<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {C}=(\mathcal {C},\mathbb {E},\mathfrak {s})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">C</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo>,</mo> <mi mathvariant="double-struck">E</mi> <mo>,</mo> <mi mathvariant="fraktur">s</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be an extriangulated category with a proper class <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${{\mathbb {E}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">E</mi> </math></EquationSource> </InlineEquation>-triangles. In this paper, we introduce and study quasi-resolving subcategories in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation>. More precisely, we first introduce the notion of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">X</mi> </math></EquationSource> </InlineEquation>-resolution dimensions for a quasi-resolving subcategory <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">X</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> and then give some equivalent characterizations of objects which have finite <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">X</mi> </math></EquationSource> </InlineEquation>-resolution dimensions. As an application, we introduce Gorenstein quasi-resolving subcategories, denoted by <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {GQP_X}(\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">GQP</mi> <mi mathvariant="script">X</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, in terms of a quasi-resolving subcategory <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">X</mi> </math></EquationSource> </InlineEquation>, and prove that <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {GQP_X}(\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">GQP</mi> <mi mathvariant="script">X</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is also a quasi-resolving subcategory of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation>. Moreover, some classical known results are generalized in <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathcal {GQP_X}(\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">GQP</mi> <mi mathvariant="script">X</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Quasi-Resolving Subcategories and Dimensions in Extriangulated Categories

期刊论文
发表日期：

2026年7月23日

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Zhenggang He,
Longfu Shi,
Shuangyan Li

摘要

Let $\mathcal {C}=(\mathcal {C},\mathbb {E},\mathfrak {s})$ $C = (C, E, s)$ be an extriangulated category with a proper class $\xi $ $ξ$ of ${{\mathbb {E}}}$ $E$ -triangles. In this paper, we introduce and study quasi-resolving subcategories in $\mathcal {C}$ $C$ . More precisely, we first introduce the notion of $\mathcal {X}$ $X$ -resolution dimensions for a quasi-resolving subcategory $\mathcal {X}$ $X$ of $\mathcal {C}$ $C$ and then give some equivalent characterizations of objects which have finite $\mathcal {X}$ $X$ -resolution dimensions. As an application, we introduce Gorenstein quasi-resolving subcategories, denoted by $\mathcal {GQP_X}(\xi )$ ${GQP}_{X} (ξ)$ , in terms of a quasi-resolving subcategory $\mathcal {X}$ $X$ , and prove that $\mathcal {GQP_X}(\xi )$ ${GQP}_{X} (ξ)$ is also a quasi-resolving subcategory of $\mathcal {C}$ $C$ . Moreover, some classical known results are generalized in $\mathcal {GQP_X}(\xi )$ ${GQP}_{X} (ξ)$ .