We prove that, if GProj is the class of all Gorenstein projective modules over a ring R, then <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2826_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="170" /> </InlineMediaObject> <EquationSource Format="TEX">${\frak{G}\frak{B}}=(\text{GProj},\text{GProj}^{\bot})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="fraktur">G</mi> </mrow> <mrow> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mtext>GProj</mtext> <mo>,</mo> <msup> <mtext>GProj</mtext> <mrow> <mi mathvariant="normal">⊥</mi> </mrow> </msup> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> is a cotorsion pair. Moreover, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2826_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">${\frak{G}\frak{B}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="fraktur">G</mi> </mrow> <mrow> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> is complete when all projective modules are λ-pure-injective for some infinite regular cardinal λ (in particular, if R is right Σ-pure-injective); the latter condition is shown to be consistent with the axioms of ZFC modulo the existence of strongly compact cardinals.We also thoroughly study λ-pure-injective modules for an arbitrary infinite regular cardinal λ, proving along the way that: any cosyzygy module in an injective coresolution of a λ-pure-injective module is λ-pure-injective; the cotorsion pair cogenerated by a class of λ-pure-injective modules is cogenerated by a set and, under an additional technical assumption, generated by a set.Finally, assuming the set-theoretic hypothesis that 0♯ does not exist, we prove that the category of right R-modules has enough λ-pure-injective objects if and only if the ring R is right pure-semisimple. This, in turn, follows from a rather surprising result that λ-pure-injectivity amounts to pure-injectivity in the absence of 0♯.

The cotorsion pair generated by the Gorenstein projective modules and λ-pure-injective modules

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2025年9月29日

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Manuel Cortés-Izurdiaga,
Jan Šaroch

摘要

We prove that, if GProj is the class of all Gorenstein projective modules over a ring R, then ${\frak{G}\frak{B}}=(\text{GProj},\text{GProj}^{\bot})$ $G B = (GProj, {GProj}^{⊥})$ is a cotorsion pair. Moreover, ${\frak{G}\frak{B}}$ $G B$ is complete when all projective modules are λ-pure-injective for some infinite regular cardinal λ (in particular, if R is right Σ-pure-injective); the latter condition is shown to be consistent with the axioms of ZFC modulo the existence of strongly compact cardinals.

We also thoroughly study λ-pure-injective modules for an arbitrary infinite regular cardinal λ, proving along the way that: any cosyzygy module in an injective coresolution of a λ-pure-injective module is λ-pure-injective; the cotorsion pair cogenerated by a class of λ-pure-injective modules is cogenerated by a set and, under an additional technical assumption, generated by a set.

Finally, assuming the set-theoretic hypothesis that 0^♯ does not exist, we prove that the category of right R-modules has enough λ-pure-injective objects if and only if the ring R is right pure-semisimple. This, in turn, follows from a rather surprising result that λ-pure-injectivity amounts to pure-injectivity in the absence of 0^♯.