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We provide a formula for the change of the dimension of the firstHoch\-schild cohomology vector space of bound quiver algebras when adding newarrows. For this purpose we show that there exists a short exact sequence whichrelates the first cohomology vector spaces of the algebras to the firstrelative cohomology. Moreover, we show that the first Hochschild homologies areisomorphic when adding new arrows.

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