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For a coherent sheaf â„° over a ringed space, for every point y in the base space X there is a neighborhood V such that the O_X(V)-module â„°(V) of sections of â„° over V is finitely presented. On a noetherian scheme the notions of finitely presented and coherent sheaves of O-modules agree, but this is not
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A complex space is a generalization of a complex manifold. You make one by patching together analytic sets, that is subsets of C^n which are locally the zero set of a finite number of holomorphic functions. This differs from a submanifold in C^n because we do not insist that the differentials of the
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is a noetherian scheme. This is true. Now, in example II.5.2.1 of
Hartshorne, he claims that the structure sheaf is always coherent, which is true under his definition. He uses a different definition than EGA, but the two definitions agree for noetherian schemes, so he doesn't care. It's worth
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If I remember correctly, a theorem of Oka says O is coherent. I don't know the proof, but it is a deep theorem. So, in the analytic context, coherency cannot be trivial. Maybe this article should become a disk big page. I'm not sure if "coherent" in D-modules fit here. Just a comment. --
166:, (1971 edition, 0.5.3.1). The definition currently on this page is what Grothendieck calls a sheaf of finite presentation (0.5.2.5). The advantage of coherent sheaves is that they form a full exact abelian subcategory of the category of sheaves, while finitely presented ones do not.
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is finitely presented, but not coherent. If it were coherent, then by (0.5.3.4) (stating that the kernel of a morphism of coherent sheaves is coherent), the annihilator of
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You are right. I have corrected the definition and expanded the article a bit. A lot stillremains to be done here ; I added some to do items on the comments page.
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252:"Ideal sheaves: If Z is a closed complex subspace of a complex space X, the sheaf IZ of all holomorphic functions vanishing on Z is coherent."
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I think the definition given for a coherent sheaf on a ringed space is wrong, or at least, disagrees with
Grothendieck's definition in
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on cohomology of coherent sheaves (now links here); the previous should also mention coherent duality theorems
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structure sheaf O itself is a counterexample (not coherent while finitely presented)
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On the other hand, our article says that O is always coherent over itself.
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I'm no expert, but want to draw attention to the correspondig nLab article
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should be a finitely generated ideal. Namely, consider the morphism from
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true on a general scheme or general analytic space; sometimes even the
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To see that these notions are not equivalent, take a commutative ring
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functions which define our set are linearly independent.
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to itself given on global sections by multiplication by
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whose annihilator is not finitely generated. Then if
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explain the permanence properties in exact sequences
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Oops, forgot to log in. The last comment was by me:
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