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<title>Additiver Funktor</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Additiver Funktor</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><b>Additiver Funktor</b> ist ein Begriff aus dem <a href="Teilgebiete_der_Mathematik" title="Teilgebiete der Mathematik">mathematischen Teilgebiet</a> der <a href="Kategorientheorie" title="Kategorientheorie">Kategorientheorie</a>. Es handelt sich dabei um <a href="Funktor_(Mathematik)" title="Funktor (Mathematik)">Funktoren</a> zwischen <a href="Pr%C3%A4additive_Kategorie" class="mw-redirect" title="Präadditive Kategorie">präadditiven Kategorien</a>, die <a href="Gruppenhomomorphismus" title="Gruppenhomomorphismus">Gruppenhomomorphismen</a> zwischen den Morphismengruppen definieren.

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

<p>Es seien <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 {\mathfrak {C}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {C}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8cfb4d0d9067ed01e04e20160d7f644bf4955ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.425ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {C}}}" loading="lazy"></span> 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 {\mathfrak {D}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {D}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46c2461a0bd159fa416eeb2bd7a4ac0fed0262ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.934ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {D}}}" loading="lazy"></span> präadditive Kategorien.

Ein Funktor <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 F:{\mathfrak {C}}\rightarrow {\mathfrak {D}}}">

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<annotation encoding="application/x-tex">{\displaystyle F:{\mathfrak {C}}\rightarrow {\mathfrak {D}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2456b40b73d8fe348e309730569ddc6a4ef8bec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.65ex; height:2.176ex;" alt="{\displaystyle F:{\mathfrak {C}}\rightarrow {\mathfrak {D}}}" loading="lazy"></span> heißt additiv, falls die Abbildungen <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 \mathrm {Mor} _{\mathfrak {C}}(X,Y)\rightarrow \mathrm {Mor} _{\mathfrak {D}}(FX,FY);\,f\mapsto Ff}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Mor} _{\mathfrak {C}}(X,Y)\rightarrow \mathrm {Mor} _{\mathfrak {D}}(FX,FY);\,f\mapsto Ff}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fdc60364f84c13a72df0fa6141b893f66670b28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.871ex; height:2.843ex;" alt="{\displaystyle \mathrm {Mor} _{\mathfrak {C}}(X,Y)\rightarrow \mathrm {Mor} _{\mathfrak {D}}(FX,FY);\,f\mapsto Ff}" loading="lazy"></span> für je zwei Objekte <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}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> 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 Y}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> aus <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 {\mathfrak {C}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {C}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8cfb4d0d9067ed01e04e20160d7f644bf4955ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.425ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {C}}}" loading="lazy"></span> Gruppenhomomorphismen sind.

</p><p>Häufig betrachtet man additive Funktoren auf <a href="Additive_Kategorie" class="mw-redirect" title="Additive Kategorie">additiven</a> oder <a href="Abelsche_Kategorie" title="Abelsche Kategorie">abelschen Kategorien</a>, da diese auf solchen Kategorien weitere Eigenschaften haben. Die meisten natürlich auftretenden Funktoren zwischen präadditiven Kategorien sind additiv.

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

<p>Für Funktoren zwischen abelschen Kategorien hat man folgende Charakterisierung:<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> Ein Funktor <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 F:{\mathfrak {A}}\rightarrow {\mathfrak {B}}}">

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<annotation encoding="application/x-tex">{\displaystyle F:{\mathfrak {A}}\rightarrow {\mathfrak {B}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50dd8d4198ac08bdbb45c5a65d7ec193abe908f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.015ex; height:2.176ex;" alt="{\displaystyle F:{\mathfrak {A}}\rightarrow {\mathfrak {B}}}" loading="lazy"></span> ist genau dann additiv, wenn <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 F(A_{1}\oplus A_{2})=F(A_{1})\oplus F(A_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle F(A_{1}\oplus A_{2})=F(A_{1})\oplus F(A_{2})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c2a06775c09f8b09b69fbc0f3a5544981ea242e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.619ex; height:2.843ex;" alt="{\displaystyle F(A_{1}\oplus A_{2})=F(A_{1})\oplus F(A_{2})}" loading="lazy"></span> für alle Objekte <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 A_{1},A_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{1},A_{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/250743c2a3f816df26a8bc3216d4fd50171f420f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.629ex; height:2.509ex;" alt="{\displaystyle A_{1},A_{2}}" loading="lazy"></span> aus <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 {\mathfrak {A}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}" loading="lazy"></span>, wobei die Gleichheit folgendes bedeuten soll: Ist <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 (\iota _{j}:A_{j}\rightarrow A_{1}\oplus A_{2})_{j=1,2}}">

