Talk:Coherent sheaf

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Mathematics rating:	Start Class	Mid Priority	Field: Geometry
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. 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)

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)

[edit] 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)