Coherent sheaf Totally Explained

Everything about Coherent Sheaf totally explained

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometrical information. In addition, there's a related concept of quasi-coherent sheaves. Many results and properties in algebraic geometry and complex analytic geometry are both formulated in terms of coherent sheaves and their cohomology.
Coherent sheaves can be seen as a generalization of (sheaves of sections of) vector bundles. They form a category closed under usual operations such as taking kernels, cokernels and finite direct sums. In addition, under suitable compactness conditions they're preserved under maps of the underlying spaces and have finite dimensional cohomology spaces.

Definition

A coherent sheaf on a ringed space

(X,mathcal_X

. This makes consideration of them natural from the perspective of homological algebra.

Examples of coherent sheaves

On noetherian schemes, the structure sheaf O_X itself.

Sheaves of sections in vector bundles.

Ideal sheaves: If Z is a closed complex subspace of a complex analytic space X, the sheaf I_Z of all holomorphic functions vanishing on Z is coherent.

Structure sheaves of subspaces.

Coherent cohomology

The sheaf cohomology theory of coherent sheaves is called coherent cohomology. It is one of the major and most fruitful applications of sheaves, and its results connect quickly with classical theories.
   Using previous works of Schwartz, Cartan and Serre proved that compact complex manifolds have the property that their sheaf cohomology for any coherent sheaf consists of vector spaces of finite dimension. This result had been proved previously by Kodaira for the particular case of locally free sheaves. According to testimonies, it seems that at that time the usefulness of such a result was rather unclear. An algebraic version of this theorem was proved by Serre. Relative versions of this result for a proper morphism were proved, by Grothendieck in the algebraic case, and by Grauert and Remmert in the analytic case. For example Grothendieck's result concerns the functor ť Rf_*

or push-forward, in sheaf cohomology. (It is the right derived functor of the direct image of a sheaf.) For a proper morphism in the sense of scheme theory, it was shown that this functor sends coherent sheaves to coherent sheaves. The Serre result is the case of a morphism to a point.
   The duality theory in scheme theory that extends Serre duality is called coherent duality (also called Grothendieck duality). Under some mild conditions of finiteness, the sheaf of Kähler differentials on an algebraic variety is a coherent sheaf Ω¹. When the variety is non-singular its 'top' exterior power acts as the dualising object; and it's locally free (effectively it's the sheaf of sections of the cotangent bundle, when working over the complex numbers, but that's a statement that requires more precision since only holomorphic 1-forms count as sections). The successful extension of the theory beyond this case was a major step.

Further Information

Get more info on 'Coherent Sheaf'.

External Link Exchanges

Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:

<a href="http://coherent_sheaf.totallyexplained.com">Coherent sheaf Totally Explained</a>

Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned.