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<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Stein factorization</span></span>
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In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, the <b>Stein factorization</b>, introduced by <a href="Karl_Stein_(mathematician)" title="Karl Stein (mathematician)">Karl Stein</a> (<a href="#CITEREFStein1956">1956</a>) for the case of complex spaces, states that a <a href="Proper_morphism" title="Proper morphism">proper morphism</a> of <a href="Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> can be factorized as a composition of a <a href="Finite_morphism" title="Finite morphism">finite mapping</a> and a proper morphism with <a href="Connected_space" title="Connected space">connected</a> fibers. Roughly speaking, Stein factorization contracts the connected components of the fibers of a mapping to points.
</p>
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<div class="mw-heading mw-heading2"><h2 id="Statement">Statement</h2></div>
<p>One version for schemes states the following: (<a href="#CITEREFEGA">EGA</a>, III.4.3.1)
</p>
<blockquote>
<p>Let <i>X</i> be a <a href="Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a>, <i>S</i> a <a href="Locally_noetherian_scheme" class="mw-redirect" title="Locally noetherian scheme">locally noetherian scheme</a> 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 f:X\to S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to S}</annotation>
</semantics>
</math></span><img src="./a21a941b35354b73f5a62849067deac7ee9794dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.309ex; height:2.509ex;" alt="{\displaystyle f:X\to S}" loading="lazy"></span> a <a href="Proper_morphism" title="Proper morphism">proper morphism</a>. Then one can write
</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 f=g\circ f'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>=</mo>
<mi>g</mi>
<mo>∘<!-- ∘ --></mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f=g\circ f'}</annotation>
</semantics>
</math></span><img src="./434513ddd4044dc7f60d88311e859a31855ccfc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.693ex; height:2.843ex;" alt="{\displaystyle f=g\circ f'}" 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 g\colon S'\to S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>:<!-- : --></mo>
<msup>
<mi>S</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g\colon S'\to S}</annotation>
</semantics>
</math></span><img src="./1573d32c28195519a3588947fafd77d15027e71a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.47ex; height:2.843ex;" alt="{\displaystyle g\colon S'\to S}" loading="lazy"></span> is a <a href="Finite_morphism" title="Finite morphism">finite morphism</a> 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 f'\colon X\to S'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo>:<!-- : --></mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>S</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'\colon X\to S'}</annotation>
</semantics>
</math></span><img src="./8b261a7b1ff4a355db6eebbd9d05dcaa410c343b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.84ex; height:2.843ex;" alt="{\displaystyle f'\colon X\to S'}" loading="lazy"></span> is a proper morphism so that <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'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
<mo>′</mo>
</msubsup>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">O</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">O</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi>S</mi>
<mo>′</mo>
</msup>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}.}</annotation>
</semantics>
</math></span><img src="./afc4af2cd7a21f55437d34b5e6c89c19cc14bcfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.111ex; height:2.676ex;" alt="{\displaystyle f'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}.}" loading="lazy"></span>
</p>
</blockquote>
<p>The existence of this decomposition itself is not difficult. See below. But, by <a href="Zariski's_connectedness_theorem" title="Zariski's connectedness theorem">Zariski's connectedness theorem</a>, the last part in the above says that the fiber <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'^{-1}(s)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow>
<mo class="MJX-variant">′</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'^{-1}(s)}</annotation>
</semantics>
</math></span><img src="./084cd55c35fbe962503a51d5efa76ba4f8516b3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.006ex; height:3.176ex;" alt="{\displaystyle f'^{-1}(s)}" loading="lazy"></span> is connected for any <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 s\in S'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>S</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s\in S'}</annotation>
</semantics>
</math></span><img src="./c61620caa8408cc39405ede67842e3efcc543a4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.137ex; height:2.509ex;" alt="{\displaystyle s\in S'}" loading="lazy"></span>. It follows:
</p><p><b>Corollary</b>: For any <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 s\in S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo>∈<!-- ∈ --></mo>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s\in S}</annotation>
</semantics>
</math></span><img src="./acce52dffd84d073a24f4606a175da60148fd0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}" loading="lazy"></span>, the set of connected components of the fiber <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^{-1}(s)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{-1}(s)}</annotation>
</semantics>
</math></span><img src="./464cc2fd8eb7cb156223b620d778925bf081f291.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.553ex; height:3.176ex;" alt="{\displaystyle f^{-1}(s)}" loading="lazy"></span> is in bijection with the set of points in the fiber <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 g^{-1}(s)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g^{-1}(s)}</annotation>
