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9 - Curves, Sheaves, and Cohomology

Published online by Cambridge University Press:  06 July 2010

Summary

In this chapter we give a quick introduction to sheaves, Čech cohomology, and divisors on curves. The first main point is that many objects, in mathematics and in life, are defined by local information–imagine a road atlas where each page shows a state and a tiny fraction of the adjacent states. If you have two different local descriptions, how can you relate them? In the road map analogy, when you switch pages, where are you on the new page? Roughly speaking, a sheaf is a collection of local data, and cohomology is the mechanism for “gluing” local information together. The second main point is that geometric objects do not necessarily live in a fixed place. They have a life of their own, and we can embed the same object in different spaces. For an algebraic curve C, it turns out that the ways in which we can map C to ℙn are related to studying sets of points (divisors) on the curve. If the ground field is ℂ, the maximum principle tells us that there are no global holomorphic functions on C, so it is natural to consider meromorphic functions. Hence, we'll pick a bunch of points on the curve, and study functions on C with poles only at the points. Sheaves and cohomology enter the picture because, while it is easy to describe a meromorphic function locally, it is hard to get a global understanding of such things.

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Publisher: Cambridge University Press
Print publication year: 2003

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