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Proper morphism

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a proper morphism between <a href="Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> is an analog of a <a href="Proper_map" title="Proper map">proper map</a> between <a href="Complex_analytic_space" class="mw-redirect" title="Complex analytic space">complex analytic spaces</a>.

Some authors call a proper <a href="Algebraic_variety" title="Algebraic variety">variety</a> over a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> a <a href="Complete_variety" title="Complete variety">complete variety</a>. For example, every <a href="Projective_variety" title="Projective variety">projective variety</a> over a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is proper over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. A scheme <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><img src="./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"> of <a href="Morphism_of_finite_type" title="Morphism of finite type">finite type</a> over the <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> (for example, a variety) is proper over C if and only if the space <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><img src="./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">(C) of complex points with the classical (Euclidean) topology is <a href="Compact_space" title="Compact space">compact</a> and <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a>.

A <a href="Closed_immersion" title="Closed immersion">closed immersion</a> is proper. A morphism is <a href="Finite_morphism" title="Finite morphism">finite</a> if and only if it is proper and <a href="Quasi-finite_morphism" title="Quasi-finite morphism">quasi-finite</a>.

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

A <a href="Morphism_of_schemes" title="Morphism of schemes">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

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

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</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> of schemes is called universally closed if for every scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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

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</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> with a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\to Y}">

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

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</math><img src="./a2f836de5648fddbae09be6abce0cc80eb9ba3f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.068ex; height:2.176ex;" alt="{\displaystyle Z\to Y}" loading="lazy">, the projection from the <a href="Fiber_product_of_schemes" title="Fiber product of schemes">fiber product</a>

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<annotation encoding="application/x-tex">{\displaystyle X\times _{Y}Z\to Z}</annotation>

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</math><img src="./85c26cb650d45687d627150b18a099a487703fdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.281ex; height:2.509ex;" alt="{\displaystyle X\times _{Y}Z\to Z}" loading="lazy"></dd></dl>

is a <a href="Closed_map" class="mw-redirect" title="Closed map">closed map</a> of the underlying <a href="Topological_space" title="Topological space">topological spaces</a>. A morphism of schemes is called proper if it is <a href="Separated_morphism" class="mw-redirect" title="Separated morphism">separated</a>, of <a href="Morphism_of_finite_type" title="Morphism of finite type">finite type</a>, and universally closed ([EGA] II, 5.4.1 <a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20051108184937/http://modular.fas.harvard.edu/scans/papers/grothendieck/PMIHES_1961__8__5_0.pdf">[1]</a>). One also says that <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><img src="./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"> is proper over <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><img src="./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">. In particular, a variety <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><img src="./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"> over a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is said to be proper over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> if the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to \operatorname {Spec} (k)}">

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<annotation encoding="application/x-tex">{\displaystyle X\to \operatorname {Spec} (k)}</annotation>

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</math><img src="./99b4d7605f999dd15711416dee6a18f7570bff29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.264ex; height:2.843ex;" alt="{\displaystyle X\to \operatorname {Spec} (k)}" loading="lazy"> is proper.

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

For any natural number n, <a href="Projective_space" title="Projective space">projective space</a> Pn over a <a href="Commutative_ring" title="Commutative ring">commutative ring</a> R is proper over R. <a href="Projective_morphism" class="mw-redirect" title="Projective morphism">Projective morphisms</a> are proper, but not all proper morphisms are projective. For example, there is a <a href="Smooth_scheme" title="Smooth scheme">smooth</a> proper complex variety of dimension 3 which is not projective over C.<a href="#cite_note-1">[1]</a> <a href="Affine_variety" title="Affine variety">Affine varieties</a> of positive dimension over a field k are never proper over k. More generally, a proper <a href="Affine_morphism" class="mw-redirect" title="Affine morphism">affine morphism</a> of schemes must be finite.<a href="#cite_note-2">[2]</a> For example, it is not hard to see that the <a href="Affine_line" class="mw-redirect" title="Affine line">affine line</a> A1 over a field k is not proper over k, because the morphism A1 → Spec(k) is not universally closed. Indeed, the pulled-back morphism

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{1}\times _{k}\mathbb {A} ^{1}\to \mathbb {A} ^{1}}</annotation>

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</math><img src="./8f25cf296fff79ca65cf6b9340dade4083a68610.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.74ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} ^{1}\times _{k}\mathbb {A} ^{1}\to \mathbb {A} ^{1}}" loading="lazy"></dd></dl>

(given by (x,y) ↦ y) is not closed, because the image of the closed subset xy = 1 in A1 × A1 = A2 is A1 − 0, which is not closed in A1.

