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We prove that if A is an artin algebra, G is a finite group acting on A such that |G| is invertible in A, and R = A[G]b is a basic algebra associated with the skew group algebra, then A is left supported (or right supported, or laura, or left glued, or right glued, or weakly shod, or shod) if and only if so is R. [ABSTRACT FROM AUTHOR]

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree. or if and only if its first Hochschild cohomology group H²(A) with coefficients in AAA vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H¹ (A) = 0 and it contains no full convex subcategory which is hereditary of type A. or if and only lilt is separated and contains no full convex subcategory which is hereditary of type A. [ABSTRACT FROM AUTHOR]

Let be a component of the Auslander-Reiten quiver of an Artin algebra containing projective and injective modules. Assume that the length of any path from an injective in to a projective in is bounded by some fixed number. We prove here that such component has no oriented cycles and is generalized standard, so containing only finitely many -orbits. Components of the Auslander-Reiten quiver of an Artin algebra containing paths in ind from injective to projective modules have appeared naturally in some classes of algebras such as quasitilted or shod. In both cases, these paths can be refined to paths of irreducible maps and any such path has either none hooks (in the case of a quasitilted algebra) or at most two of them (in the case of a shod algebra). See below for definitions.Here, we are interested in studying the components of such that there exists a number such that any path from an injective to a projective lying on it has at most hooks. We shall see that this is equivalent to the existence of a number such that any path in ind from an injective to a projective lying on it has length at most (Theorem 4.1). Such a component does not have oriented cycles (Corollary 3.4) and it is generalized standard (Theorem 4.3), hence containing only finitely many -orbits.In fact, when considering the existence of oriented cycles in such components, we shall prove the following more general result. If is a component of such that the number of hooks in any path of irreducible maps from an injective in to a projective in is bounded, then has no oriented cycles (Theorem 3.1). This generalizes Liu's main result in, where he shows that any component of such that any path of irreducible maps from an injective in to a projective in is sectional (that is, with no hooks) has no oriented cycles. Observe that Li has also considered such components and has shown that the above condition on the paths from injectives to projectives characterizes the existence of a section in . The... [ABSTRACT FROM AUTHOR]