Given a module X and a regular cardinal <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2817_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>, we study various notions of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2817_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$(\kappa ,\textrm{Add}(X))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>κ</mi> <mo>,</mo> <mtext>Add</mtext> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-freeness and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2817_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$(\kappa ,\textrm{Add}(X))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>κ</mi> <mo>,</mo> <mtext>Add</mtext> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-separability. Bearing on appropriate set-theoretic assumptions, we construct a non-trivial <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2817_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\kappa ^+$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>κ</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>-generated, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2817_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">$(\kappa ^+,\textrm{Add}(X))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>κ</mi> <mo>+</mo> </msup> <mo>,</mo> <mtext>Add</mtext> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-free and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2817_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">$(\kappa ^+,\textrm{Add}(X))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>κ</mi> <mo>+</mo> </msup> <mo>,</mo> <mtext>Add</mtext> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-separable module. Our construction allows <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2817_Article_IEq7.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation> to be singular thus extending (Guil et al. in Forum Math 22(3):485–507, 2010, Theorem 4.7). Bearing on similar set-theoretic assumptions, we characterize when every module X has a perfect decomposition. As a subproduct, we show that Enochs’ conjecture for classes <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2817_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{Add}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Add</mtext> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is consistent with ZFC—a fact first proved by Šaroch (Isr J Math 255(1):401–415, 2023).

Almost Free Modules, Perfect Decomposition and Enochs’ Conjecture

期刊论文
发表日期：

2025年2月28日

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Manuel Cortés-Izurdiaga,
Alejandro Poveda

摘要

Given a module X and a regular cardinal $\kappa $ $κ$ , we study various notions of $(\kappa ,\textrm{Add}(X))$ $(κ, Add (X))$ -freeness and $(\kappa ,\textrm{Add}(X))$ $(κ, Add (X))$ -separability. Bearing on appropriate set-theoretic assumptions, we construct a non-trivial $\kappa ^+$ $κ^{+}$ -generated, $(\kappa ^+,\textrm{Add}(X))$ $(κ^{+}, Add (X))$ -free and $(\kappa ^+,\textrm{Add}(X))$ $(κ^{+}, Add (X))$ -separable module. Our construction allows $\kappa $ $κ$ to be singular thus extending (Guil et al. in Forum Math 22(3):485–507, 2010, Theorem 4.7). Bearing on similar set-theoretic assumptions, we characterize when every module X has a perfect decomposition. As a subproduct, we show that Enochs’ conjecture for classes $\textrm{Add}(X)$ $Add (X)$ is consistent with ZFC—a fact first proved by Šaroch (Isr J Math 255(1):401–415, 2023).