Derived Delooping Levels of One-Point Extensions and Finitistic Dimensions

Abstract

Let $B$ be a finite-dimensional algebra, $M\in\modu B$, and $A=B[M]$ the one-point extension. We give a stable two-step extension criterion for the new simple $A$-module to have derived delooping level at most one. For a concrete algebra $B$ with a self-injective Nakayama quotient $R$, we determine the derived and classical delooping levels of $B[M]$ for every nonzero $M\in\modu R$. In particular, we obtain an explicit algebra $A$ satisfying \[ \Findim(A^{\mathrm{op}})=1<2=\ddell A=\dell A, \] which gives a negative answer to a question of Guo and Igusa.

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