View source for Proper morphism

A morphism of schemes that is separated, universally closed and of finite type. A morphism of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754501.png" /> is called closed if for any closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754502.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754503.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754504.png" />, and universally closed if for any [[Base change|base change]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754505.png" /> the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754506.png" /> is closed. The property of being a proper morphism is preserved under composition, base change and taking Cartesian products. Proper morphisms are closely related to projective morphisms: any projective morphism is proper, and a proper quasi-projective morphism is projective. Any proper morphism is dominated by a projective one (Chow's lemma). See also [[Complete algebraic variety|Complete algebraic variety]]; [[Projective scheme|Projective scheme]].

Proper morphisms have a number of good cohomological properties. 1) If a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754507.png" /> is proper and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754508.png" /> is a [[Coherent sheaf|coherent sheaf]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754509.png" />-modules, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545010.png" /> the sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545011.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545012.png" /> are coherent (the finiteness theorem). A similar fact holds for étale cohomology. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545013.png" /> is a complete scheme over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545014.png" />, then the cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545015.png" /> are finite-dimensional. 2) For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545016.png" />, the completion of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545017.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545018.png" /> coincides with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545019.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545020.png" /> is the ideal of the subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545022.png" /> (the comparison theorem). 3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545023.png" /> is a proper scheme over a complete local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545024.png" />, then the categories of coherent sheaves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545025.png" /> and on its formal completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545026.png" /> are equivalent (the algebraization theorem). There are analytic analogues of the first and third properties. For example (see [[#References|[3]]]), for a complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545027.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545028.png" /> any [[Coherent analytic sheaf|coherent analytic sheaf]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545029.png" /> is algebraizable and

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4) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545031.png" /> be a proper morphism, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545032.png" /> be a sheaf of finite Abelian groups in the étale topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545033.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545034.png" /> be a geometric point of the [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545035.png" />. Then the fibre of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545036.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545037.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545038.png" /> (the base-change theorem, see [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   J. Dieudonné,   "Eléments de géometrie algébrique"  ''Publ. Math. IHES'' , '''2–3'''  (1961–1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  "Théorie des topos et cohomologie étale des schémas"  M. Artin (ed.)  A. Grothendieck (ed.)  J.-L. Verdier (ed.) , ''SGA 4'' , ''Lect. notes in math.'' , '''269; 270; 305''' , Springer  (1972–1973)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Grothendieck (ed.)  et al. (ed.) , ''Revêtements étales et groupe fondamental. SGA 1'' , ''Lect. notes in math.'' , '''224''' , Springer  (1971)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)</TD></TR></table>



====Comments====
A morphism of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545039.png" /> is locally of finite type if there exists a covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545040.png" /> by affine open subschemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545041.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545042.png" /> there is an open covering by affine subschemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545044.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545045.png" /> is a finitely-generated algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545046.png" /> (with respect to the homomorphism of rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545047.png" /> which defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545048.png" />). The morphism is of finite type if the coverings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545049.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545050.png" /> can be taken finite for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545051.png" />.

A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545052.png" /> is finite if there exists an affine open covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545054.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545056.png" /> is affine for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545057.png" />, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545058.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545059.png" /> is a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545060.png" />-module.

The analytic analogue of property 1) above is called Grauert's finiteness theorem, see [[Finiteness theorems|Finiteness theorems]].

In topology a mapping of topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545061.png" /> is said to be proper it for each topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545062.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545063.png" /> is closed. It follows that for every continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545064.png" /> the base-change mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545066.png" />, is closed, so that a proper mapping of topological spaces is the same thing as a universally closed mapping. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545067.png" /> is locally compact, a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545068.png" /> is proper if and only if the inverse image of each compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545069.png" /> is compact. Sometimes this last property is taken as a definition.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545070.png" /> be a Noetherian ring which is complete (and separated) with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545071.png" />-adic topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545072.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545073.png" />. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545074.png" /> one defines a sheaf of topological rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545075.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545076.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545077.png" />. The ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545078.png" /> is called the formal spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545079.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545080.png" />). It is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545081.png" />. A locally Noetherian formal scheme is, by definition, a topologically ringed space which is locally isomorphic to formal spectra of a Noetherian ring. Morphisms of formal schemes are morphisms of the corresponding topologically ringed spaces.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545082.png" /> be a (locally) Noetherian scheme and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545083.png" /> a closed subscheme defined by a sheaf of ideas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545084.png" />. The formal completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545085.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545086.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545087.png" />, is the topologically ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545088.png" />. It is a (locally) Noetherian formal scheme.

All this serves to state the following theorem, which is sometimes called the fundamental theorem on proper morphisms: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545089.png" /> be a proper morphism of locally Noetherian schemes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545090.png" /> a closed subscheme, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545091.png" /> the inverse image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545092.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545094.png" /> be the formal completions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545096.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545098.png" />, respectively. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545099.png" /> be the induced morphism of formal schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p075450100.png" />. Then, for every coherent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p075450101.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p075450102.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p075450103.png" />, there are canonical isomorphisms

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This theorem can be used to prove the Zariski connectedness theorem (cf. [[Zariski theorem|Zariski theorem]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. General topology" , '''Chapt. I, §10''' , Addison-Wesley  (1966)  (Translated from French)</TD></TR></table>