arXiv 2104.086022021-04math.RTmath.CT

由Gorenstein投射模与λ-纯内射模生成的余挠对

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

Manuel Cortés-Izurdiaga, Jan Šaroch

备注：: 18 pages

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本文研究由Gorenstein投射模与λ-纯内射模生成的余挠对，证明其在特定纯内射条件下的完备性，并揭示在0^♯不存在时λ-纯内射性等同于纯内射性，从而刻画右纯半单环。

摘要：

我们证明，若GProj是环R上所有Gorenstein投射模的类，则GP=(GProj,GProj^⊥)是一个余挠对。此外，当所有投射模对某个无穷正则基数λ都是λ-纯内射时（特别地，若R是右Σ-纯内射环），GP是完备的；后一条件被证明与ZFC公理在强紧基数存在性的意义上相一致。我们还对任意无穷正则基数λ的λ-纯内射模进行了深入研究，并在此过程中证明：λ-纯内射模的内射余分解中的任意cosyzygy模都是λ-纯内射的；由一类λ-纯内射模余生成的余挠对可由一个集合余生成，并且在一个额外的技术假设下可由一个集合生成。最后，在集合论假设0^♯不存在的条件下，我们证明右R-模范畴具有足够多的λ-纯内射对象当且仅当环R是右纯半单环。这反过来源于一个相当令人惊讶的结果：在0^♯不存在时，λ-纯内射性等同于纯内射性。

英文摘要：

We prove that, if $\textrm{GProj}$ is the class of all Gorenstein projective modules over a ring $R$, then $\mathfrak{GP}=(\textrm{GProj},\textrm{GProj}^\perp)$ is a cotorsion pair. Moreover, $\mathfrak{GP}$ 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 $0^\sharp$ 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 $0^\sharp$.

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