The Stacks project

Lemma 29.13.3. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

  1. The morphism $f$ is quasi-affine.

  2. There exists an affine open covering $S = \bigcup W_ j$ such that each $f^{-1}(W_ j)$ is quasi-affine.

  3. There exists a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras $\mathcal{A}$ and a quasi-compact open immersion

    \[ \xymatrix{ X \ar[rr] \ar[rd] & & \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) \ar[dl] \\ & S & } \]

    over $S$.

  4. Same as in (3) but with $\mathcal{A} = f_*\mathcal{O}_ X$ and the horizontal arrow the canonical morphism of Constructions, Lemma 27.4.7.

Proof. It is obvious that (1) implies (2) and that (4) implies (3).

Assume $S = \bigcup _{j \in J} W_ j$ is an affine open covering such that each $f^{-1}(W_ j)$ is quasi-affine. By Schemes, Lemma 26.19.2 we see that $f$ is quasi-compact. By Schemes, Lemma 26.21.6 we see the morphism $f$ is quasi-separated. Hence by Schemes, Lemma 26.24.1 the sheaf $\mathcal{A} = f_*\mathcal{O}_ X$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras. Thus we have the scheme $g : Y = \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) \to S$ over $S$. The identity map $\text{id} : \mathcal{A} = f_*\mathcal{O}_ X \to f_*\mathcal{O}_ X$ provides, via the definition of the relative spectrum, a morphism $can : X \to Y$ over $S$, see Constructions, Lemma 27.4.7. By assumption, the lemma just cited, and Properties, Lemma 28.18.4 the restriction $can|_{f^{-1}(W_ j)} : f^{-1}(W_ j) \to g^{-1}(W_ j)$ is a quasi-compact open immersion. Thus $can$ is a quasi-compact open immersion. We have shown that (2) implies (4).

Assume (3). Choose any affine open $U \subset S$. By Constructions, Lemma 27.4.6 we see that the inverse image of $U$ in the relative spectrum is affine. Hence we conclude that $f^{-1}(U)$ is quasi-affine (note that quasi-compactness is encoded in (3) as well). Thus (3) implies (1). $\square$

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