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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 22nd 2013
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   Author: Urs
   Format: MarkdownItexbrief note _[[comparison theorem (étale cohomology)]]_

   brief note comparison theorem (étale cohomology)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 22nd 2013
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   Author: Urs
   Format: MarkdownItexadded a few more lines as such: *** Specifically, let $A$ be either of * a [[finite group|finite]] [[field]]; * the [[p-adic integers]] $\mathbb{Z}_p$ for some [[prime]] $p \geq 2$; * the [[p-adic numbers]] $\mathbb{A}_{p}$ (but _not_ for instance the [[integers]] themselves). Then for $X$ a [[variety]] over the [[complex numbers]] and $X(\mathbb{C})$ its [[topological space]] of complex point there is an [[isomorphism]] $$ H^\bullet(X_{et}, A) \simeq H^\bullet(X(\mathbb{C}), A) $$ between the [[étale cohomology]] of $X$ and the [[ordinary cohomology]] of $X(\mathbb{C})$. Notice that on the other hand for instance if instead $X = Spec(k)$ is the [[spectrum of a commutative ring|spectrum]] of a [[field]], then its [[étale cohomology]] coincides with the [[Galois cohomology]] of $k$. In this way [[étale cohomology]] interpolates between "topological" and "number theoretic" notions of cohomology.

   added a few more lines as such:

   ---

   Specifically, let $A$ be either of

   • a finite field;

   • the p-adic integers $\mathbb{Z}_p$ for some prime $p \geq 2$;

   • the p-adic numbers $\mathbb{A}_{p}$

   (but not for instance the integers themselves).

   Then for $X$ a variety over the complex numbers and $X(\mathbb{C})$ its topological space of complex point there is an isomorphism

   $$H^\bullet(X_{et}, A) \simeq H^\bullet(X(\mathbb{C}), A)$$

   between the étale cohomology of $X$ and the ordinary cohomology of $X(\mathbb{C})$.

   Notice that on the other hand for instance if instead $X = Spec(k)$ is the spectrum of a field, then its étale cohomology coincides with the Galois cohomology of $k$. In this way étale cohomology interpolates between “topological” and “number theoretic” notions of cohomology.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 22nd 2013
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   Author: David_Corfield
   Format: MarkdownItexI wonder if the Coq-formulation of the p-adics [here](http://arxiv.org/abs/1302.1207) is going to be important. They say they're looking to move towards p-adic symplectic geometry.

   I wonder if the Coq-formulation of the p-adics here is going to be important. They say they’re looking to move towards p-adic symplectic geometry.

4. • CommentRowNumber4.
   • CommentAuthorZhen Lin
   • CommentTimeNov 22nd 2013
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   Author: Zhen Lin
   Format: MarkdownItexIt is quite misleading to talk about étale cohomology with coefficients in $\mathbb{F}_p$, $\mathbb{Z}_p$, and $\mathbb{Q}_p$ all in the same breath, when each one is constructed in a different way!

   It is quite misleading to talk about étale cohomology with coefficients in $\mathbb{F}_p$, $\mathbb{Z}_p$, and $\mathbb{Q}_p$ all in the same breath, when each one is constructed in a different way!

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 22nd 2013
   • (edited Nov 22nd 2013)
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   Author: Urs
   Format: MarkdownItex@David, I am still lacking a good feeling for how to think of $p$-adic stuff in the grand scheme of things. Also the real numbers have been formalized in HoTT, in * [[Univalent Foundations Project]], chapter 11 of _[[Homotopy Type Theory – Univalent Foundations of Mathematics]]_ Does considering these two formalizations carry any message beyond that it can be done? Is one somehow "more fundamental" or "more natural" from the perspective of HoTT? Does anyone have an idea?

   @David,

   I am still lacking a good feeling for how to think of $p$-adic stuff in the grand scheme of things.

   Also the real numbers have been formalized in HoTT, in

   • Univalent Foundations Project, chapter 11 of Homotopy Type Theory – Univalent Foundations of Mathematics

   Does considering these two formalizations carry any message beyond that it can be done? Is one somehow “more fundamental” or “more natural” from the perspective of HoTT?

   Does anyone have an idea?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 22nd 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItex@Zhen Lin, not sure what you are trying me to do. But you can hit the edit button and do it yourself!

   @Zhen Lin,

   not sure what you are trying me to do. But you can hit the edit button and do it yourself!