<div class="mw-heading mw-heading2"><h2 id="Properties_and_characterizations_of_proper_morphisms">Properties and characterizations of proper morphisms</h2></div>

In the following, let f: X → Y be a morphism of schemes.

<ul><li>The composition of two proper morphisms is proper.</li>

<li>Any <a href="Fiber_product_of_schemes" title="Fiber product of schemes">base change</a> of a proper morphism f: X → Y is proper. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×Y Z → Z is proper.</li>

<li>Properness is a <a href="Local_property" title="Local property">local property</a> on the base (in the <a href="Zariski_topology" title="Zariski topology">Zariski topology</a>). That is, if Y is covered by some open subschemes Yi and the restriction of f to all f−1(Yi) is proper, then so is f.</li>

<li>More strongly, properness is local on the base in the <a href="Fpqc_topology" class="mw-redirect" title="Fpqc topology">fpqc topology</a>. For example, if X is a scheme over a field k and E is a field extension of k, then X is proper over k if and only if the base change XE is proper over E.<a href="#cite_note-3">[3]</a></li>

<li><a href="Closed_immersion" title="Closed immersion">Closed immersions</a> are proper.</li>

<li>More generally, finite morphisms are proper. This is a consequence of the <a href="Going_up_and_going_down" title="Going up and going down">going up</a> theorem.</li>

<li>By <a href="Pierre_Deligne" title="Pierre Deligne">Deligne</a>, a morphism of schemes is finite if and only if it is proper and quasi-finite.<a href="#cite_note-4">[4]</a> This had been shown by <a href="Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck</a> if the morphism f: X → Y is <a href="Glossary_of_algebraic_geometry#finite_presentation" title="Glossary of algebraic geometry">locally of finite presentation</a>, which follows from the other assumptions if Y is <a href="Noetherian_scheme" title="Noetherian scheme">noetherian</a>.<a href="#cite_note-5">[5]</a></li>

<li>For X proper over a scheme S, and Y separated over S, the image of any morphism X → Y over S is a closed subset of Y.<a href="#cite_note-6">[6]</a> This is analogous to the theorem in topology that the image of a continuous map from a compact space to a Hausdorff space is a closed subset.</li>

<li>The <a href="Stein_factorization" title="Stein factorization">Stein factorization</a> theorem states that any proper morphism to a locally noetherian scheme can be factored as X → Z → Y, where X → Z is proper, surjective, and has geometrically connected fibers, and Z → Y is finite.<a href="#cite_note-7">[7]</a></li>

<li><a href="Chow's_lemma" title="Chow's lemma">Chow's lemma</a> says that proper morphisms are closely related to <a href="Projective_morphism" class="mw-redirect" title="Projective morphism">projective morphisms</a>. One version is: if X is proper over a <a href="Quasi-compact" class="mw-redirect" title="Quasi-compact">quasi-compact</a> scheme Y and X has only finitely many irreducible components (which is automatic for Y noetherian), then there is a projective surjective morphism g: W → X such that W is projective over Y. Moreover, one can arrange that g is an isomorphism over a dense open subset U of X, and that g−1(U) is dense in W. One can also arrange that W is integral if X is integral.<a href="#cite_note-8">[8]</a></li>

<li><a href="Nagata's_compactification_theorem" title="Nagata's compactification theorem">Nagata's compactification theorem</a>, as generalized by Deligne, says that a separated morphism of finite type between quasi-compact and <a href="Quasi-separated_morphism" title="Quasi-separated morphism">quasi-separated</a> schemes factors as an open immersion followed by a proper morphism.<a href="#cite_note-9">[9]</a></li>

<li>Proper morphisms between locally noetherian schemes preserve <a href="Coherent_sheaf" title="Coherent sheaf">coherent sheaves</a>, in the sense that the <a href="Higher_direct_image" class="mw-redirect" title="Higher direct image">higher direct images</a> Rif∗(F) (in particular the <a href="Direct_image" class="mw-redirect" title="Direct image">direct image</a> f∗(F)) of a coherent sheaf F are coherent (EGA III, 3.2.1). (Analogously, for a proper map between complex analytic spaces, <a href="Hans_Grauert" title="Hans Grauert">Grauert</a> and <a href="Reinhold_Remmert" title="Reinhold Remmert">Remmert</a> showed that the higher direct images preserve coherent analytic sheaves.) As a very special case: the ring of regular functions on a proper scheme X over a field k has finite dimension as a k-vector space. By contrast, the ring of regular functions on the affine line over k is the polynomial ring k[x], which does not have finite dimension as a k-vector space.</li>

