help pls

[Anonymous No.8349401]

Let [math]X[/math] be a smooth projective variety over [math]\mathbb{C}[/math] cut out by polynomials with integral coefficients.

Let [math]{{\bar X}_p}[/math] be the corresponding variety over the algebraic closure of [math]{\mathbb{F}_p}[/math] defined by taking the polynomials mod p.

We know the weil cohomology theories of the two varieties are related by [math]H_{et}^i\left( {{{\bar X}_p},{\mathbb{Q}_\ell }} \right) \otimes \mathbb{C} \cong {H^i}\left( {X,\mathbb{C}} \right)[/math].

I was wondering if we could find relations between other cohomology groups of these varieties, particularly cohomology in the tangent sheaf.

The reasoning behind this is I want to see if there are any interesting relation between [math]{T_{{{\operatorname{Def} }_X}}}[/math] and [math]{T_{{{\operatorname{Def} }_{{{\bar X}_p}}}}}[/math].

[Anonymous No.8350405]

?

[Anonymous No.8350431]

>>8349401
>>>/mathoverflow/
I'd love to know what you're talking about, I'm so jealous

[Anonymous No.8350444]

>>8349401
How much Algebraic Geo do I have to learn to understand this?

[Anonymous No.8350447]

>>8350444
some complex geometry and etale cohomology

so quite a bit

[Anonymous No.8350476]

>>8350431
>>>>/mathoverflow/
I think my question it too vague for there.

[Anonymous No.8351214]

lol this is iq and pretentious intellect board, not actual science and math what are you doing here

[Anonymous No.8351220]

>>8351214
I have been on 4chan since high school and can't leave.

[Anonymous No.8351851]

This question is way over wrought. All you're asking given a proj C variety and a variety over F_p, are there some other cohomology theories that relate them? I didn't know about this result for the etale topology, but given that holds, what other topologies would you intend to use? Fidelement plat pleine finie? Fidelement plat quasi compacte? I dont even think there cohomology results that are well known for these finer topologies, I just think you asked the most basic question that you could understand to use what sound like big words. You're surely right to believe this would be closed immediately on MO.

[Anonymous No.8351859]

>>8350431
>>8350444
Why do people on this board want to know algebraic geometry if they don't know what it's about ? Why not literally any other area of math ?

[Anonymous No.8351872]

>>8351859
Why do people seem to want to know about algebraic geometry on \sci? Overwhemingly the emphasis here is on applied mathematics and most posters seem to scorn any topic that doesn't immediately lead to 300k starting. AG is arguably the most robust ediface in mathematics. When people talk about extending gemetric intuition to unknown lands, they are usually appealing to AG, not differential geometry and especially not calculus. Even where AG is seemingly deficient, especially so with respect to homotopy theory, the subject is highly interesting because after consideration one can usually manage the appropriate extension to AG. Consider Hopkins and Lurie in this regard, or Voevodsky and ultimately the great Grothendieck as well. Grothendiecks final love letter to mathematics was on this very topic. Tldr: people would want to know about AG because if you're interested in pure mathematics, it is THE subject to care about.

[Anonymous No.8352408]

>>8351872
>AG is arguably the most robust ediface in mathematics
>if you're interested in pure mathematics, it is THE subject to care about

>AG faggots actually believe this

AG is only cool because Grothendieck set up a nice categorical foundations for it. You can literally do this for any area of math. But no, you fucking bandwagoners just imitate, you don't see the thinking behind it.

[Anonymous No.8352430]

>>8351851
>but given that holds, what other topologies would you intend to use? Fidelement plat pleine finie? Fidelement plat quasi compacte? I dont even think there cohomology results that are well known for these finer topologies,

I'm not talking about different topologies, I mean different sheaves. Particularly the relative tangent sheaves.

As I stated, the reason I am interested in this is because of deformations.

The space of first order deformations (over some base R), i.e. the tangent space [math]{T_{{{\operatorname{Def} }_X}}}[/math], is isomorphic to [math]{H^1}\left( {X,{\mathcal{T}_{X/R}}} \right)[/math].

I want to see if there are any interesting ways I can relate [math]{H^1}\left( {X,{\mathcal{T}_{X/\mathbb{C}}}} \right)[/math] and [math]{H^1}\left( {{{\bar X}_p},{\mathcal{T}_{X/{{\bar \mathbb{F}}_p}}}} \right)[/math].

[Anonymous No.8352438]

>>8352430

** [math]{H^1}\left( {{{\bar X}_p},{\mathcal{T}_{X/{{\bar{ \mathbb{F}}_p}}}}} \right)[/math]