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

Claude Cibils (1) , Marcelo Lanzilotta (2) , Eduardo N. Marcos (3) , Sibylle Schroll (4) , Andrea Solotar

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 IMERL - Instituto de Matemática y Estadística Rafael Laguardia [Montevideo]
3 IME-USP - Institute of Mathematics and Statistics [Sao Paulo]
4 University of Leicester

We provide a formula for the change of the dimension of the first Hochschild 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.

Mots clés: Hochschild; Arrow; Quiver; Relative; Homology; Cohomology

Domaines: Physique mathématique [math-ph]

Claude Cibils, Marcelo Lanzilotta, Eduardo N. Marcos, Sibylle Schroll, Andrea Solotar. The first Hochschild (co)homology when adding arrows to a bound quiver algebra. J.Algebra, 2019, 540, pp.63-77. ⟨10.1016/j.jalgebra.2019.08.029⟩. ⟨hal-02338429⟩