<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12188_2025_289_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {T}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">T</mi> </math></EquationSource> </InlineEquation> be an algebraic triangulated category and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12188_2025_289_Article_IEq2.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> an extension-closed subcategory with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12188_2025_289_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="135" /> </InlineMediaObject> <EquationSource Format="TEX">${{\,\textrm{Hom}\,}}({\mathcal {C}}, \Sigma ^{<0} {\mathcal {C}})=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>Hom</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo>,</mo> <msup> <mi mathvariant="normal">Σ</mi> <mrow> <mo><</mo> <mn>0</mn> </mrow> </msup> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Then <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12188_2025_289_Article_IEq2.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> has an exact structure induced from exact triangles in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12188_2025_289_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {T}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">T</mi> </math></EquationSource> </InlineEquation>. Keller and Vossieck say that there exists a triangle functor <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12188_2025_289_Article_IEq6.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$\operatorname {D}^{b}({\mathcal {C}}) \rightarrow {\mathcal {T}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mo>D</mo> <mi>b</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="script">T</mi> </mrow> </math></EquationSource> </InlineEquation> extending the inclusion <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12188_2025_289_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {C}} \subseteq {\mathcal {T}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">C</mi> <mo>⊆</mo> <mi mathvariant="script">T</mi> </mrow> </math></EquationSource> </InlineEquation>. We provide the missing details for a complete proof.</p>

Realization functors in algebraic triangulated categories

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2025年4月1日

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Janina C. Letz,
Julia Sauter

摘要

Let ${\mathcal {T}}$ $T$ be an algebraic triangulated category and ${\mathcal {C}}$ $C$ an extension-closed subcategory with ${{\,\textrm{Hom}\,}}({\mathcal {C}}, \Sigma ^{<0} {\mathcal {C}})=0$ $Hom (C, Σ^{< 0} C) = 0$ . Then ${\mathcal {C}}$ $C$ has an exact structure induced from exact triangles in ${\mathcal {T}}$ $T$ . Keller and Vossieck say that there exists a triangle functor $\operatorname {D}^{b}({\mathcal {C}}) \rightarrow {\mathcal {T}}$ $D^{b} (C) \to T$ extending the inclusion ${\mathcal {C}} \subseteq {\mathcal {T}}$ $C \subseteq T$ . We provide the missing details for a complete proof.