The first Hochschild (co)homology when adding arrows to a bound quiver algebra

Abstract: 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.