<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation> be an abelian category with enough projective objects and enough injective objects and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {A}=\mathcal {B}\ltimes _\eta \textsf{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <mo>=</mo> <mi mathvariant="script">B</mi> <msub> <mo>⋉</mo> <mi>η</mi> </msub> <mi mathvariant="sans-serif">F</mi> </mrow> </math></EquationSource> </InlineEquation> be an <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-extension of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation>. Given a cotorsion pair <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(\mathcal {X},\;\mathcal {Y})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">X</mi> <mo>,</mo> <mspace width="0.277778em" /> <mi mathvariant="script">Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation>, we construct a cotorsion pair <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40840_2026_2057_IEq7_HTML.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="120" Type="Linedraw" Width="137" /> </InlineMediaObject> </InlineEquation> in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> and a cotorsion pair <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(\Delta (\mathcal {X}),\;\Delta (\mathcal {X})^\perp )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">X</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.277778em" /> <mi mathvariant="normal">Δ</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">X</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊥</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\textsf{F}^2=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="sans-serif">F</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In addition, the heredity and completeness of these cotorsion pairs are studied. We also state the dual versions of the main results for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\zeta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation>-coextensions of abelian categories. Finally, we give some applications and examples in comma categories, some Morita context rings and trivial extensions of rings to illustrate our main results.</p>

Cotorsion Pairs in Extensions of Abelian Categories

期刊论文
发表日期：

2026年2月10日

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Dongdong Hu

摘要

Let $\mathcal {B}$ $B$ be an abelian category with enough projective objects and enough injective objects and let $\mathcal {A}=\mathcal {B}\ltimes _\eta \textsf{F}$ $A = B ⋉_{η} F$ be an $\eta $ $η$ -extension of $\mathcal {B}$ $B$ . Given a cotorsion pair $(\mathcal {X},\;\mathcal {Y})$ $(X, Y)$ in $\mathcal {B}$ $B$ , we construct a cotorsion pair in $\mathcal {A}$ $A$ and a cotorsion pair $(\Delta (\mathcal {X}),\;\Delta (\mathcal {X})^\perp )$ $(Δ (X), Δ {(X)}^{⊥})$ in $\mathcal {A}$ $A$ for $\textsf{F}^2=0$ $F^{2} = 0$ . In addition, the heredity and completeness of these cotorsion pairs are studied. We also state the dual versions of the main results for $\zeta $ $ζ$ -coextensions of abelian categories. Finally, we give some applications and examples in comma categories, some Morita context rings and trivial extensions of rings to illustrate our main results.