It is shown that every $\omega$-graded module over $k\left[X\right]$ is a direct sum of cyclics. The invariants for such modules are exactly the smooth invariants of valuated abelian $p$-groups.

Let A be a special biserial algebra over an algebraically closed field. We show that the first Hohchshild cohomology group of A with coefficients in the bimodule A vanishes if and only if A is representation-finite and simply connected (in the sense of Bongartz and Gabriel), if and only if the Euler characteristic of Q equals the number of indecomposable non-uniserial projective-injective A-modules (up to isomorphism). Moreover, if this is the case, then all the higher Hochschild cohomology groups...

Let $R$ be a fusion ring and $R_{ℂ}:=R{\otimes }_{\mathbb{Z}}ℂ$ be the corresponding fusion algebra. We first show that the algebra $R_{ℂ}$ has only one left (right, two-sided) cell and the corresponding left (right, two-sided) cell module. Then we prove that, up to isomorphism, $R_{ℂ}$ admits a unique special module, which is 1-dimensional and given by the Frobenius-Perron homomorphism FPdim. Moreover, as an example, we explicitly determine the special module of the interpolated fusion algebra $R\left(\mathrm{PSL}(2,q)\right):=r\left(\mathrm{PSL}(2,q)\right){\otimes }_{\mathbb{Z}}ℂ$ up to isomorphism, where $r\left(\mathrm{PSL}\right(2,q\left)\right)$ is the interpolated...

Let $$𝕂$$ be an algebraically closed field. Consider a finite dimensional monomial relations algebra $$\Lambda =𝕂\Gamma /\phantom{𝕂\Gamma I}I$$ of finite global dimension, where Γ is a quiver and I an admissible ideal generated by a set of paths from the path algebra $$𝕂\Gamma$$ . There are many modules over Λ which may be represented graphically by a tree with respect to a top element, of which the indecomposable projectives are the most natural example. These trees possess branches which correspond to right subpaths of cycles in the quiver. A pattern...

In the first part, we study algebras A such that A = R ⨿ I, where R is a subalgebra and I a two-sided nilpotent ideal. Under certain conditions on I, we show that A is standardly stratified if and only if R is standardly stratified. Next, for $A=\left[\begin{array}{cc}U& 0\\ M& V\end{array}\right]$, we show that A is standardly stratified if and only if the algebra R = U × V is standardly stratified and ${}_{V}M$ is a good V-module.

The class of n-fundamental algebras is introduced. It is a subclass of string algebras. For n-fundamental algebras we study the problem of when the Auslander-Reiten quiver contains, at the beginning or at the end, a component which is not generalized standard.

Let Λ be an artinian ring and let 𝔯 denote its Jacobson radical. We show that a simple module of finite projective dimension has no self-extensions when Λ is graded by its radical, with at most two simple modules and 𝔯⁴ = 0, in particular, when Λ is a finite-dimensional algebra over an algebraically closed field with at most two simple modules and 𝔯³ = 0.

We study the simple connectedness and strong simple connectedness of the following classes of algebras: (tame) coil enlargements of tame concealed algebras and n-iterated coil enlargement algebras.

We show that there is a one-to-one correspondence between basic cotilting complexes and certain contravariantly finite subcategories of the bounded derived category of an artin algebra. This is a triangulated version of a result by Auslander and Reiten. We use this to find an existence criterion for complements to exceptional complexes.

Si supponga che l'anello $𝐑$ ammetta una decomposizione come prodotto subdiretto $𝐑=\underset{\widetilde{\alpha \in 𝚲}}{\times }{𝐑}_{\alpha }$ di anelli ${𝐑}_{\alpha }\ne \mathrm{𝟎}$, tali che per ${𝐒}_{\alpha }=𝐑\cap {𝐑}_{\alpha }$ si abbia ${Ann}_{{𝐑}_{\alpha }}{𝐒}_{\alpha }=\mathrm{𝟎}$ ($\forall \alpha \in A$), e sia $𝐒={\oplus }_{\alpha \in 𝐀}{𝐒}_{\alpha }$. Si scelga un $𝐑$-modulo (destro) $𝐌$ che sia libero da torsione rispetto ad $𝐒$, cioè ${Ann}_{𝐌}𝐒=\mathrm{𝟎}$; allora $𝐌$ può essere rappresentato come prodotto subdiretto irridondante $𝐌\cong \underset{\widetilde{\alpha \in 𝚲}}{\times }{𝐌}_{\alpha }$ degli ${𝐑}_{\alpha }$-moduli ${𝐌}_{\alpha }$ liberi da torsione rispetto ad ${𝐒}_{\alpha }$. Si fa uno studio di un subprodotto generale di una classe $𝐂$ di $𝐑$-moduli ${𝐌}^{(𝐢)}$$...$

Let A = kQ/I be a finite dimensional basic algebra over an algebraically closed field k presented by its quiver Q with relations I. A fundamental problem in the representation theory of algebras is to decide whether or not A is of tame or wild type. In this paper we consider triangular algebras A whose quiver Q has no oriented paths. We say that A is essentially sincere if there is an indecomposable (finite dimensional) A-module whose support contains all extreme vertices of Q. We prove that if...

Soient ${𝔑}_n$ (resp. ${ℰ}_n$) l’anneau des germes de fonctions de Nash (resp. l’anneau des germes de fonctions $C^{\infty }$) à l’origine de ${\mathbf{R}}^n$: ${ℬ}_n$ (resp. ${ℬ}_n^{\text{'}}$) le module sur ${𝔑}_n$ des germes de fonctions de Bernstein $C^{\infty }$ (resp. le module sur ${𝔑}_n$ des germes de distributions de Bernstein) à l’origine de ${\mathbf{R}}^n$. Les deux résultats principaux de l’article sont les suivants : ${ℬ}_n^{\text{'}}$ est un module injectif sur ${𝔑}_n$ et ${ℰ}_n/{ℬ}_n$ est un module plat sur ${𝔑}_n$.

By an extension algebra of a finite-dimensional K-algebra A we mean a Hochschild extension algebra of A by the dual A-bimodule $Ho{m}_K(A,K)$. We study the problem of when extension algebras of a K-algebra A are symmetric. (1) For an algebra A= KQ/I with an arbitrary finite quiver Q, we show a sufficient condition in terms of a 2-cocycle for an extension algebra to be symmetric. (2) Let L be a finite extension field of K. By using a given 2-cocycle of the K-algebra L, we construct a 2-cocycle of the K-algebra...

We classify (up to Morita equivalence) all symmetric special biserial algebras of Euclidean type, by algebras arising from Brauer graphs.