Proper Morphism - Properties and Characterizations of Proper Morphisms

Properties and Characterizations of Proper Morphisms

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

• Properness is a local property on the base, i.e. 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.
• Proper morphisms are stable under base change and composition.
• Closed immersions are proper.
• More generally, finite morphisms are proper. This is a consequence of the going up theorem.
• Conversely, every quasi-finite, locally of finite presentation and proper morphism is finite. (EGA III, 4.4.2 in the noetherian case and EGA IV, 8.11.1 for the general case)
• Stein factorization theorem states that any proper morphism to a locally noetherian scheme can be factorized into, where the first morphism has geometrically connected fibers and the second on is finite.
• Proper morphisms are closely related to projective morphisms: If f is proper over a noetherian base Y, then there is a morphism: g: X' →X which is an isomorphism when restricted to a suitable open dense subset: g-1(U) ≅ U, such that f' := fg is projective. This statement is called Chow's lemma.
• Nagata's compactification theorem says that a separated morphism of finite type between quasi-compact and quasi-separated schemes (e.g., noetherian schemes) factors as an open immersion followed by a proper morphism.
• Proper morphisms between locally noetherian schemes or complex analytic spaces preserve coherent sheaves, in the sense that the higher direct images Rif_∗(F) (in particular the direct image f_∗(F)) of a coherent sheaf F are coherent (EGA III, 3.2.1). This boils down to the fact that the cohomology groups of projective space over some field k with respect to coherent sheaves are finitely generated over k, a statement which fails for non-projective varieties: consider C∗, the punctured disc and its sheaf of holomorphic functions . Its sections is the ring of Laurent polynomials, which is infinitely generated over C.
• There is also a slightly stronger statement of this:(EGA III, 3.2.4) let be a morphism of finite type, S locally noetherian and a -module. If the support of F is proper over S, then for each the higher direct image is coherent.:
• (SGA 1, XII) If X, Y are schemes of locally of finite type over the field of complex numbers, f induces a morphism of complex-analytic spaces

between their sets of complex points with their complex topology. (This is an instance of GAGA). Then f is a proper morphism defined above if and only if is a proper map in the sense of Bourbaki and is separated.

• 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