</semantics>
</math></span><img src="./218bcaaf73d3c135fa3046b225ef3fb44f4de878.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.351ex; height:3.176ex;" alt="{\displaystyle g^{-1}(s)}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Proof">Proof</h2></div>
<p>Set:
</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 S'=\operatorname {Spec} _{S}f_{*}{\mathcal {O}}_{X}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mo>′</mo>
</msup>
<mo>=</mo>
<msub>
<mi>Spec</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msub>
<mo><!-- --></mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">O</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S'=\operatorname {Spec} _{S}f_{*}{\mathcal {O}}_{X}}</annotation>
</semantics>
</math></span><img src="./fa5e4e300561bdf7f73e5f109442a390c5643906.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.31ex; height:3.009ex;" alt="{\displaystyle S'=\operatorname {Spec} _{S}f_{*}{\mathcal {O}}_{X}}" loading="lazy"></span></dd></dl>
<p>where Spec<sub><i>S</i></sub> is the <a href="Relative_Spec" class="mw-redirect" title="Relative Spec">relative <b>Spec</b></a>. The construction gives the natural map <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 g\colon S'\to S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>:<!-- : --></mo>
<msup>
<mi>S</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g\colon S'\to S}</annotation>
</semantics>
</math></span><img src="./1573d32c28195519a3588947fafd77d15027e71a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.47ex; height:2.843ex;" alt="{\displaystyle g\colon S'\to S}" loading="lazy"></span>, which is finite since <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}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">O</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>
</semantics>
</math></span><img src="./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> is <a href="Coherent_sheaf" title="Coherent sheaf">coherent</a> and <i>f</i> is proper. The morphism <i>f</i> factors through <i>g</i> and one gets <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'\colon X\to S'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo>:<!-- : --></mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>S</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'\colon X\to S'}</annotation>
</semantics>
</math></span><img src="./8b261a7b1ff4a355db6eebbd9d05dcaa410c343b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.84ex; height:2.843ex;" alt="{\displaystyle f'\colon X\to S'}" loading="lazy"></span>, which is proper. By construction, <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'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
<mo>′</mo>
</msubsup>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">O</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">O</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi>S</mi>
<mo>′</mo>
</msup>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}}</annotation>
</semantics>
</math></span><img src="./22cffb78d498585eab248fc25a2c84b6896b7e15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.464ex; height:2.676ex;" alt="{\displaystyle f'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}}" loading="lazy"></span>. One then uses the <a href="Theorem_on_formal_functions" title="Theorem on formal functions">theorem on formal functions</a> to show that the last equality implies <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'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'}</annotation>
</semantics>
</math></span><img src="./258eaada38956fb69b8cb1a2eef46bcb97d3126b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}" loading="lazy"></span> has connected fibers. (This part is sometimes referred to as Zariski's connectedness theorem.)
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Contraction_morphism" title="Contraction morphism">Contraction morphism</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFHartshorne1977" class="citation cs2"><a href="Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977), <i><a href="Algebraic_Geometry_(book)" title="Algebraic Geometry (book)">Algebraic Geometry</a></i>, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 52, New York: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90244-9</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463157">0463157</a></cite></li>
<li><cite id="CITEREFEGA"><cite id="CITEREFGrothendieckDieudonné1961" class="citation journal cs1"><a href="Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck, Alexandre</a>; <a href="Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (1961). <a rel="nofollow" class="external text" href="http://www.numdam.org/item/PMIHES_1961__11__5_0">"Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie"</a>. <i><a href="Publications_Math%C3%A9matiques_de_l'IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a></i>. <b>11</b>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02684274">10.1007/bf02684274</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0217085">0217085</a>.</cite></cite></li>
<li><cite id="CITEREFStein1956" class="citation cs2"><a href="Karl_Stein_(mathematician)" title="Karl Stein (mathematician)">Stein, Karl</a> (1956), "Analytische Zerlegungen komplexer Räume", <i><a href="Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>, <b>132</b>: <span class="nowrap">63–</span>93, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01343331">10.1007/BF01343331</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5831">0025-5831</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0083045">0083045</a></cite></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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