Journal of the Mathematical Society of Japan

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

Ayako ITABA, Diego A. MEJÍA, and Teruyuki YORIOKA

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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. 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, $\mathbf{MA}_{\aleph_1}$ (Martin's axiom for $\aleph_1$ many dense sets, which is a combinatorial axiom in set theory) implies that ${\rm Ext}^1_{KQ}(M_{\omega_1}, KQ) = 0$.

Article information

Source
J. Math. Soc. Japan, Volume 72, Number 2 (2020), 413-433.

Dates
Received: 13 February 2018
First available in Project Euclid: 16 January 2020

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1579165217

Digital Object Identifier
doi:10.2969/jmsj/79857985

Mathematical Reviews number (MathSciNet)
MR4090342

Zentralblatt MATH identifier
07196908

Subjects
Primary: 16G20: Representations of quivers and partially ordered sets
Secondary: 16G10: Representations of Artinian rings 03E35: Consistency and independence results 03E50: Continuum hypothesis and Martin's axiom [See also 03E57]

Keywords
path algebras quiver representations non-projective modules Martin's axiom

Citation

ITABA, Ayako; MEJÍA, Diego A.; YORIOKA, Teruyuki. Some infinitely generated non-projective modules over path algebras and their extensions under Martin's axiom. J. Math. Soc. Japan 72 (2020), no. 2, 413--433. doi:10.2969/jmsj/79857985. https://projecteuclid.org/euclid.jmsj/1579165217


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