<p>Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(\mathcal {A'},\mathcal {A},\mathcal {A''})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="script">A</mi> <mo>′</mo> </msup> <mo>,</mo> <mi mathvariant="script">A</mi> <mo>,</mo> <msup> <mi mathvariant="script">A</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a recollement of abelian categories. We establish a bijection between certain ICE-closed subcategories of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> and those of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {A''}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="script">A</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> </math></EquationSource> </InlineEquation>. As an application, when <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Lambda ', \Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">Λ</mi> <mo>′</mo> </msup> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Lambda ''$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> are artin algebras such that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(\operatorname {mod}\Lambda ', \operatorname {mod}\Lambda , \operatorname {mod}\Lambda '')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <msup> <mi mathvariant="normal">Λ</mi> <mo>′</mo> </msup> <mo>,</mo> <mo>mod</mo> <mi mathvariant="normal">Λ</mi> <mo>,</mo> <mo>mod</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a recollement of abelian categories, we establish a bijection between certain doubly functorially finite ICE-closed subcategories of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\operatorname {mod}\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>mod</mo> <mi mathvariant="normal">Λ</mi> </mrow> </math></EquationSource> </InlineEquation> and those of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\operatorname {mod}\Lambda ''$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>mod</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we provide some constructions of wide <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-tilting modules in a recollement.</p>

ICE-Closed Subcategories, Wide $\tau $-Tilting Modules and Recollements

期刊论文
发表日期：

2026年7月1日

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Weili Gu,
Zhaoyong Huang,
Xin Ma

摘要

Let $(\mathcal {A'},\mathcal {A},\mathcal {A''})$ $(A^{'}, A, A^{''})$ be a recollement of abelian categories. We establish a bijection between certain ICE-closed subcategories of $\mathcal {A}$ $A$ and those of $\mathcal {A''}$ $A^{''}$ . As an application, when $\Lambda ', \Lambda $ $Λ^{'}, Λ$ and $\Lambda ''$ $Λ^{''}$ are artin algebras such that $(\operatorname {mod}\Lambda ', \operatorname {mod}\Lambda , \operatorname {mod}\Lambda '')$ $(mod Λ^{'}, mod Λ, mod Λ^{''})$ is a recollement of abelian categories, we establish a bijection between certain doubly functorially finite ICE-closed subcategories of $\operatorname {mod}\Lambda $ $mod Λ$ and those of $\operatorname {mod}\Lambda ''$ $mod Λ^{''}$ . Furthermore, we provide some constructions of wide $\tau $ $τ$ -tilting modules in a recollement.

ICE-Closed Subcategories, Wide \(\tau \)-Tilting Modules and Recollements

摘要