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<annotation encoding="application/x-tex">{\displaystyle (\iota _{j}:A_{j}\rightarrow A_{1}\oplus A_{2})_{j=1,2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c0ab588c14aeaa8d6fef8be7669cd817680c1ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.471ex; height:3.009ex;" alt="{\displaystyle (\iota _{j}:A_{j}\rightarrow A_{1}\oplus A_{2})_{j=1,2}}" loading="lazy"></span> eine <a href="Direkte_Summe" title="Direkte Summe">direkte Summe</a>, so auch <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 (F\iota _{j}:FA_{j}\rightarrow F(A_{1}\oplus A_{2}))_{j=1,2}}">

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<annotation encoding="application/x-tex">{\displaystyle (F\iota _{j}:FA_{j}\rightarrow F(A_{1}\oplus A_{2}))_{j=1,2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06b56ab91f59706142f50c4264a8d76ba38db70f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.503ex; height:3.009ex;" alt="{\displaystyle (F\iota _{j}:FA_{j}\rightarrow F(A_{1}\oplus A_{2}))_{j=1,2}}" loading="lazy"></span>.

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

<ul><li>Die <a href="Hom-Funktor" title="Hom-Funktor">Hom-Funktoren</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 \mathrm {Hom} _{R}(A,-)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} _{R}(A,-)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22df3cc5b7b6eddb3aa081001092af8efc6bdeb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.716ex; height:2.843ex;" alt="{\displaystyle \mathrm {Hom} _{R}(A,-)}" loading="lazy"></span> von 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 {\mathfrak {M}}_{R}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {M}}_{R}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e1351a2d49909be509219878c1280357d530202.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.919ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {M}}_{R}}" 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 R}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>-<a href="Modul_(Mathematik)" title="Modul (Mathematik)">Moduln</a> über einem <a href="Ring_(Algebra)" title="Ring (Algebra)">Ring</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 R}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> in die 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 {\mathfrak {Ab}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {Ab}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2bbf9e944a8fb34401313ebc60e4ab309cd0e64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.861ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {Ab}}}" loading="lazy"></span> der <a href="Abelsche_Gruppe" title="Abelsche Gruppe">abelschen Gruppen</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 A}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> ein fester <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 R}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>-Modul, ist additiv. Das Gleiche gilt für die Funktoren <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 \mathrm {Hom} _{R}(-,A):{\mathfrak {M}}_{R}\rightarrow {\mathfrak {Ab}}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} _{R}(-,A):{\mathfrak {M}}_{R}\rightarrow {\mathfrak {Ab}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d0e45530b1a0b2f1727cff24a1627ae282196ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.048ex; height:2.843ex;" alt="{\displaystyle \mathrm {Hom} _{R}(-,A):{\mathfrak {M}}_{R}\rightarrow {\mathfrak {Ab}}}" loading="lazy"></span></li>

<li>Die <a href="Tensorprodukt" title="Tensorprodukt">Tensorfunktoren</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 (A\otimes _{R}-):{\mathfrak {M}}_{R}\rightarrow {\mathfrak {Ab}}}">

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<annotation encoding="application/x-tex">{\displaystyle (A\otimes _{R}-):{\mathfrak {M}}_{R}\rightarrow {\mathfrak {Ab}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/983e6162fa0b965ef1b3db5fb4e2066791757223.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.013ex; height:2.843ex;" alt="{\displaystyle (A\otimes _{R}-):{\mathfrak {M}}_{R}\rightarrow {\mathfrak {Ab}}}" loading="lazy"></span> sind additiv, ebenso <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 (-\otimes _{R}A):{\mathfrak {M}}_{R}\rightarrow {\mathfrak {Ab}}}">

