An Intersection-Dimension Formula for Preprojective Modules of Type $\widetilde{D}_n$

Abstract

This paper proves the existence of an intersection-dimension formula for preprojective modules over path algebras of type $\widetilde{D}_n$. Identical intersection-dimension formulas have previously been provided for modules over path algebras of type $A_n, D_n,$ and $\widetilde{A}_n$ due to Schiffler as well as He, Zhou, and Zhu. These modules can be represented geometrically by some set of curves on special surfaces. The intersection-dimension formula is an equality of the intersection number between two curves and the dimensions of the first extension spaces between the two modules they represent. This paper takes a direct approach to proving the formula utilizing the known structure of the Auslander-Reiten quiver of type $\widetilde{D}_n$. Future work will extend the formula to the entire module category (not just the preprojective modules) over path algebras of type $\widetilde{D}_n$.