<li>There is also a slightly stronger statement of this:(<a href="#CITEREFEGA_III">EGA III</a>, 3.2.4) let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to S}">

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

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</math><img src="./27e8d3408b4c1cf2726538fe710475aad1dda9ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.406ex; height:2.509ex;" alt="{\displaystyle f\colon X\to S}" loading="lazy"> be a morphism of finite type, S locally noetherian and <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><img src="./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"> a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

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

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</math><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">-module. If the support of F is proper over S, then for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle i\geq 0}</annotation>

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</math><img src="./405e1424cb9c4fc171c433a8e8f04b3e5938e366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.063ex; height:2.343ex;" alt="{\displaystyle i\geq 0}" loading="lazy"> the <a href="Higher_direct_image" class="mw-redirect" title="Higher direct image">higher direct image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{i}f_{*}F}">

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<annotation encoding="application/x-tex">{\displaystyle R^{i}f_{*}F}</annotation>

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</math><img src="./0f2bfdcabc69c17e4c4b9dcba6ccdd12e835b03b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.498ex; height:3.009ex;" alt="{\displaystyle R^{i}f_{*}F}" loading="lazy"> is coherent.</li>

<li>For a scheme X of finite type over the complex numbers, the set X(C) of complex points is a <a href="Complex_analytic_space" class="mw-redirect" title="Complex analytic space">complex analytic space</a>, using the classical (Euclidean) topology. For X and Y separated and of finite type over C, a morphism f: X → Y over C is proper if and only if the continuous map f: X(C) → Y(C) is proper in the sense that the inverse image of every compact set is compact.<a href="#cite_note-10">[10]</a></li>

<li>If f: X→Y and g: Y→Z are such that gf is proper and g is separated, then f is proper. This can for example be easily proven using the following criterion.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Valuative_criterion_of_properness">Valuative criterion of properness</h2></div>

There is a very intuitive criterion for properness which goes back to <a href="Claude_Chevalley" title="Claude Chevalley">Chevalley</a>. It is commonly called the valuative criterion of properness. Let f: X → Y be a morphism of finite type of <a href="Noetherian_scheme" title="Noetherian scheme">Noetherian schemes</a>. Then f is proper if and only if for all <a href="Discrete_valuation_ring" title="Discrete valuation ring">discrete valuation rings</a> R with <a href="Field_of_fractions" title="Field of fractions">fraction field</a> K and for any K-valued point x ∈ X(K) that maps to a point f(x) that is defined over R, there is a unique lift of x to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {x}}\in X(R)}">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {x}}\in X(R)}</annotation>

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</math><img src="./9d1dcf3bbd6117721b77427119e1997409fb84be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.839ex; height:2.843ex;" alt="{\displaystyle {\overline {x}}\in X(R)}" loading="lazy">. (EGA II, 7.3.8). More generally, a quasi-separated morphism f: X → Y of finite type (note: finite type includes quasi-compact) of 'any' schemes X, Y is proper if and only if for all <a href="Valuation_ring" title="Valuation ring">valuation rings</a> R with <a href="Field_of_fractions" title="Field of fractions">fraction field</a> K and for any K-valued point x ∈ X(K) that maps to a point f(x) that is defined over R, there is a unique lift of x to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {x}}\in X(R)}">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {x}}\in X(R)}</annotation>

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</math><img src="./9d1dcf3bbd6117721b77427119e1997409fb84be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.839ex; height:2.843ex;" alt="{\displaystyle {\overline {x}}\in X(R)}" loading="lazy">. (Stacks project Tags 01KF and 01KY). Noting that Spec K is the <a href="Generic_point" title="Generic point">generic point</a> of Spec R and discrete valuation rings are precisely the <a href="Regular_local_ring" title="Regular local ring">regular</a> <a href="Local_ring" title="Local ring">local</a> one-dimensional rings, one may rephrase the criterion: given a regular curve on Y (corresponding to the morphism s: Spec R → Y) and given a lift of the generic point of this curve to X, f is proper if and only if there is exactly one way to complete the curve.

