Episode 323
Cohomological descent
Fri, 2018-Mar-23 00:34 UTC
Length - 2:07
Direct Link
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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