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Episode 322             Episode 324
Episode 323

Cohomological descent
Fri, 2018-Mar-23 00:34 UTC
Length - 2:07

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Welcome to random Wiki of the Day where we read the summary of a random Wikipedia page every day.

The random article for Friday, 23 March 2018 is Cohomological descent.

In algebraic geometry, a cohomological descent is, roughly, a "derived" version of a fully faithful descent in the classical descent theory. This point is made precise by the below: the following are equivalent: in an appropriate setting, given a map a from a simplicial space X to a space S,











a









:



D



+





(

S

)





D



+





(

X

)





{\displaystyle a^{*}:D^{+}(S)\to D^{+}(X)}

is fully faithful.

The natural transformation









id





D



+





(

S

)







R



a













a













{\displaystyle \operatorname {id} _{D^{+}(S)}\to Ra_{*}\circ a^{*}}

is an isomorphism.

The map a is then said to be a morphism of cohomological descent.

The treatment in SGA uses a lot of topos theory. Conrad's notes gives a more down-to-earth exposition.

This recording reflects the Wikipedia text as of 00:34 UTC on Friday, 23 March 2018.

For the full current version of the article, go to http://en.wikipedia.org/wiki/Cohomological_descent.

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