Tuesday, June 25, 15:15, 7.527

Odysseas Giatagantzidis (Thessaloniki)
The generalized arrow removal operation and the arrow reduced version of an algebra.

Abstract:
We determine conditions for a set of arrows A of a bound quiver algebra Λ = kQ/I such that the little or big finitistic dimension of Λ is finite if and only if the respective dimension of the quotient algebra over the ideal generated by the arrows of A is finite, generalizing thus the arrow removal operation of Green-Psaroudakis-Solberg (2021). Moreover, we prove that if Λ=Λ₀, Λ₁,... , Λ_n is a sequence of algebras such that (i) each Λ_j is the quotient of Λ_j-1 over the ideal generated by a set of arrows in Λ_j as above and (ii) Λ_n is arrow reduced (i.e. no set of arrows satisfies the aforementioned conditions in Λ_n) then Λ_n is uniquely defined. We call the latter algebra the arrow reduced version of Λ. Finally, we illustrate with a concrete example that the arrow reduced version of an algebra depends on the choice of the admissible ideal I in its representation as a bound quiver algebra.