Similarly, f is separated if and only if in every such diagram, there is at most one lift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {x}}\in X(R)}">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {x}}\in X(R)}</annotation>

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</math><img src="./9d1dcf3bbd6117721b77427119e1997409fb84be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.839ex; height:2.843ex;" alt="{\displaystyle {\overline {x}}\in X(R)}" loading="lazy">.

For example, given the valuative criterion, it becomes easy to check that projective space Pn is proper over a field (or even over Z). One simply observes that for a discrete valuation ring R with fraction field K, every K-point [x0,...,xn] of projective space comes from an R-point, by scaling the coordinates so that all lie in R and at least one is a unit in R.

<div class="mw-heading mw-heading3"><h3 id="Geometric_interpretation_with_disks">Geometric interpretation with disks</h3></div>

One of the motivating examples for the valuative criterion of properness is the interpretation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}(\mathbb {C} [[t]])}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}(\mathbb {C} [[t]])}</annotation>

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</math><img src="./fc33c099b18393b811b4bc4b36b47da7e99dfdf7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.564ex; height:2.843ex;" alt="{\displaystyle {\text{Spec}}(\mathbb {C} [[t]])}" loading="lazy"> as an infinitesimal disk, or complex-analytically, as the disk <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =\{x\in \mathbb {C} :|x|<1\}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \Delta =\{x\in \mathbb {C} :|x|<1\}}</annotation>

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</math><img src="./bcaae8217ee02fa85b4219dec9940552fc51ddc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.029ex; height:2.843ex;" alt="{\displaystyle \Delta =\{x\in \mathbb {C} :|x|<1\}}" loading="lazy">. This comes from the fact that every power series<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=\sum _{n=0}^{\infty }a_{n}t^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle f(t)=\sum _{n=0}^{\infty }a_{n}t^{n}}</annotation>

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</math><img src="./2a0aeba72d55e62b7a80027ae72840ee162a3b2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.274ex; height:6.843ex;" alt="{\displaystyle f(t)=\sum _{n=0}^{\infty }a_{n}t^{n}}" loading="lazy"></blockquote>converges in some disk of radius <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><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> around the origin. Then, using a change of coordinates, this can be expressed as a power series on the unit disk. Then, if we invert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, this is the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} [[t]][t^{-1}]=\mathbb {C} ((t))}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [[t]][t^{-1}]=\mathbb {C} ((t))}</annotation>

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</math><img src="./c613740aad26856289780a3dc2ae2c48ff8d57f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.806ex; height:3.176ex;" alt="{\displaystyle \mathbb {C} [[t]][t^{-1}]=\mathbb {C} ((t))}" loading="lazy"> which are the power series which may have a pole at the origin. This is represented topologically as the open disk <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{*}=\{x\in \mathbb {C} :0<|x|<1\}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Delta ^{*}=\{x\in \mathbb {C} :0<|x|<1\}}</annotation>

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</math><img src="./463659039e0f69384323b6e66ece7fafe7b25a47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.344ex; height:2.843ex;" alt="{\displaystyle \Delta ^{*}=\{x\in \mathbb {C} :0<|x|<1\}}" loading="lazy"> with the origin removed. For a morphism of schemes over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}(\mathbb {C} )}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}(\mathbb {C} )}</annotation>

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</math><img src="./866f765542cf4a7d539c38512286e7ed9a5d7bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.137ex; height:2.843ex;" alt="{\displaystyle {\text{Spec}}(\mathbb {C} )}" loading="lazy">, this is given by the commutative diagram<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\Delta ^{*}&\to &X\\\downarrow &&\downarrow \\\Delta &\to &Y\end{matrix}}}">

<mtr>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\Delta ^{*}&\to &X\\\downarrow &&\downarrow \\\Delta &\to &Y\end{matrix}}}</annotation>

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</math><img src="./244fa6700ac10d5f95d71e364f8fd645e27efda5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:12.691ex; height:9.176ex;" alt="{\displaystyle {\begin{matrix}\Delta ^{*}&\to &X\\\downarrow &&\downarrow \\\Delta &\to &Y\end{matrix}}}" loading="lazy"></blockquote>Then, the valuative criterion for properness would be a filling in of the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in \Delta }">

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<annotation encoding="application/x-tex">{\displaystyle 0\in \Delta }</annotation>

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</math><img src="./5afe183fd73c2f4251dcea5218ea4b4d53dfc716.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.939ex; height:2.176ex;" alt="{\displaystyle 0\in \Delta }" loading="lazy"> in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{*}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Delta ^{*}}</annotation>

