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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

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  • [1] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, second edition, Grad. Texts in Math., 13, Springer-Verlag, New York, 1992.
  • [2] M. Auslander, I. Reiten and S. Smalø, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., 36, Cambridge Univ. Press, Cambridge, 1995.
  • [3] I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory, London Math. Soc. Stud. Texts, 65, Cambridge Univ. Press, Cambridge, 2006.
  • [4] D. J. Benson, Representations and Cohomology. I. Basic Representation Theory of Finite Groups and Associative Algebras, Cambridge Stud. Adv. Math., 30, Cambridge Univ. Press, Cambridge, 1991.
  • [5] H. Brune, Some left pure semisimple ringoids which are not right pure semisimple, Comm. Algebra, 7 (1979), 1795–1803.
  • [6] K. Devlin and S. Shelah, A weak version of $\diamondsuit$ which follows from $2^{\aleph_0} < 2^{\aleph_1}$, Israel J. Math., 29 (1978), 239–247.
  • [7] E. E. Enochs and S. Estrada, Projective representations of quivers, Comm. Algebra, 33 (2005), 3467–3478.
  • [8] E. E. Enochs, S. Estrada, J. R. García Rozas and L. Oyonarte, Flat covers of representations of the quiver $A_\infty$, Int. J. Math. Math. Sci., 2003 (2003), 4409–4419.
  • [9] E. E. Enochs, L. Oyonarte and B. Torrecillas, Flat covers and flat representations of quivers, Comm. Algebra, 32 (2004), 1319–1338.
  • [10] P. C. Eklof, Whitehead's problem is undecidable, Amer. Math. Monthly, 83 (1976), 775–788.
  • [11] P. C. Eklof and A. H. Mekler, Almost Free Modules. Set-Theoretic Methods, revised edition, North-Holland Math. Library, 65, North-Holland Publishing Co., Amsterdam, 2002.
  • [12] D. H. Fremlin, Consequences of Martin's Axiom, Cambridge Tracts in Math., 84, Cambridge Univ. Press, Cambridge, 1984.
  • [13] P. Gabriel, Auslander–Reiten sequences and representation-finite algebras, In: Representation Theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., 831, Springer, Berlin, 1980, 1–71.
  • [14] P. Gabriel and A. V. Roiter, Representations of finite-dimensional algebras, With a chapter by B. Keller, Encyclopaedia Math. Sci., 73, Algebra, VIII, Springer, Berlin, 1992, 1–177.
  • [15] D. Herbera and J. Trlifaj, Almost free modules and Mittag-Leffler conditions, Adv. Math., 229 (2012), 3436–3467.
  • [16] H. L. Hiller and S. Shelah, Singular cohomology in $L$, Israel J. Math., 26 (1977), 313–319.
  • [17] K. Kunen, Set Theory, Stud. Log. (London), 34, College Publications, London, 2011.
  • [18] D. Martin and R. Solovay, Internal Cohen extensions, Ann. Math. Logic, 2 (1970), 143–178.
  • [19] H. Minamoto, Ampleness of two-sided tilting complexes, Int. Math. Res. Not. IMRN, 2012 (2012), 67–101.
  • [20] S. Shelah, Infinite abelian groups, Whitehead problem and some constructions, Israel J. Math., 18 (1974), 243–256.
  • [21] R. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin's problem, Ann. of Math. (2), 94 (1971), 201–245.
  • [22] J. Trlifaj, Non-perfect rings and a theorem of Eklof and Shelah, Comment. Math. Univ. Carolin., 32 (1991), 27–32.