Posts tagged ‘local’

Extensions of sheaves and the local-global spectral sequence

GIven a topological space X, algebraic topologists would sometimes be interested in sheaves over this space. In most cases, these sheaves are sheaves of functions with some special property, or sheaves of modules over these rings. We could casually say thet the notion of sheaf is some mix of topology and algebra (more generally, categories), which allows to remember where are localized objects. For example, the set of continuous functions over a compact space can sometimes allow to recover an isomorphic space, but its mere normed algebra structure does not immediately say things like: “i am greater here than there”, “i am positive here and negative there”…

So a sheaf is not only determined by a set of fonctions (the commonly used term is section) defined over the whole space, but also by sections over open sets, and by the restriction maps between these various sets of sections. A sheaf has also a gluing theory : if we choose sections over open sets with equal restrictions on the intersections, they must glue to a section over the union of the open sets (if not, our object does not deserve to be a sheaf, so it is only a presheaf).

Virtually anything which can be restricted to open sets and defined locally defines a sheaf: continuous functions form a sheaf, but functions with ||f||_{\infty} = 1 do not usually form a sheaf (this property is not local). A sheaf of abelian groups is a sheaf of things which can be (locally) added and substracted, a sheaf of rings has sections which can be multiplied, and we can also define sheaves of modules over a sheaf of rings.

A morphism of sheaves f: F \to G is a map which is determined by the image of localized sections. In the case of abelian groups, the kernel sheaf is the sheaf of sections of F which are locally in the kernel of f, while the quotient sheaf (cokernel) is something looking like G where sections are identified if locally it should be so. A morphism is thus injective or surjective if locally it is so. Sheaves of abelian groups (or sheaves of modules over some sheaf of rings) form an abelian category, and it makes sense to speak of the extension group \mathrm{Ext}^1(F,G) for two such sheaves F and G. Suppose we are working with sheaves of modules (especially coherent sheaves) over a sheaf of rings \mathcal O_X. How should we compute this group ?


10 January 2009 at 5:02 pm Leave a comment