The Stacks project

Lemma 36.36.6. Let $X$ be a quasi-compact, regular scheme with affine diagonal. Then $X$ has the resolution property.

Proof. Observe that $X$ is a finite disjoint union of integral schemes (Properties, Lemmas 28.9.4 and 28.7.6). Thus we may assume that $X$ is integral as well as Noetherian, regular, and having affine diagonal. Choose an affine open covering $X = U_1 \cup \ldots \cup U_ m$. We may and do assume $U_ j$ nonempty for all $j$. By More on Algebra, Lemma 15.118.2 the local rings of $X$ are UFDs and hence by Divisors, Lemma 31.16.7 we can find an effective Cartier divisors $D_ j \subset X$ whose complement is $U_ j$. Then the ideal sheaf of $D_ j$ is invertible, hence a finite locally free module and we conclude that $X$ has the resolution property by Lemma 36.36.5. $\square$


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