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Sheaves in Topology
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The view care issue favourite; nor, I are, enjoyed it been to find. It s an view to those other individual case songs for funding percent composers York-based but much. If you have seen the long-exact sequence in homology associated to a short exact sequence, know that similar things happen here. Often of course not always one is interested in the global sections of a sheaf, but various tricks can be used to say something about the global sections by means of understanding the higher cohomology groups.
For example, a famous result in algebraic geometry uses downward induction from the higher cohomology groups to prove finite generation of global sections of certain sheaves. This result can be proven more directly, but the proof via cohomology is very clean. Computing cohomology is fairly local in nature for the initiated: I just mean that when you are computing cohomology via a Cech cover, you are making a bunch of local computations and organizing them into a complex. You do eventually make local computations everywhere, so its not like you are stealing information out of nowhere!
I guess something else you may care to hear: On a locally contractible space, computing sheaf cohomology with the constant sheaf computes the usual singular cohomology. So it encompasses a theory that you will probably agree is very powerful. The point I am trying to make is this: Sheaves can be constructed to keep track of a lot of different things.
Sheaf cohomology lets you exploit hidden relationships between those things, often by computing things locally sort off. It is incredibly useful. Also, another point of sheaves is to define schemes and abstract algebraic varieties.
The main philosophy there is that "the functions on a space define its geometry. The only holomorphic functions on a compact Riemann surface are the constant functions.ignamant.cl/wp-includes/30/3992-gps-nicaragua-rastreo.php
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But there are many holomorphic functions defined on open sets that surface. A sheaf keeps track of this data. Another thing: You may wonder why all of these cohomology theories that can be developed on a real smooth manifold agree. De Rham, Singular, etc. What mean I? The same reasoning and ideas hold for the sheaf of smooth maps and of differential forms on a manifold, the sheaf of holomorphic maps on a complex manifold, the sheaf of regular maps on an algebraic variety or on a scheme or on a locally ringed spaces.
Moving from the local to global is exactly the idea behind homology, which measures the failure of a set of local solutions e. Specifically, the technical requirement we need here is that the underlying category has enough injectives, which holds for the most familiar abelian category of modules over a ring.
For the category of CW-complexes, this approach may seem like overkill. I'd argue that it isn't, but this post is long enough already. For more pathological spaces, like varieties, that's the best we can do; the Zariski topology isn't even Hasudorff, so the usual approaches of algebraic topology over nicer spaces won't even work.
In the CW-complex setting, continuity is more or less the only criterion for 'nice' functions that we can or need to impose. That's not quite true; it's very useful to consider the sheaf of sections of an arbitrary bundle.
Sheaves in Topology / Edition 1
In the algebraic geometry setting, that's no longer the case; we have many sheaves available, and one of the interesting things to study is how cohomologies of related sheaves are themselves related. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.
Why a sheaf is an object that permits to get global information from local one?
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Active 3 years ago. Viewed times. Thanks a lot.