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

## TL;DR

This paper derives a formula for how the first Hochschild cohomology dimension changes when adding arrows to a bound quiver algebra, revealing invariance in the first Hochschild homology.

## Contribution

It introduces a formula relating the first Hochschild cohomology change to relative cohomology and proves the invariance of the first Hochschild homology under arrow addition.

## Key findings

- A short exact sequence connecting first Hochschild cohomologies.
- The first Hochschild homology remains unchanged when adding arrows.
- A formula for the change in the dimension of the first Hochschild cohomology.

## 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.