Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_918_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> be a k-linear Hom-finite Krull–Schmidt triangulated category with a cluster-tilting object T. We introduce <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_918_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$T[-1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo stretchy="false">[</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>-cluster tilting objects in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_918_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation>, which generalize cluster tilting objects. Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_918_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$A=$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> </mrow> </math></EquationSource> </InlineEquation> End<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_918_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$_{\mathcal {C}}^{op}(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mrow> <mi mathvariant="script">C</mi> </mrow> <mrow> <mi mathvariant="italic">op</mi> </mrow> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the opposite algebra of the endomorphism algebra of T. We show that there is a bijection between <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_918_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$T[-1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo stretchy="false">[</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>-cluster tilting objects in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_918_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> and support <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_918_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau ^{-}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>τ</mi> <mo>-</mo> </msup> </math></EquationSource> </InlineEquation>-tilting A-modules. Subsequently, it induces a bijection between support <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_918_Article_IEq9.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-tilting A-modules and support <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_918_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau ^{-}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>τ</mi> <mo>-</mo> </msup> </math></EquationSource> </InlineEquation>-tilting A-modules, which coincides with the bijection in Adachi et al. (Compos Math 150:415–452, 2014).

T[-1]-Cluster Tilting Objects in Triangulated Categories

期刊论文
发表日期：

2025年1月3日

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Jifen Liu,
Jiaqun Wei

摘要

Let $\mathcal {C}$ $C$ be a k-linear Hom-finite Krull–Schmidt triangulated category with a cluster-tilting object T. We introduce $T[-1]$ $T [- 1]$ -cluster tilting objects in $\mathcal {C}$ $C$ , which generalize cluster tilting objects. Let $A=$ $A =$ End $_{\mathcal {C}}^{op}(T)$ $_{C}^{op} (T)$ be the opposite algebra of the endomorphism algebra of T. We show that there is a bijection between $T[-1]$ $T [- 1]$ -cluster tilting objects in $\mathcal {C}$ $C$ and support $\tau ^{-}$ $τ^{-}$ -tilting A-modules. Subsequently, it induces a bijection between support $\tau $ $τ$ -tilting A-modules and support $\tau ^{-}$ $τ^{-}$ -tilting A-modules, which coincides with the bijection in Adachi et al. (Compos Math 150:415–452, 2014).