<p>Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> be a Krull–Schmidt triangulated category with shift functor [1] and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> be a rigid subcategory of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {C}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">C</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We are concerned with the mutation of two-term weak <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {R}[1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>-cluster tilting subcategories. We show that any almost complete two-term weak <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {R}[1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>-cluster tilting subcategory has exactly two completions. Then we apply the results on relative cluster tilting subcategories to the domain of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-tilting theory in functor categories and abelian categories.</p>

Relative cluster tilting theory and $\tau $-tilting theory

期刊论文
发表日期：

2026年5月23日

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Yu Liu,
Jixing Pan,
Panyue Zhou

摘要

Let $\mathcal {C}$ $C$ be a Krull–Schmidt triangulated category with shift functor [1] and $\mathcal {R}$ $R$ be a rigid subcategory of $\mathcal {C}.$ $C .$ We are concerned with the mutation of two-term weak $\mathcal {R}[1]$ $R [1]$ -cluster tilting subcategories. We show that any almost complete two-term weak $\mathcal {R}[1]$ $R [1]$ -cluster tilting subcategory has exactly two completions. Then we apply the results on relative cluster tilting subcategories to the domain of $\tau $ $τ$ -tilting theory in functor categories and abelian categories.

Relative cluster tilting theory and \(\tau \)-tilting theory

摘要