Proper morphism

In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.

Some authors call a proper variety over a field k a complete variety. For example, every projective variety over a field k is proper over k. A scheme X of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space X(C) of complex points with the classical (Euclidean) topology is compact and Hausdorff.

A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite.

The composition of two proper morphisms is proper.

Any 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.

base change

Properness is a on the base (in the Zariski topology). That is, if Y is covered by some open subschemes Y_i and the restriction of f to all f⁻¹(Y_i) is proper, then so is f.

local property

More strongly, properness is local on the base in the . 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 X_E is proper over E.^[3]

fpqc topology

are proper.

Closed immersions

More generally, finite morphisms are proper. This is a consequence of the theorem.

going up

By , a morphism of schemes is finite if and only if it is proper and quasi-finite.^[4] This had been shown by Grothendieck if the morphism f: X → Y is locally of finite presentation, which follows from the other assumptions if Y is noetherian.^[5]

Deligne

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. 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.

[6]

The 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.^[7]

Stein factorization

says that proper morphisms are closely related to projective morphisms. One version is: if X is proper over a quasi-compact 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.^[8]

Chow's lemma

as generalized by Deligne, says that a separated morphism of finite type between quasi-compact and quasi-separated schemes factors as an open immersion followed by a proper morphism.^[9]

Nagata's compactification theorem

Proper morphisms between locally noetherian schemes preserve coherent sheaves, in the sense that the Rⁱf_∗(F) (in particular the direct image f_∗(F)) of a coherent sheaf F are coherent (EGA III, 3.2.1). (Analogously, for a proper map between complex analytic spaces, Grauert and Remmert 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.

higher direct images

There is also a slightly stronger statement of this:(, 3.2.4) let $f\colon X\to S$ be a morphism of finite type, S locally noetherian and $F$ a ${\mathcal {O}}_{X}$ -module. If the support of F is proper over S, then for each $i\geq 0$ the higher direct image $R^{i}f_{*}F$ is coherent.

EGA III

For a scheme X of finite type over the complex numbers, the set X(C) of complex points is 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.^[10]

complex analytic space

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.

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

Proper morphism of formal schemes[edit]

Let $f\colon {\mathfrak {X}}\to {\mathfrak {S}}$ be a morphism between locally noetherian formal schemes. We say f is proper or ${\mathfrak {X}}$ is proper over ${\mathfrak {S}}$ if (i) f is an adic morphism (i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map $f_{0}\colon X_{0}\to S_{0}$ is proper, where $X_{0}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/I),S_{0}=({\mathfrak {S}},{\mathcal {O}}_{\mathfrak {S}}/K),I=f^{*}(K){\mathcal {O}}_{\mathfrak {X}}$ and K is the ideal of definition of ${\mathfrak {S}}$ .(EGA III, 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 ${\widehat {g}}\colon Y_{/Y_{0}}\to Z_{/Z_{0}}$ on formal completions is a proper morphism of formal schemes.

Grothendieck proved the coherence theorem in this setting. Namely, let $f\colon {\mathfrak {X}}\to {\mathfrak {S}}$ be a proper morphism of locally noetherian formal schemes. If F is a coherent sheaf on ${\mathfrak {X}}$ , then the higher direct images $R^{i}f_{*}F$ are coherent.^[11]

Proper base change theorem

Stein factorization

SGA1 Revêtements étales et groupe fondamental, 1960–1961 (Étale coverings and the fundamental group), Lecture Notes in Mathematics 224, 1971

(2007), "Deligne's notes on Nagata compactifications" (PDF), Journal of the Ramanujan Mathematical Society, 22: 205–257, MR 2356346

Conrad, Brian

; Dieudonné, Jean (1961). "Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS. 8: 5–222. doi:10.1007/bf02699291. MR 0217084., section 5.3. (definition of properness), section 7.3. (valuative criterion of properness)

Grothendieck, Alexandre

; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11: 5–167. doi:10.1007/bf02684274. MR 0217085.

Grothendieck, Alexandre

; Dieudonné, Jean (1966). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie". Publications Mathématiques de l'IHÉS. 28: 5–255. doi:10.1007/bf02684343. MR 0217086., section 15.7. (generalizations of valuative criteria to not necessarily noetherian schemes)

Grothendieck, Alexandre

; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32: 5–361. doi:10.1007/bf02732123. MR 0238860.

Grothendieck, Alexandre

(1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

Hartshorne, Robin

Liu, Qing (2002), Algebraic geometry and arithmetic curves, Oxford: , ISBN 9780191547805, MR 1917232

Proper morphism

base change

local property

fpqc topology

Closed immersions

going up

Deligne

[6]

Stein factorization

Chow's lemma

Nagata's compactification theorem

higher direct images

EGA III

complex analytic space

Proper morphism of formal schemes[edit]

Proper base change theorem

Stein factorization

Conrad, Brian

Grothendieck, Alexandre

Grothendieck, Alexandre

Grothendieck, Alexandre

Grothendieck, Alexandre

Hartshorne, Robin

Oxford University Press

"Proper morphism"

The Stacks Project