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<p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Definition'>Definition</a></li> <ul> <li><a href='#InAbelianCategory'>In an abelian category</a></li> <li><a href='#in_a_semiabelian_category'>In a semi-abelian category</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#RelationToChainHomotopy'>Relation to chain homotopy</a></li> <li><a href='#OfVectorSpaces'>Of free modules and vector spaces</a></li> <li><a href='#InvolvingInjectiveObjects'>Involving injective/projective objects</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Definition">Definition</h2> <h3 id="InAbelianCategory">In an abelian category</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>.</p> <div class="num_defn" id="SplitnessInAbelianCategory"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mi>B</mi><mover><mo>→</mo><mi>p</mi></mover><mi>C</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0\to A \stackrel{i}{\to} B \stackrel{p}{\to} C\to 0</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> is called <strong>split</strong> if either of the following equivalent conditions hold</p> <ol> <li> <p>There exists a <a class="existingWikiWord" href="/nlab/show/section">section</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, hence a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">s \colon C\to B</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>s</mi><mo>=</mo><msub><mi>id</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">p \circ s = id_C</annotation></semantics></math>.</p> </li> <li> <p>There exists a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, hence a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">r \colon B\to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∘</mo><mi>i</mi><mo>=</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">r \circ i = id_A</annotation></semantics></math>.</p> </li> <li> <p>There exists an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of sequences with the sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo>⊕</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0\to A\to A\oplus C\to C\to 0 </annotation></semantics></math></div> <p>given by the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> and its canonical injection/projection morphisms.</p> </li> </ol> </div> <div class="num_lemma" id="SplittingLemma"> <h6 id="lemma">Lemma</h6> <p><strong>(splitting lemma)</strong></p> <p>The three conditions in def. <a class="maruku-ref" href="#SplitnessInAbelianCategory"></a> are indeed <a class="existingWikiWord" href="/nlab/show/equivalence">equivalent</a>.</p> </div> <p>(e.g. <a href="#Hatcher02">Hatcher (2002)</a>, p. 147)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>It is clear that the third condition implies the first two: take the section/retract to be given by the canonical injection/projection maps that come with a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>.</p> <p>Conversely, suppose we have a retract <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">r \colon B \to A</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i \colon A \to B</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mover><mo>→</mo><mi>r</mi></mover><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">P \colon B \stackrel{r}{\to} A \stackrel{i}{\to} B</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/idempotent">idempotent</a>.</p> <p>Then every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math> can be decomposed as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>=</mo><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b = (b - P(b)) + P(b)</annotation></semantics></math> hence with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b - P(b) \in ker(r)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>im</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(b) \in im(i)</annotation></semantics></math>. Moreover this decomposition is unique since if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>=</mo><mi>i</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b = i(a)</annotation></semantics></math> while at the same time <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">r(b) = 0</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>=</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">0 = r(i(a)) = a</annotation></semantics></math>. This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>≃</mo><mi>im</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \simeq im(i) \oplus ker(r)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i \colon A \to B</annotation></semantics></math> is the canonical inclusion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(i)</annotation></semantics></math>. By exactness it then follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>im</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ker(r) \simeq im(p)</annotation></semantics></math> and hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>≃</mo><mi>A</mi><mo>⊕</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">B \simeq A \oplus C</annotation></semantics></math> with the canonical inclusion and projection.</p> <p>The implication that the second condition also implies the third is formally dual to this argument.</p> </div> <h3 id="in_a_semiabelian_category">In a semi-abelian category</h3> <p>There is a nonabelian analog of split exact sequences in <a class="existingWikiWord" href="/nlab/show/semiabelian+categories">semiabelian categories</a>. See there.</p> <h2 id="properties">Properties</h2> <h3 id="RelationToChainHomotopy">Relation to chain homotopy</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C_\bullet</annotation></semantics></math> is <em>split exact</em> precisely if the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> from the 0-chain complex, namely the <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">0 \to C_\bullet</annotation></semantics></math> is actually a <a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a> <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">equivalence</a>, in that the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C_\bullet</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/null+homotopy">null homotopy</a>.</p> </div> <h3 id="OfVectorSpaces">Of free modules and vector spaces</h3> <p>Assuming the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>:</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/free+abelian+groups">free abelian groups</a> is split.</p> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>Every exact sequence of <a class="existingWikiWord" href="/nlab/show/free+modules">free modules</a> which is <a class="existingWikiWord" href="/nlab/show/bounded-below+chain+complex">bounded below</a> is split.</p> </div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/field">field</a> and denote by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>≔</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} \coloneqq k</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> <div class="num_cor" id="SESOfVectorSpacesSplits"> <h6 id="corollary">Corollary</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of vector spaces is split.</p> </div> <p>(Essentially by the <a class="existingWikiWord" href="/nlab/show/basis+theorem">basis theorem</a>, for exposition see for instance <a href="https://unapologetic.wordpress.com/2008/06/26/exact-sequences-split">here</a>.)</p> <h3 id="InvolvingInjectiveObjects">Involving injective/projective objects</h3> <div class="num_lemma"> <h6 id="lemma_2">Lemma</h6> <p>If in a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mo>→</mo><mi>B</mi><mo>→</mo><mi>C</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to A \to B \to C \to 0</annotation></semantics></math> in an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> the first object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a> or the last object is a <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a> then the sequence is split exact.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Consider the first case. The other is formally dual.</p> <p>By the properties of a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \to B</annotation></semantics></math> here is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>. By definition of <a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is injective then it has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> and so there is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">q : B \to A</annotation></semantics></math> that makes the following <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commute</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>id</mi> <mi>A</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mi>q</mi></msub></mtd></mtr> <mtr><mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\stackrel{id_A}{\to}& A \\ \downarrow & \nearrow_{q} \\ B } \,. </annotation></semantics></math></div> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> as in def. <a class="maruku-ref" href="#SplitnessInAbelianCategory"></a>.</p> </div> <h2 id="references">References</h2> <p>For instance</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Charles Weibel</a>, Section 1.4 of: <em><a class="existingWikiWord" href="/nlab/show/An+Introduction+to+Homological+Algebra">An Introduction to Homological Algebra</a></em> (1994)</p> </li> <li id="Hatcher02"> <p><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, pp. 147 of: <em>Algebraic Topology</em>, Cambridge University Press (2002) [<a href="https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401">ISBN:9780521795401</a>, <a href="https://pi.math.cornell.edu/~hatcher/AT/ATpage.html">webpage</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 23, 2023 at 09:35:45. 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