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<annotation encoding="application/x-tex">{\displaystyle (-\otimes _{R}A):{\mathfrak {M}}_{R}\rightarrow {\mathfrak {Ab}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0789f88098615967821c6fde61fef77d06b64bb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.013ex; height:2.843ex;" alt="{\displaystyle (-\otimes _{R}A):{\mathfrak {M}}_{R}\rightarrow {\mathfrak {Ab}}}" loading="lazy"></span></li>

<li><a href="Halbexakt" class="mw-redirect" title="Halbexakt">Halbexakte</a> Funktoren sind additiv.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li>

<li>Der Funktor <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 F:{\mathfrak {M}}_{R}\rightarrow {\mathfrak {M}}_{R}}">

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<annotation encoding="application/x-tex">{\displaystyle F:{\mathfrak {M}}_{R}\rightarrow {\mathfrak {M}}_{R}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88ba69b3beaa9bebe74ab82834ad03aa4752f08a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.131ex; height:2.509ex;" alt="{\displaystyle F:{\mathfrak {M}}_{R}\rightarrow {\mathfrak {M}}_{R}}" 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 FA=A\oplus R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle FA=A\oplus R}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4879c128ad130d4722fc0eb62c713e4ec437f248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.93ex; height:2.343ex;" alt="{\displaystyle FA=A\oplus R}" loading="lazy"></span> für jeden Modul <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 A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> 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 Ff=f\oplus \mathrm {id} _{R}}">

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<annotation encoding="application/x-tex">{\displaystyle Ff=f\oplus \mathrm {id} _{R}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/affcff94e3bda12bd2256209ea6f525ac94923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.656ex; height:2.509ex;" alt="{\displaystyle Ff=f\oplus \mathrm {id} _{R}}" loading="lazy"></span> für jeden Morphismus <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 f}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> ist nicht additiv.</li></ul>

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

<p>Additive Funktoren zwischen abelschen Kategorien haben folgende Eigenschaften:

</p>

<ul><li>Additive Funktoren überführen <a href="Anfangsobjekt%2C_Endobjekt_und_Nullobjekt" title="Anfangsobjekt, Endobjekt und Nullobjekt">Nullobjekte</a> in Nullobjekte.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li>

<li>Additive Funktoren überführen endliche direkte Summen in direkte Summen.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 0\rightarrow A\rightarrow A^{'}\rightarrow A^{''}\rightarrow 0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53162020d2d1499f1aaf57f1e4e136e699c01745.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.906ex; height:2.843ex;" alt="{\displaystyle 0\rightarrow A\rightarrow A^{'}\rightarrow A^{''}\rightarrow 0}" loading="lazy"></span> eine kurze exakte Sequenz 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 F}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> ein additiver Funktor, so hat man eine lange exakte Sequenz</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \ldots \rightarrow L_{n}FA\rightarrow L_{n}FA^{'}\rightarrow L_{n}FA^{''}\rightarrow \ldots \rightarrow L_{0}FA\rightarrow L_{0}FA^{'}\rightarrow L_{0}FA^{''}\rightarrow 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4669f45dc739f09eb8477f54056b113fdfdb369.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:76.53ex; height:3.176ex;" alt="{\displaystyle \ldots \rightarrow L_{n}FA\rightarrow L_{n}FA^{'}\rightarrow L_{n}FA^{''}\rightarrow \ldots \rightarrow L_{0}FA\rightarrow L_{0}FA^{'}\rightarrow L_{0}FA^{''}\rightarrow 0}" loading="lazy"></span>,</dd>

<dd>wobei <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 L_{n}}">

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle L_{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebec334cb04f246db1139e2ca6be0b957d2ef520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.801ex; height:2.509ex;" alt="{\displaystyle L_{n}}" loading="lazy"></span> für die <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 n}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-te <a href="Abgeleiteter_Funktor" title="Abgeleiteter Funktor">Linksableitung</a> stehe.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Insbesondere ist die 0-te Linksableitung eines additiven Funktors <a href="Rechtsexakt" class="mw-redirect" title="Rechtsexakt">rechtsexakt</a>.</dd></dl>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle F{\xrightarrow {\rho }}F^{'}{\xrightarrow {\sigma }}F^{''}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f69e2945c825c00c72bf1429a4d03a4a663ecd3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.408ex; width:11.913ex; height:3.509ex;" alt="{\displaystyle F\xrightarrow {\rho } F^{'}\xrightarrow {\sigma } F^{''}}" loading="lazy"></span> eine Folge additiver Funktoren und <a href="Nat%C3%BCrliche_Transformation" title="Natürliche Transformation">natürlicher Transformationen</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 \rho }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> 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 \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span> und ist für jeden <a href="Projektiver_Modul" class="mw-redirect" title="Projektiver Modul">projektiven Modul</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 P}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> die Sequenz</li></ul>