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</math><img src="./fd5dc90a345f2d07efe43fabca8968eea534212a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.99ex; height:2.343ex;" alt="{\displaystyle \Delta ^{*}}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Example">Example</h4></div>

It's instructive to look at a counter-example to see why the valuative criterion of properness should hold on spaces analogous to closed compact manifolds. If we take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\mathbb {P} ^{1}-\{x\}}">

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<annotation encoding="application/x-tex">{\displaystyle X=\mathbb {P} ^{1}-\{x\}}</annotation>

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</math><img src="./9594f7164c58456a9ebf04153365d31f233aaae9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.048ex; height:3.176ex;" alt="{\displaystyle X=\mathbb {P} ^{1}-\{x\}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y={\text{Spec}}(\mathbb {C} )}">

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<annotation encoding="application/x-tex">{\displaystyle Y={\text{Spec}}(\mathbb {C} )}</annotation>

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</math><img src="./f0f7c6436a04cf9da4f97c584c09b67747e96811.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.009ex; height:2.843ex;" alt="{\displaystyle Y={\text{Spec}}(\mathbb {C} )}" loading="lazy">, then a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}(\mathbb {C} ((t)))\to X}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}(\mathbb {C} ((t)))\to X}</annotation>

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</math><img src="./9d2266df75ea7c21d128df3b46e4c8d3089fe13d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.189ex; height:2.843ex;" alt="{\displaystyle {\text{Spec}}(\mathbb {C} ((t)))\to X}" loading="lazy"> factors through an affine chart of <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><img src="./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">, reducing the diagram to<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{Spec}}(\mathbb {C} ((t)))&\to &{\text{Spec}}(\mathbb {C} [t,t^{-1}])\\\downarrow &&\downarrow \\{\text{Spec}}(\mathbb {C} [[t]])&\to &{\text{Spec}}(\mathbb {C} )\end{matrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{Spec}}(\mathbb {C} ((t)))&\to &{\text{Spec}}(\mathbb {C} [t,t^{-1}])\\\downarrow &&\downarrow \\{\text{Spec}}(\mathbb {C} [[t]])&\to &{\text{Spec}}(\mathbb {C} )\end{matrix}}}</annotation>

</semantics>

</math><img src="./c90137402723316fe7a1968b3e824759155f5242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:34.793ex; height:9.509ex;" alt="{\displaystyle {\begin{matrix}{\text{Spec}}(\mathbb {C} ((t)))&\to &{\text{Spec}}(\mathbb {C} [t,t^{-1}])\\\downarrow &&\downarrow \\{\text{Spec}}(\mathbb {C} [[t]])&\to &{\text{Spec}}(\mathbb {C} )\end{matrix}}}" loading="lazy"></blockquote>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}(\mathbb {C} [t,t^{-1}])=\mathbb {A} ^{1}-\{0\}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}(\mathbb {C} [t,t^{-1}])=\mathbb {A} ^{1}-\{0\}}</annotation>

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</math><img src="./9131c02b021a792e30c207d22c47ecb6dc7cac31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.635ex; height:3.176ex;" alt="{\displaystyle {\text{Spec}}(\mathbb {C} [t,t^{-1}])=\mathbb {A} ^{1}-\{0\}}" loading="lazy"> is the chart centered around <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><img src="./a120eeb8a091b516595765bd08b306f2394e7721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</mrow>

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

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</math><img src="./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">. This gives the commutative diagram of commutative algebras<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\mathbb {C} ((t))&\leftarrow &\mathbb {C} [t,t^{-1}]\\\uparrow &&\uparrow \\\mathbb {C} [[t]]&\leftarrow &\mathbb {C} \end{matrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\mathbb {C} ((t))&\leftarrow &\mathbb {C} [t,t^{-1}]\\\uparrow &&\uparrow \\\mathbb {C} [[t]]&\leftarrow &\mathbb {C} \end{matrix}}}</annotation>

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</math><img src="./f76b0e37164b97cc222831de9f43381d60bd05e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:21.874ex; height:9.509ex;" alt="{\displaystyle {\begin{matrix}\mathbb {C} ((t))&\leftarrow &\mathbb {C} [t,t^{-1}]\\\uparrow &&\uparrow \\\mathbb {C} [[t]]&\leftarrow &\mathbb {C} \end{matrix}}}" loading="lazy"></blockquote>Then, a lifting of the diagram of schemes, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}(\mathbb {C} [[t]])\to {\text{Spec}}(\mathbb {C} [t,t^{-1}])}">

