2020 年 72 巻 2 号 p. 413-433
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 𝒳 of 𝑄, if Ext1𝐾𝑄(𝒳, 𝐾𝑄) = 0 then 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 𝑀𝜔_1 such that, when 𝐾 is a countable field, 𝐌𝐀ℵ_1 (Martin's axiom for ℵ1 many dense sets, which is a combinatorial axiom in set theory) implies that Ext1𝐾𝑄(𝑀𝜔_1, 𝐾𝑄) = 0.
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