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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Split exact sequence</span></span>

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<p>The term <b>split exact sequence</b> is used in two different ways by different people. Some people mean a <a href="Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> that right-splits (thus corresponding to a semidirect product) and some people mean a <a href="Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> that left-splits (which implies it right-splits, and corresponds to a direct product). This article takes the latter approach, but both are in common use. When reading a book or paper, it is important to note precisely which of the two meanings is in use.

</p><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>split exact sequence</b> is a <a href="Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> in which the middle term is built out of the two outer terms in the simplest possible way.

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<div class="mw-heading mw-heading2"><h2 id="Equivalent_characterizations">Equivalent characterizations</h2></div>

<p>A short exact sequence of <a href="Abelian_group" title="Abelian group">abelian groups</a> or of <a href="Module_(mathematics)" title="Module (mathematics)">modules</a> over a fixed <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>, or more generally of objects in an <a href="Abelian_category" title="Abelian category">abelian category</a>

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<annotation encoding="application/x-tex">{\displaystyle 0\to A\mathrel {\stackrel {a}{\to }} B\mathrel {\stackrel {b}{\to }} C\to 0}</annotation>

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</math></span><img src="./e90917113cf8319b3df86b4ee16193c1d517760a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.055ex; height:3.676ex;" alt="{\displaystyle 0\to A\mathrel {\stackrel {a}{\to }} B\mathrel {\stackrel {b}{\to }} C\to 0}" loading="lazy"></span></dd></dl>

<p>is called split exact if it is isomorphic to the exact sequence where the middle term is the <a href="Direct_sum" title="Direct sum">direct sum</a> of the outer ones:

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<annotation encoding="application/x-tex">{\displaystyle 0\to A\mathrel {\stackrel {i}{\to }} A\oplus C\mathrel {\stackrel {p}{\to }} C\to 0}</annotation>

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</math></span><img src="./0c13e168eccb44963a678066148239c75263a1f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.64ex; height:3.676ex;" alt="{\displaystyle 0\to A\mathrel {\stackrel {i}{\to }} A\oplus C\mathrel {\stackrel {p}{\to }} C\to 0}" loading="lazy"></span></dd></dl>

<p>The requirement that the sequence is isomorphic means that there is an <a href="Isomorphism" title="Isomorphism">isomorphism</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 f:B\to A\oplus C}">

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<annotation encoding="application/x-tex">{\displaystyle f:B\to A\oplus C}</annotation>

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</math></span><img src="./d28e85786ef8b5da40166754cb25d0fdc6e7f3ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.944ex; height:2.509ex;" alt="{\displaystyle f:B\to A\oplus C}" loading="lazy"></span> such that the composite <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\circ a}">

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

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</math></span><img src="./b7bd250a28273a3f8ef6410a031a9f0442a9852b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.703ex; height:2.509ex;" alt="{\displaystyle f\circ a}" loading="lazy"></span> is the natural <a href="Inclusion_map" title="Inclusion map">inclusion</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 i:A\to A\oplus C}">

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<annotation encoding="application/x-tex">{\displaystyle i:A\to A\oplus C}</annotation>

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</math></span><img src="./b706f38ef3b7978f7111d3b35618d0f50993fd78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.447ex; height:2.343ex;" alt="{\displaystyle i:A\to A\oplus C}" loading="lazy"></span> and such that the composite <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\circ f}">

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

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</math></span><img src="./d28e34f4c6ecfb577a55bce6e3cee686945bb2f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.732ex; height:2.509ex;" alt="{\displaystyle p\circ f}" loading="lazy"></span> equals <i>b</i>. This can be summarized by a <a href="Commutative_diagram" title="Commutative diagram">commutative diagram</a> as:

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</p><p>The <a href="Splitting_lemma" title="Splitting lemma">splitting lemma</a> provides further equivalent characterizations of split exact sequences.

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

<p>A trivial example of a split short exact sequence is

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<annotation encoding="application/x-tex">{\displaystyle 0\to M_{1}\mathrel {\stackrel {q}{\to }} M_{1}\oplus M_{2}\mathrel {\stackrel {p}{\to }} M_{2}\to 0}</annotation>

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</math></span><img src="./aaaf574eb7098e8e1f9517d648d5e057457ca87c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.855ex; height:3.676ex;" alt="{\displaystyle 0\to M_{1}\mathrel {\stackrel {q}{\to }} M_{1}\oplus M_{2}\mathrel {\stackrel {p}{\to }} M_{2}\to 0}" loading="lazy"></span></dd></dl>

<p>where <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 M_{1},M_{2}}">

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

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</math></span><img src="./b66e3a3dcb1a11e2a9ac7b65d40c059db38ab8c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.651ex; height:2.509ex;" alt="{\displaystyle M_{1},M_{2}}" loading="lazy"></span> are <i>R</i>-modules, <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 q}">

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

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</math></span><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span> is the canonical injection and <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}">

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

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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> is the canonical projection.

</p><p>Any short exact sequence of <a href="Vector_space" title="Vector space">vector spaces</a> is split exact. This is a rephrasing of the fact that any <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> vectors in a vector space can be extended to a <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>.

</p><p>The exact sequence <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 0\to \mathbf {Z} \mathrel {\stackrel {2}{\to }} \mathbf {Z} \to \mathbf {Z} /2\mathbf {Z} \to 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0\to \mathbf {Z} \mathrel {\stackrel {2}{\to }} \mathbf {Z} \to \mathbf {Z} /2\mathbf {Z} \to 0}</annotation>

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</math></span><img src="./849b39642d6df53187ac2ab55fbee5890741b5b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.642ex; height:4.009ex;" alt="{\displaystyle 0\to \mathbf {Z} \mathrel {\stackrel {2}{\to }} \mathbf {Z} \to \mathbf {Z} /2\mathbf {Z} \to 0}" loading="lazy"></span> (where the first map is multiplication by 2) is not split exact.

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<div class="mw-heading mw-heading2"><h2 id="Related_notions">Related notions</h2></div>

<p><a href="Pure_exact_sequence" class="mw-redirect" title="Pure exact sequence">Pure exact sequences</a> can be characterized as the <a href="Filtered_colimit" class="mw-redirect" title="Filtered colimit">filtered colimits</a> of split exact sequences.<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="References">References</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFFuchs2015">Fuchs (2015</a>, Ch. 5, Thm. 3.4)</span>

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

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</style><cite id="CITEREFFuchs2015" class="citation cs2"><a href="L%C3%A1szl%C3%B3_Fuchs" title="László Fuchs">Fuchs, László</a> (2015), <i>Abelian Groups</i>, Springer Monographs in Mathematics, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783319194226</bdi></cite></li>

<li><cite id="CITEREFSharp,_R._Y.2001" class="citation cs2">Sharp, R. Y., Rodney (2001), <i>Steps in Commutative Algebra, 2nd ed.</i>, London Mathematical Society Student Texts, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0521646235</bdi></cite></li></ul></div><div><div class="zim-footer">

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