Some infinitely generated non projective modules over path algebras and their extensions under Martin's Axiom

TL;DR

This paper explores the properties of modules over path algebras, showing conditions under which modules are projective and constructing specific non-projective modules whose extension groups vanish under Martin's Axiom.

Contribution

It establishes a link between module projectivity and Ext groups over path algebras, and demonstrates the influence of set-theoretic axioms on module properties.

Findings

Proves that certain finite-dimensional modules are projective if Ext^1 vanishes.

Constructs an infinitely generated non-projective module with vanishing Ext^1 under Martin's Axiom.

Shows set-theoretic axioms can affect module extension properties.

Abstract

In this paper it is proved that, when $Q$ is a quiver that admits some closure, for any algebraically closed field $K$ and any finite dimensional $K$-linear representation $\mathcal{X}$ of $Q$, if ${\rm Ext}^1_{KQ}(\mathcal{X},KQ)=0$ then $\mathcal{X}$ is projective (Theorem 1.10). In contrast, we show that if $Q$ is a specific quiver of the type above, then there is an infinitely generated non-projective $KQ$-module $M_{\omega_1}$ such that, when $K$ is a countable field, $\operatorname{\sf MA}_{\aleph_1}$ (Martin's Axiom for $\aleph_1$ many dense sets, which is a combinatorial axiom in set theory) implies that ${\rm Ext}^1_{KQ}(M_{\omega_1},KQ)=0$ (Theorem 2.11).