</mrow>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}(\mathbb {C} [[t]])\to {\text{Spec}}(\mathbb {C} [t,t^{-1}])}</annotation>

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</math><img src="./ddbfa815e4443f9d485c98ebcbd9666192b259e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.655ex; height:3.176ex;" alt="{\displaystyle {\text{Spec}}(\mathbb {C} [[t]])\to {\text{Spec}}(\mathbb {C} [t,t^{-1}])}" loading="lazy">, would imply there is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} [t,t^{-1}]\to \mathbb {C} [[t]]}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [t,t^{-1}]\to \mathbb {C} [[t]]}</annotation>

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</math><img src="./bf19229e3d790d3f24abb9b84f0cf192c83d3199.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.737ex; height:3.176ex;" alt="{\displaystyle \mathbb {C} [t,t^{-1}]\to \mathbb {C} [[t]]}" loading="lazy"> sending <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto t}">

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

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</math><img src="./651d7383566646d4c9916878d7b2539344335e01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.293ex; height:2.009ex;" alt="{\displaystyle t\mapsto t}" loading="lazy"> from the commutative diagram of algebras. This, of course, cannot happen. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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

</semantics>

</math><img src="./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"> is not proper over <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><img src="./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">.

<div class="mw-heading mw-heading3"><h3 id="Geometric_interpretation_with_curves">Geometric interpretation with curves</h3></div>

There is another similar example of the valuative criterion of properness which captures some of the intuition for why this theorem should hold. Consider a curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

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

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> and the complement of a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C-\{p\}}">

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

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</math><img src="./d0d0968b3bf92a70676e1446be11f724a20739d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.101ex; height:2.843ex;" alt="{\displaystyle C-\{p\}}" loading="lazy">. Then the valuative criterion for properness would read as a diagram<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}C-\{p\}&\rightarrow &X\\\downarrow &&\downarrow \\C&\rightarrow &Y\end{matrix}}}">

<mtr>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}C-\{p\}&\rightarrow &X\\\downarrow &&\downarrow \\C&\rightarrow &Y\end{matrix}}}</annotation>

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</math><img src="./7ea7e7c3dcc222f00b30e024223fa6c8e4e63d00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:17.801ex; height:9.509ex;" alt="{\displaystyle {\begin{matrix}C-\{p\}&\rightarrow &X\\\downarrow &&\downarrow \\C&\rightarrow &Y\end{matrix}}}" loading="lazy"></blockquote>with a lifting of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\to X}">

</mstyle>

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

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</math><img src="./3d6ea90cf8d86ce73fbaf06f16eb97ff8862922a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.36ex; height:2.176ex;" alt="{\displaystyle C\to X}" loading="lazy">. Geometrically this means every curve in the scheme <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><img src="./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"> can be completed to a compact curve. This bit of intuition aligns with what the scheme-theoretic interpretation of a morphism of topological spaces with compact fibers, that a sequence in one of the fibers must converge. Because this geometric situation is a problem locally, the diagram is replaced by looking at the local ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{C,{\mathfrak {p}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{C,{\mathfrak {p}}}}</annotation>

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</math><img src="./422b1f1a19c11f678c6c2c2ae03687defd654088.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.611ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{C,{\mathfrak {p}}}}" loading="lazy">, which is a DVR, and its fraction field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}})}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}})}</annotation>

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</math><img src="./24febdc740eba41d4530e77fd2aa756135b60ae8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.044ex; height:3.009ex;" alt="{\displaystyle {\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}})}" loading="lazy">. Then, the lifting problem then gives the commutative diagram<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{Spec}}({\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}}))&\rightarrow &X\\\downarrow &&\downarrow \\{\text{Spec}}({\mathcal {O}}_{C,{\mathfrak {p}}})&\rightarrow &Y\end{matrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{Spec}}({\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}}))&\rightarrow &X\\\downarrow &&\downarrow \\{\text{Spec}}({\mathcal {O}}_{C,{\mathfrak {p}}})&\rightarrow &Y\end{matrix}}}</annotation>

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</math><img src="./bbc9ba77dc6529d405ec52652dc89fdfee37d43e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:27.204ex; height:9.843ex;" alt="{\displaystyle {\begin{matrix}{\text{Spec}}({\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}}))&\rightarrow &X\\\downarrow &&\downarrow \\{\text{Spec}}({\mathcal {O}}_{C,{\mathfrak {p}}})&\rightarrow &Y\end{matrix}}}" loading="lazy"></blockquote>where the scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}({\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}}))}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}({\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}}))}</annotation>

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</math><img src="./66641c5342b55e85a1bc647545476af9a18316e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.503ex; height:3.009ex;" alt="{\displaystyle {\text{Spec}}({\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}}))}" loading="lazy"> represents a local disk around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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

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</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> with the closed point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> removed.