<mover>

<msup>

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</mover>

</mrow>

<msup>

<msup>

</msup>

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<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\rightarrow FP{\xrightarrow {\rho ^{P}}}F^{'}P{\xrightarrow {\sigma ^{P}}}F^{''}P\rightarrow 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b340e0850e35b12a8cf04d28aa2a2445e2e04ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.053ex; margin-top: -0.294ex; margin-bottom: -0.451ex; width:28.825ex; height:4.343ex;" alt="{\displaystyle 0\rightarrow FP\xrightarrow {\rho ^{P}} F^{'}P\xrightarrow {\sigma ^{P}} F^{''}P\rightarrow 0}" loading="lazy"></span></dd>

<dd>exakt, so hat man für beliebige Moduln <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 A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> eine lange exakte Sequenz<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></dd>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \ldots \rightarrow L_{n}FA\rightarrow L_{n}F^{'}A\rightarrow L_{n}F^{''}A\rightarrow \ldots \rightarrow L_{0}FA\rightarrow L_{0}F^{'}A\rightarrow L_{0}F^{''}A\rightarrow 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11f78d8cfadc22182279e37de46d9edf20d62dc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:76.826ex; height:3.176ex;" alt="{\displaystyle \ldots \rightarrow L_{n}FA\rightarrow L_{n}F^{'}A\rightarrow L_{n}F^{''}A\rightarrow \ldots \rightarrow L_{0}FA\rightarrow L_{0}F^{'}A\rightarrow L_{0}F^{''}A\rightarrow 0}" loading="lazy"></span>.</dd></dl>

<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">Peter Hilton: <i>Lectures in Homological Algebra.</i> American Mathematical Society, 2005, ISBN 0-8218-3872-5, Satz 3.1.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Peter Hilton: <i>Lectures in Homological Algebra.</i> American Mathematical Society, 2005, ISBN 0-8218-3872-5, Satz 3.2.</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Götz Brunner: <i>Homologische Algebra.</i> B.I.-Wissenschaftsverlag, 1973, <span class="falsche-isbn">ISBN 3-411-014420-2</span>, Kapitel III, Satz 23.</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Götz Brunner: <i>Homologische Algebra.</i> B.I.-Wissenschaftsverlag, 1973, <span class="falsche-isbn">ISBN 3-411-014420-2</span>, Kapitel III, Satz 24.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Peter Hilton: <i>Lectures in Homological Algebra.</i> American Mathematical Society, 2005, ISBN 0-8218-3872-5, Theorem 3.6.</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Peter Hilton: <i>Lectures in Homological Algebra.</i> American Mathematical Society, 2005, ISBN 0-8218-3872-5, Theorem 3.8.</span>

</li>

</ol>

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<div class="klappleiste-kopf"><a href="Kategorientheorie" title="Kategorientheorie">Kategorientheorie</a><div class="erweiterte-navigationsleiste-quicklinks" style="float:left; font-weight:normal; font-size:75%; margin-left:1em; margin-right:2em; display:none;"><span title="Vorlage anzeigen">V</span> </div></div>

<td class="erw-nav-gruppe" style="white-space: nowrap;text-align: right;border: 1px solid transparent;border-top: 1px solid #FFF;border-bottom: 2px solid #FFF;padding: 0 1em;"><b>Einordnung</b>

</td>

<td class="erw-nav-gruppe" style="white-space: nowrap;text-align: right;border: 1px solid transparent;border-top: 1px solid #FFF;border-bottom: 2px solid #FFF;padding: 0 1em;"><b>Typen von Kategorien</b>

