PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (192)
Orphanage
Unclass'd
Unproven (496)
Corrections (23)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: Low
coherent sheaf (Definition)

Let $ R$ be a ring with unity, and $ X = \mathrm{Spec}\, R$ be its prime spectrum. Given an $ R$-module $ M$, one can define a presheaf on $ X$ by defining its sections on an open set $ U$ to be $ \O _X(U)\otimes_R M$. We call the sheafification of this $ \tilde M$, and a sheaf of this form on $ X$ is called quasi-coherent. If $ M$ is a finitely generated module, then $ \tilde{M}$ is called coherent. A sheaf on an arbitrary scheme $ X$ is called (quasi-)coherent if it is (quasi-)coherent on each open affine subset of $ X$.



"coherent sheaf" is owned by PrimeFan. [ full author list (2) | owner history (2) ]
(view preamble)

View style:

Other names:  quasi-coherent sheaf
Log in to rate this entry.
(view current ratings)

Cross-references: subset, open, scheme, finitely generated module, sheaf, sheafification, open set, sections, presheaf, prime spectrum, ring with unity
There are 6 references to this entry.

This is version 8 of coherent sheaf, born on 2003-08-15, modified 2007-01-16.
Object id is 4596, canonical name is CoherentSheaf.
Accessed 4088 times total.

Classification:
AMS MSC: 14A15 (Algebraic geometry :: Foundations :: Schemes and morphisms)
Pending Errata and Addenda
None.
[ View all 4 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)