<div class="mw-heading mw-heading2"><h2 id="Proper_morphism_of_formal_schemes">Proper morphism of formal schemes</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon {\mathfrak {X}}\to {\mathfrak {S}}}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon {\mathfrak {X}}\to {\mathfrak {S}}}</annotation>

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</math><img src="./bb33576e243e6d4cc0a1ecde12a764d84f1875e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.524ex; height:2.509ex;" alt="{\displaystyle f\colon {\mathfrak {X}}\to {\mathfrak {S}}}" loading="lazy"> be a morphism between <a href="Locally_noetherian_formal_scheme" class="mw-redirect" title="Locally noetherian formal scheme">locally noetherian formal schemes</a>. We say f is proper or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {X}}}">

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</math><img src="./073868db564a5e806f61f8b2d526ba975cb5c077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.671ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {X}}}" loading="lazy"> is proper over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {S}}}">

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

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</math><img src="./91d521f12ee3c00ee2fc7ab16af9ea17d915a750.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {S}}}" loading="lazy"> if (i) f is an <a href="Adic_morphism" class="mw-redirect" title="Adic morphism">adic morphism</a> (i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{0}\colon X_{0}\to S_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{0}\colon X_{0}\to S_{0}}</annotation>

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</math><img src="./7a4bdd17d37c512ef56d0e5b272a06b32dfca68b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.299ex; height:2.509ex;" alt="{\displaystyle f_{0}\colon X_{0}\to S_{0}}" loading="lazy"> is proper, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/I),S_{0}=({\mathfrak {S}},{\mathcal {O}}_{\mathfrak {S}}/K),I=f^{*}(K){\mathcal {O}}_{\mathfrak {X}}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{0}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/I),S_{0}=({\mathfrak {S}},{\mathcal {O}}_{\mathfrak {S}}/K),I=f^{*}(K){\mathcal {O}}_{\mathfrak {X}}}</annotation>

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</math><img src="./c39a86ca21770884f383307f5ae4a1da991ffe20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.062ex; height:2.843ex;" alt="{\displaystyle X_{0}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/I),S_{0}=({\mathfrak {S}},{\mathcal {O}}_{\mathfrak {S}}/K),I=f^{*}(K){\mathcal {O}}_{\mathfrak {X}}}" loading="lazy"> and K is the ideal of definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {S}}}">

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

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</math><img src="./91d521f12ee3c00ee2fc7ab16af9ea17d915a750.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {S}}}" loading="lazy">.(<a href="#CITEREFEGA_III">EGA III</a>, 3.4.1) The definition is independent of the choice of K.

For example, if g: Y → Z is a proper morphism of locally noetherian schemes, Z0 is a closed subset of Z, and Y0 is a closed subset of Y such that g(Y0) ⊂ Z0, then the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {g}}\colon Y_{/Y_{0}}\to Z_{/Z_{0}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {g}}\colon Y_{/Y_{0}}\to Z_{/Z_{0}}}</annotation>

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</math><img src="./68b8659d80e00b63138fd6dc491fa2295ea38c26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.667ex; height:3.009ex;" alt="{\displaystyle {\widehat {g}}\colon Y_{/Y_{0}}\to Z_{/Z_{0}}}" loading="lazy"> on formal completions is a proper morphism of formal schemes.

Grothendieck proved the coherence theorem in this setting. Namely, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon {\mathfrak {X}}\to {\mathfrak {S}}}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon {\mathfrak {X}}\to {\mathfrak {S}}}</annotation>

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</math><img src="./bb33576e243e6d4cc0a1ecde12a764d84f1875e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.524ex; height:2.509ex;" alt="{\displaystyle f\colon {\mathfrak {X}}\to {\mathfrak {S}}}" loading="lazy"> be a proper morphism of locally noetherian formal schemes. If F is a coherent sheaf on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {X}}}">

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

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</math><img src="./073868db564a5e806f61f8b2d526ba975cb5c077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.671ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {X}}}" loading="lazy">, then the higher direct images <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{i}f_{*}F}">

