Global obstruction property and its applications

Abstract

We introduce the global obstruction property (GOP) for Artin algebras: for every finitely generated module of infinite projective dimension, two distinct syzygies have a common nonprojective indecomposable direct summand. We prove that GOP implies the generalized Auslander--Reiten condition, and hence the Auslander--Reiten conjecture. We give criteria for GOP to hold for pullbacks over semisimple algebras and construct an example which is not syzygy-finite. For each Artin algebra $A$, we construct an algebra $\mathcal{G}(A)$ satisfying GOP by using triangular matrix extensions. We show that $\text{fin.dim}\mathcal{G}(A)=\text{fin.dim} A+2$ and that $A$ and $\mathcal{G}(A)$ are singularly equivalent. Finally, we give a pair of derived equivalent finite-dimensional algebras for which GOP holds on only one side.