A Homological Bridge Between Finite and Infinite Dimensional Representations of Algebras

TL;DR

This paper explores a homological connection between finite and infinite dimensional representations of algebras, providing conditions under which modules of finite projective dimension are expressible as direct limits, impacting the finitistic dimension conjecture.

Contribution

It establishes that contravariant finiteness of certain module categories implies modules of finite projective dimension are direct limits, offering a new criterion related to the finitistic dimension conjecture.

Findings

Contravariant finiteness leads to modules being direct limits of finite projective dimension modules.

Provides a sufficient condition for the finitistic dimension conjecture to hold.

Connects finite and infinite dimensional representation theories through homological conditions.

Abstract

Given a finite dimensional algebra $\Lambda$, we show that a frequently satisfied finiteness condition for the category ${\cal P}^{\infty}(\Lambda\rm{-mod})$ of all finitely generated (left) $\Lambda$-modules of finite projective dimension, namely contravariant finiteness of ${\cal P}^{\infty}(\Lambda\rm{-mod})$ in $\Lambda\rm{-mod}$, forces arbitrary modules of finite projective dimension to be direct limits of objects in ${\cal P}^{\infty}(\Lambda\rm{-mod})$. Among numerous applications, this yields an encompassing sufficient condition for the validity of the first finitistic dimension conjecture, that is, for the little finitistic dimension of $\Lambda$ to coincide with the big (this is well-known to fail over finite dimensional algebras in general).