</td>

<p><a href="Duale_Kategorie" title="Duale Kategorie">dual</a> | <a href="Diskrete_Kategorie" title="Diskrete Kategorie">diskret</a> | <a href="Kleine_Kategorie" class="mw-redirect" title="Kleine Kategorie">klein</a> | <a href="Lokal_kleine_Kategorie" title="Lokal kleine Kategorie">lokal klein</a> | <a href="Monoidale_Kategorie" title="Monoidale Kategorie">monoidal</a> | <a href="Symmetrische_monoidale_Kategorie" title="Symmetrische monoidale Kategorie">symmetrisch monoidal</a> | <a href="Angereicherte_Kategorie" title="Angereicherte Kategorie">angereichert</a> | <a href="Ausgeglichene_Kategorie" title="Ausgeglichene Kategorie">ausgeglichen</a> | <a href="Erreichbare_Kategorie" title="Erreichbare Kategorie">erreichbar</a> | <a href="Vollst%C3%A4ndige_Kategorie" title="Vollständige Kategorie">vollständig</a> | <a href="Kovollst%C3%A4ndige_Kategorie" class="mw-redirect" title="Kovollständige Kategorie">kovollständig</a>

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<td class="erw-nav-gruppe" style="white-space: nowrap;text-align: right;border: 1px solid transparent;border-top: 1px solid #FFF;border-bottom: 2px solid #FFF;padding: 0 1em;"><b>Typen von Objekten</b>

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<p><a href="Anfangsobjekt%2C_Endobjekt_und_Nullobjekt" title="Anfangsobjekt, Endobjekt und Nullobjekt">initial</a> | <a href="Anfangsobjekt%2C_Endobjekt_und_Nullobjekt" title="Anfangsobjekt, Endobjekt und Nullobjekt">terminal</a> | <a href="Anfangsobjekt%2C_Endobjekt_und_Nullobjekt" title="Anfangsobjekt, Endobjekt und Nullobjekt">null</a> | <a href="Injektives_Objekt" title="Injektives Objekt">injektiv</a> | <a href="Projektives_Objekt" title="Projektives Objekt">projektiv</a> | <a href="Generator_und_Kogenerator" title="Generator und Kogenerator">Generator</a> | <a href="Kogenerator" class="mw-redirect" title="Kogenerator">Kogenerator</a> | <a href="Ind-Objekte_und_Pro-Objekte" title="Ind-Objekte und Pro-Objekte">Pro</a> | <a href="Ind-Objekte_und_Pro-Objekte" title="Ind-Objekte und Pro-Objekte">Ind</a> | <a href="Gruppenobjekt" title="Gruppenobjekt">Gruppe</a> | <a href="Monoid-Objekt" title="Monoid-Objekt">Monoid</a> | <a href="Exponentiales_Objekt" title="Exponentiales Objekt">exponential</a> | <a href="Freies_Objekt" title="Freies Objekt">frei</a> | <a href="Kompaktes_Objekt" title="Kompaktes Objekt">kompakt</a>

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<td class="erw-nav-gruppe" style="white-space: nowrap;text-align: right;border: 1px solid transparent;border-top: 1px solid #FFF;border-bottom: 2px solid #FFF;padding: 0 1em;"><b>Typen von Morphismen</b>

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<p><a href="Monomorphismus" title="Monomorphismus">Mono</a> | <a href="Epimorphismus" title="Epimorphismus">Epi</a> | <a href="Bimorphismus" title="Bimorphismus">Bi</a> | <a href="Retraktion_und_Koretraktion" title="Retraktion und Koretraktion">Retraktion</a> | <a href="Koretraktion" class="mw-redirect" title="Koretraktion">Koretraktion</a> | <a href="Injektive_Aufl%C3%B6sung" title="Injektive Auflösung">Injektive Auflösung</a> | <a href="Projektive_Aufl%C3%B6sung" title="Projektive Auflösung">Projektive Auflösung</a>

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<td class="erw-nav-gruppe" style="white-space: nowrap;text-align: right;border: 1px solid transparent;border-top: 1px solid #FFF;border-bottom: 1px solid #FFF;padding: 0 1em;"><b>Typen von Funktoren</b>

