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<title>Invertierbare Garbe</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Invertierbare Garbe</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Eine <b>invertierbare Garbe</b> ist im <a href="Mathematik" title="Mathematik">mathematischen</a> Teilgebiet der Garbentheorie eine <a href="Modulgarbe" title="Modulgarbe">Modulgarbe</a> über einem <a href="Geringter_Raum" title="Geringter Raum">geringten Raum</a>, welche bezüglich des <a href="Tensorprodukt" title="Tensorprodukt">Tensorproduktes</a> von Modulgarben eine inverse Modulgarbe besitzt. Unter der Verallgemeinerung von <a href="Vektorb%C3%BCndel" title="Vektorbündel">Vektorbündeln</a> durch <a href="Koh%C3%A4rente_Garbe" title="Kohärente Garbe">kohärente Garben</a> entsprechen die invertierbaren Garben genau den Geradenbündeln, für welche komplett analog bezüglich des Tensorproduktes von Vektorbündeln ein inverses Geradenbündel existiert.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {O}}_{X})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46716c1fec068ffe9981107a5a215fa1fc9e7d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.306ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {O}}_{X})}" loading="lazy"></span> ein <a href="Geringter_Raum" title="Geringter Raum">geringter Raum</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> eine <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy"></span>-<a href="Modulgarbe" title="Modulgarbe">Modulgarbe</a>. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> wird <i>invertierbar</i> genannt, wenn die folgenden äquivalenten Bedingungen erfüllt sind:

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<ul><li>Es gibt eine kohärente <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy"></span>-Modulgarbe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a980c59d42c003fd07fdf3646e1fb95ff82f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {G}}\cong {\mathcal {O}}_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {G}}\cong {\mathcal {O}}_{X}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21184dc5462df2d12f473edcf3a5aeb40a63e216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.581ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {G}}\cong {\mathcal {O}}_{X}}" loading="lazy"></span></li>

<li>Die kanonische Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {F}}^{\vee }\rightarrow {\mathcal {O}}_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {F}}^{\vee }\rightarrow {\mathcal {O}}_{X}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77e2dc6081170b3208491161ad9278ab25ad00a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.036ex; height:3.176ex;" alt="{\displaystyle {\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {F}}^{\vee }\rightarrow {\mathcal {O}}_{X}}" loading="lazy"></span> mit der dualen Garbe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}^{\vee }={\underline {\operatorname {Hom} }}({\mathcal {F}},{\mathcal {O}}_{X})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}^{\vee }={\underline {\operatorname {Hom} }}({\mathcal {F}},{\mathcal {O}}_{X})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad5a3d0cc6e6ddd1097f460f32c0ec66070d421f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:19.526ex; height:3.509ex;" alt="{\displaystyle {\mathcal {F}}^{\vee }={\underline {\operatorname {Hom} }}({\mathcal {F}},{\mathcal {O}}_{X})}" loading="lazy"></span> ist ein Isomorphismus.</li>

<li>Der <a href="Endofunktor" class="mw-redirect" title="Endofunktor">Endofunktor</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}-\colon \mathbf {Sh} (X,{\mathcal {O}}_{X})\rightarrow \mathbf {Sh} (X,{\mathcal {O}}_{X})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}-\colon \mathbf {Sh} (X,{\mathcal {O}}_{X})\rightarrow \mathbf {Sh} (X,{\mathcal {O}}_{X})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a113f5f7d9d4ffa79c1b61f0671ad2837929eb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.617ex; height:3.009ex;" alt="{\displaystyle {\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}-\colon \mathbf {Sh} (X,{\mathcal {O}}_{X})\rightarrow \mathbf {Sh} (X,{\mathcal {O}}_{X})}" loading="lazy"></span> mit der Kategorie <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Sh} (X,{\mathcal {O}}_{X})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {Sh} (X,{\mathcal {O}}_{X})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ad434f1e82205fb0b2695f0379e6c24497f895a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.276ex; height:2.843ex;" alt="{\displaystyle \mathbf {Sh} (X,{\mathcal {O}}_{X})}" loading="lazy"></span> der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy"></span>-Modulgarben ist eine <a href="Kategorien%C3%A4quivalenz" class="mw-redirect" title="Kategorienäquivalenz">Kategorienäquivalenz</a>.</li></ul>

<p>Über lokal geringten Räumen sind die invertierbaren Garben genau die lokal freien Garben ersten Ranges.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

<ul><li>Tensorprodukte von invertierbaren Modulgarben sind invertierbare Modulgarben.</li>

<li>Invertierbare Modulgarben über Schemata sind <a href="Quasikoh%C3%A4rente_Garbe" title="Quasikohärente Garbe">quasikohärent</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Picard-Gruppe">Picard-Gruppe</h2></div>

<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Picardgruppe" title="Picardgruppe">Picardgruppe</a></i></div><p>Invertierbare Garben über einem Schema bilden eine Gruppe, die als Picardgruppe bezeichnet wird. Für ein Schema <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,{\mathcal {O}}_{X})}">

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<annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {O}}_{X})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46716c1fec068ffe9981107a5a215fa1fc9e7d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.306ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {O}}_{X})}" loading="lazy"></span> ist diese gegeben durch die erste <a href="Garbenkohomologie" title="Garbenkohomologie">Garbenkohomologie</a>:

</p><dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Pic} (X):=H^{1}(X,{\mathcal {O}}_{X}^{*}).}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Pic} (X):=H^{1}(X,{\mathcal {O}}_{X}^{*}).}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7922afc916a282b4f6bafbe960c5dea50972d42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.907ex; height:3.176ex;" alt="{\displaystyle \operatorname {Pic} (X):=H^{1}(X,{\mathcal {O}}_{X}^{*}).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Ravi_Vakil" title="Ravi Vakil">Ravi Vakil</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="The_Rising_Sea%3A_Foundations_of_Algebraic_Geometry" title="The Rising Sea: Foundations of Algebraic Geometry">The Rising Sea: Foundations of Algebraic Geometry</a></cite>. <a href="Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, Princeton 2025, ISBN 978-0-691-26867-5 (englisch, <a rel="nofollow" class="external text" href="https://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf">stanford.edu</a> [PDF]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Invertierbare+Garbe&rft.au=Ravi+Vakil&rft.btitle=The+Rising+Sea%3A+Foundations+of+Algebraic+Geometry&rft.date=2025&rft.genre=book&rft.isbn=9780691268675&rft.place=Princeton&rft.pub=Princeton+University+Press" style="display:none"> </span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/02AC">Invertible sheaves</a> auf dem Stacks Project (englisch)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><i>The Rising Sea</i>, 13.1.4. Definition</span>

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