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

抄録

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 Ext¹_𝐾𝑄(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 𝑀_{𝜔_1} such that, when 𝐾 is a countable field, 𝐌𝐀_{ℵ_1} (Martin's axiom for ℵ₁ many dense sets, which is a combinatorial axiom in set theory) implies that Ext¹_𝐾𝑄(𝑀_{𝜔_1}, 𝐾𝑄) = 0.