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<p><a href="Konstanter_Funktor" title="Konstanter Funktor">konstant</a> | <a href="Voller_Funktor" class="mw-redirect" title="Voller Funktor">voll</a> | <a href="Treuer_Funktor" title="Treuer Funktor">treu</a> | <a href="Volltreuer_Funktor" class="mw-redirect" title="Volltreuer Funktor">volltreu</a> | <a class="mw-selflink selflink">additiv</a> | <a href="Exakter_Funktor" title="Exakter Funktor">exakt</a> | <a href="Abgeleiteter_Funktor" title="Abgeleiteter Funktor">abgeleitet</a> | <a href="Glatter_Funktor" title="Glatter Funktor">glatt</a>

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<td class="erw-nav-gruppe" style="white-space: nowrap;text-align: right;border: 1px solid transparent;border-top: 1px solid #FFF;border-bottom: 2px solid #FFF;padding: 0 1em;"><b>Konstruktionen</b>

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<td class="erw-nav-gruppe" style="white-space: nowrap;text-align: right;border: 1px solid transparent;border-top: 1px solid #FFF;border-bottom: 2px solid #FFF;padding: 0 1em;"><b><a href="Limes_(Kategorientheorie)" title="Limes (Kategorientheorie)">Limes</a></b>

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<p><a href="Produkt_und_Koprodukt" title="Produkt und Koprodukt">Produkt</a> | <a href="Differenzkern" title="Differenzkern">Differenzkern</a> | <a href="Faserprodukt" title="Faserprodukt">Faserprodukt</a> | <a href="Ende_(Kategorientheorie)" title="Ende (Kategorientheorie)">Ende</a>

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<td class="erw-nav-gruppe" style="white-space: nowrap;text-align: right;border: 1px solid transparent;border-top: 1px solid #FFF;border-bottom: 1px solid #FFF;padding: 0 1em;"><b><a href="Kolimes" title="Kolimes">Kolimes</a></b>

</td>

<p><a href="Filtrierter_Kolimes" title="Filtrierter Kolimes">Filtrierter Kolimes</a> | <a href="Koprodukt" class="mw-redirect" title="Koprodukt">Koprodukt</a> | <a href="Differenzkokern" title="Differenzkokern">Differenzkokern</a> | <a href="Kofaserprodukt" class="mw-redirect" title="Kofaserprodukt">Kofaserprodukt</a>

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</tbody></table>

<p><a href="Kan-Erweiterung" title="Kan-Erweiterung">Kan-Erweiterung</a> | <a href="Monade_(Kategorientheorie)" title="Monade (Kategorientheorie)">Monade</a> | <a href="Komonade" title="Komonade">Komonade</a> | <a href="Kategorie_der_Elemente" title="Kategorie der Elemente">Kategorie der Elemente</a> | <a href="Kommakategorie" title="Kommakategorie">Kommakategorie</a> | <a href="Pfeilkategorie" title="Pfeilkategorie">Pfeilkategorie</a> | <a href="Homotopie-Kategorie" title="Homotopie-Kategorie">Homotopie-Kategorie</a>

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<td class="erw-nav-gruppe" style="white-space: nowrap;text-align: right;border: 1px solid transparent;border-top: 1px solid #FFF;border-bottom: 2px solid #FFF;padding: 0 1em;"><b>Resultate</b>

</td>

<p><a href="Lemma_von_Yoneda" title="Lemma von Yoneda">Lemma von Yoneda</a> | <a href="Fixpunktsatz_von_Lawvere" title="Fixpunktsatz von Lawvere">Fixpunktsatz von Lawvere</a> | <a href="Einbettungssatz_von_Mitchell" title="Einbettungssatz von Mitchell">Einbettungssatz von Mitchell</a>

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<td class="erw-nav-gruppe" style="white-space: nowrap;text-align: right;border: 1px solid transparent;border-top: 1px solid #FFF;border-bottom: 1px solid #FFF;padding: 0 1em;"><b>Spezielle Funktoren</b>

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<p><a href="Hom-Funktor" title="Hom-Funktor">Hom-Funktor</a> | <a href="Potenzmengenfunktor" title="Potenzmengenfunktor">Potenzmengenfunktor</a> | <a href="Diagonalfunktor" title="Diagonalfunktor">Diagonalfunktor</a> | <a href="Ext-Funktor" class="mw-redirect" title="Ext-Funktor">Ext</a> | <a href="Tor-Funktor" class="mw-redirect" title="Tor-Funktor">Tor</a>

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