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Given a variety $X$, is the constant sheaf given by the function field $K=k(X)$ a quasicoherent sheaf?

I think so because to each affine open we assign K, and then for a localization $A \to A_f$ the map $K \otimes_A A_f \to K$ is an isomorphism.

But I don’t find this stated anywhere.

usr0192
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  • Your proof seems fine to me. Is there a context you have where you want to exploit this sheaf being coherent? – Daniel Mar 21 '24 at 00:08
  • Well it has come up over the years a few times and I’ve never seen it stated. But recently I was reading Cutkosky’s Intro to algebraic geometry, he proves $K$ has no higher (meaning deg $\geq 1$) Cech cohomology, but he only states (Lemma 17.15) that sheaf cohomology in deg 1 vanishes. If $K$ is quasicoherent then since Cech is same as sheaf cohomology in this case i was wondering why he didn’t state that higher sheaf cohomology of K vanishes – usr0192 Mar 21 '24 at 01:40

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Another way you can think about this, is that one has a natural morphism $j: \operatorname{Spec} K \to X$ picking out the generic point and the sheaf in question is $j_*\mathcal{O}_{\operatorname{Spec} K}$, which is quasicoherent.

Daniel
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