Pseudo-Coherence and Quasi-Properness

Let X be a scheme. A complex F D (X) is pseudo-coherent if each X e X has a neighborhood in which F is D-isomorphic to a bounded-above complex of finite-rank free Oa -modules [I, p. 175, 2.2.10]. If X is divisorial, and either separated or noetherian, such an F is (globally) D(X)-isomorphic to a bounded-above complex of finite-rank locally free Ox-modules [ibid., p. 174, Cor. 2.2.9]. If Ox is coherent, pseudo-coherence of F means simply that F has coherent homology [ibid., p. 115, Cor. 3.5 b)]. If X is noetherian, 

A scheme-map f X — y is pseudo-coherent if it factors locally as f = poi where i U Z U open in X) is a closed immersion such that is pseudo-coherent on Z, and p Z Y is smooth [ibid., p.228, Def. 1.2]. Pseudo-coherent maps are locally finitely-presentable (smooth maps being so by definition). 

For example, any smooth map is pseudo-coherent, any regular immersion (= closed immersion corresponding to a quasi-coherent ideal generated locally by a regular sequence) is pseudo-coherent, and any composition of pseudo-coherent maps is still pseudo-coherent [ibid., p. 236, Cor. 1.14].  

If / X — y is a proper map, and is an /-ample invertible sheaf, then / is pseudo-coherent if and only if the Ov-complex R/ ( ) is pseudo-coherent for all n 0. (The proof is indicated below, in (4.3.8)). In particular, a finite map / X — y is pseudo-coherent if and only if / C x is a pseudo-coherent Oy-module. 

For noetherian Y, any finite-type map / X — y is pseudo-coherent. Pseudo-coherence persists under tor-independent base change [I, p. 233, Cor. 1.10]. Hence, by descent to the noetherian case [EGA, IV, (11.2.7) and its proof], any flat finitely-presentable scheme-map is pseudo-coherent. 

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