result to sincere algebras proving that the strong global dimension of a sincere piecewise hereditary algebra is less or equal than three.
Let T be a tilting object in a triangulated category equivalent to the bounded derived category of a hereditary abelian category with finite dimensional homomorphism spaces and split idempotents. This text investigates the strong global dimension, in the sense of Ringel, of the endomorphism algebra of T. This invariant is expressed using the infimum of the lengths of the sequences of tilting objects successively related by tilting mutations and where the last term is T and the endomorphism algebra of the first term is quasi-tilted. It is also expressed in terms of the hereditary abelian generating subcategories of the triangulated category.
We consider an 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.
If $f$ is an idempotent in a ring $Λ$, then we find sufficient \linebreak conditions which imply that the cohomology rings $\oplus_{n\ge 0}Ext^n_Λ(Λ/{\br},Λ/{\br})$ and \linebreak $\oplus_{n\ge 0}Ext^n_{fΛf}(fΛf/f{\br} f,fΛf/f{\br} f)$ are eventually isomorphic. This result allows us to compare finite generation and GK dimension of the cohomology rings $Λ$ and $fΛf$. We are also able to compare the global dimensions of $Λ$ and $fΛf$.
In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beattie's construction of a ring with local units in \cite{be}. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery in \cite{cm}, and we provide a unification with the results on coverings of quivers and relations by E. Green in \cite{g}. We obtain a confirmation in a quiver and relations free categorical setting that both constructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representation theories of a category and of a Galois cover of it.