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 O_X 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 O_X 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 [1] (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

The comment(s) below were originally left at Talk:Coherent sheaf/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

A lot remains to do here, at least: add information on the history of coherent sheaves in relation to complex and algebraic geometry expand the relationship to vector bundles (show that coherent sheaves appear as cokernels) explain the permanence properties in exact sequences treat direct and inverse images, permance for the former under properness assumption provide a brief and non-technical (if that's possible!) summary on cohomology and link to a new article on cohomology of coherent sheaves (now links here); the previous should also mention coherent duality theorems expand application and examples add references and further reading Stca74 09:43, 15 May 2007 (UTC)

Last edited at 09:43, 15 May 2007 (UTC). 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)