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We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a cohomological interpretation of the coefficients (other than the trace) of the characteristic polynomial of the Coxeter matrix of any finite dimensional algebra with finite global dimension. [ABSTRACT FROM AUTHOR]

Abstract: We show that the main results of Happel–Rickard–Schofield (1988) and Happel–Reiten–Smalø (1996) on piecewise hereditary algebras are coherent with the notion of group action on an algebra. Then, we take advantage of this compatibility and show that if is a finite group acting on a piecewise hereditary algebra over an algebraically closed field whose characteristic does not divide the order of , then the resulting skew group algebra is also piecewise hereditary. [Copyright &y& Elsevier]

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]