The Simple Connectedness of a Tame Weakly Shod Algebra#

Abstract

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 H1(A) with coefficients in A A A 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 if it is separated and contains no full convex subcategory which is hereditary of type 𝔸˜.