Talk:Coherent sheaf

Untitled
I think the definition given for a coherent sheaf on a ringed space is wrong, or at least, disagrees with Grothendieck's definition in Éléments de Géométrie Algébrique, (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.

To see that these notions are not equivalent, take a commutative ring A having an element a whose annihilator is not finitely generated. Then if X is Spec A, the sheaf OX 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 a should be a finitely generated ideal. Namely, consider the morphism from OX to itself given on global sections by multiplication by a. An example of such a pair A, a is given by the element x in Z[x,y1, y2,...]/(xy1, xy2,...). 136.152.196.72 10:05, 30 January 2007 (UTC)


 * 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. Stca74 09:28, 15 May 2007 (UTC)

complex space
"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."

What is a complex space? Does it mean a complex vector space? --Acepectif 05:49, 6 July 2007 (UTC)


 * I think what it means is complex manifold. 131.111.24.224 14:56, 6 August 2007 (UTC)
 * Oops, forgot to log in. The last comment was by me: Artie P.S. 14:57, 6 August 2007 (UTC)


 * 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 functions which define our set are linearly independent. —Preceding unsigned comment added by 85.69.129.193 (talk) 12:41, 16 May 2009 (UTC)

O_X coherent over itself?
I'm no expert, but want to draw attention to the correspondig nLab article (section "Properties") which says (emphasis mine):


 * For a coherent sheaf ℰ 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 ℰ(V) of sections of ℰ 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 true on a general scheme or general analytic space; sometimes even the structure sheaf O itself is a counterexample (not coherent while finitely presented).

On the other hand, our article says that O is always coherent over itself. — Preceding unsigned comment added by 79.219.172.73 (talk) 05:17, 19 April 2012 (UTC)


 * Our article says it's coherent over itself when X 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 mentioning this caveat in the article. RobHar (talk) 17:05, 19 April 2012 (UTC)


 * 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. -- Taku (talk) 11:03, 10 May 2012 (UTC)

Assessment comment
Substituted at 01:53, 5 May 2016 (UTC)

Links
"Algebraic vector bundle" redirects here. BTotaro (talk) 17:24, 20 May 2016 (UTC)

"Holomorphic sheaf" redirects here. BTotaro (talk) 05:57, 31 July 2016 (UTC)