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April, 2020 Some infinitely generated non-projective modules over path algebras and their extensions under Martin's axiom
Ayako ITABA, Diego A. MEJÍA, Teruyuki YORIOKA
J. Math. Soc. Japan 72(2): 413-433 (April, 2020). DOI: 10.2969/jmsj/79857985

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 ${\rm Ext}^1_{KQ}(M_{\omega_1}, KQ) = 0$.

Citation

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Ayako ITABA. Diego A. MEJÍA. Teruyuki YORIOKA. "Some infinitely generated non-projective modules over path algebras and their extensions under Martin's axiom." J. Math. Soc. Japan 72 (2) 413 - 433, April, 2020. https://doi.org/10.2969/jmsj/79857985

Information

Received: 13 February 2018; Published: April, 2020
First available in Project Euclid: 16 January 2020

zbMATH: 07196908
MathSciNet: MR4090342
Digital Object Identifier: 10.2969/jmsj/79857985

Subjects:
Primary: 16G20
Secondary: 03E35, 03E50, 16G10

Rights: Copyright © 2020 Mathematical Society of Japan

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Vol.72 • No. 2 • April, 2020
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