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<annotation encoding="application/x-tex">{\displaystyle R^{i}f_{*}F}</annotation>

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</math><img src="./0f2bfdcabc69c17e4c4b9dcba6ccdd12e835b03b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.498ex; height:3.009ex;" alt="{\displaystyle R^{i}f_{*}F}" loading="lazy"> are coherent.<a href="#cite_note-11">[11]</a>

<ul><li><a href="Proper_base_change_theorem" class="mw-redirect" title="Proper base change theorem">Proper base change theorem</a></li>

<li><a href="Stein_factorization" title="Stein factorization">Stein factorization</a></li></ul>

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

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> Hartshorne (1977), Appendix B, Example 3.4.1.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Liu (2002), Lemma 3.3.17.

</li>

/* start https://en.wikipedia.org/ */

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</style><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/tag/02YJ">Stacks Project, Tag 02YJ</a></cite>.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Grothendieck, EGA IV, Part 4, Corollaire 18.12.4; <cite class="citation cs2"><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/tag/02LQ">Stacks Project, Tag 02LQ</a></cite>.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Grothendieck, EGA IV, Part 3, Théorème 8.11.1.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite class="citation cs2"><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/tag/01W0">Stacks Project, Tag 01W0</a></cite>.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite class="citation cs2"><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/tag/03GX">Stacks Project, Tag 03GX</a></cite>.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> Grothendieck, EGA II, Corollaire 5.6.2.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> Conrad (2007), Theorem 4.1.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFSGA_1">SGA 1</a>, XII Proposition 3.2.

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> Grothendieck, EGA III, Part 1, Théorème 3.4.2.

</li>

</ol></div>

<ul><li><cite id="CITEREFSGA_1">SGA1 Revêtements étales et groupe fondamental, 1960–1961 (Étale coverings and the fundamental group), Lecture Notes in Mathematics 224, 1971</cite></li>

<li><cite id="CITEREFConrad2007" class="citation cs2"><a href="Brian_Conrad" title="Brian Conrad">Conrad, Brian</a> (2007), <a rel="nofollow" class="external text" href="http://math.stanford.edu/~conrad/papers/nagatafinal.pdf">"Deligne's notes on Nagata compactifications"</a> (PDF), Journal of the Ramanujan Mathematical Society, 22: 205–257, <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=2356346">2356346</a></cite></li>

<li><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__8__5_0">"Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes"</a>. <a href="Publications_Math%C3%A9matiques_de_l'IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a>. 8: 5–222. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02699291">10.1007/bf02699291</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=0217084">0217084</a>.</cite>, section 5.3. (definition of properness), section 7.3. (valuative criterion of properness)</li>

<li><cite id="CITEREFEGA_III"><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>. <a href="Publications_Math%C3%A9matiques_de_l'IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a>. 11: 5–167. <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="CITEREFGrothendieckDieudonné1966" 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> (1966). <a rel="nofollow" class="external text" href="http://www.numdam.org/articles/PMIHES_1966__28__5_0">"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie"</a>. <a href="Publications_Math%C3%A9matiques_de_l'IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a>. 28: 5–255. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02684343">10.1007/bf02684343</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=0217086">0217086</a>.</cite>, section 15.7. (generalizations of valuative criteria to not necessarily noetherian schemes)</li>

<li><cite id="CITEREFGrothendieckDieudonné1967" 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> (1967). <a rel="nofollow" class="external text" href="http://www.numdam.org/articles/PMIHES_1967__32__5_0">"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie"</a>. <a href="Publications_Math%C3%A9matiques_de_l'IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a>. 32: 5–361. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02732123">10.1007/bf02732123</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=0238860">0238860</a>.</cite></li>

<li><cite id="CITEREFHartshorne1977" class="citation cs2"><a href="Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977), <a href="Algebraic_Geometry_(book)" title="Algebraic Geometry (book)">Algebraic Geometry</a>, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <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="CITEREFLiu2002" class="citation cs2">Liu, Qing (2002), Algebraic geometry and arithmetic curves, Oxford: <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780191547805</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=1917232">1917232</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFV.I._Danilov2001" class="citation cs2">V.I. Danilov (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Proper_morphism">"Proper morphism"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><cite id="CITEREFThe_Stacks_Project_Authors" class="citation cs2">The <a href="Stacks_Project" title="Stacks Project">Stacks Project</a> Authors, <a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/">The Stacks Project</a></cite></li></ul></div><div><div class="zim-footer">

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