Hi, many times on books I find the following phrase: "...by (étale) descent we can reduce to the case..." (étale can be replaced by other topologies). What does really means and why "they" can reduce proving only the "easy part" without giving other explanations? Thank you!!
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1$\begingroup$ Your question is not very precice. Are you asking what etale topology is? $\endgroup$– J.C. OttemCommented Jan 20, 2011 at 16:44
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2$\begingroup$ It's hard to give a very specific answer to your question. But descent usually means that to do something on your space $X$, it's equivalent to pick a cover $U$ of your space in whichever topology you're using, and verify that you can do that something on your cover $U$ such that it agrees on the pullback $U \underset{X}{\times} U$ and satisfies a cocycle condition on the iterated pullback (or possibly even higher coherence relations, depending on the context you're working in). For example, the sheaf condition says exactly that sections of the sheaf can be created via descent. $\endgroup$– Mike SkirvinCommented Jan 20, 2011 at 16:45
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$\begingroup$ Right! The point is why nobody check that these other conditions are verified and they just limit themselves to prove something only on the various $U$'s? It is obvious that for certain structure these "cocycle agreements" are satisfied? $\endgroup$– unknownCommented Jan 20, 2011 at 16:58
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1$\begingroup$ For a general introd. to descent theory, Vitoli's notes may be helpful homepage.sns.it/vistoli/descent.pdf $\endgroup$– Martin BrandenburgCommented Jan 20, 2011 at 18:02
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$\begingroup$ As for your question, I think should clarify the situation ... it seems to me a little offensive to denounce proofs which you have not understood yet. $\endgroup$– Martin BrandenburgCommented Jan 20, 2011 at 18:05
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