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I'm a graduate student at UC Berkeley, studying mathematical logic. I'm especially interested in reverse mathematics and abstract model theory.
8m

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Can we construct cohomolgy theory on noetherian separated schemes without Axiom of Choice?
(In particular, specifying what you are asking about  besides just being a good idea  will show that you've put serious thought into the question, and I suspect for a variety of reasons that doubt about this is part of the reason for the votes to close.) 
19m

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Can we construct cohomolgy theory on noetherian separated schemes without Axiom of Choice?
I'll point out the (very wellknown) fact that $\Pi^1_2$ sentences don't depend on AC, so any very concrete consequence of cohomological reasoning will not require choice to prove. But presumably you are interested in abstract results specifically about cohomology, which is again why you need to specify what exactly you want to do. 
21m

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Can we construct cohomolgy theory on noetherian separated schemes without Axiom of Choice?
Tl;dr version: if you don't want to ask "how much," then you have to specify what you want to "construct." 
21m

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Can we construct cohomolgy theory on noetherian separated schemes without Axiom of Choice?
First of all, I said this seems a real and appropriateforMO question if the question is "how much?." What I'm a little confused by is exactly what you mean by "Can we construct cohomology theory . . . " What would constitute a successful "construction?" For example, without choice, we can study whatever you like  injective resolutions, or wellorderings of the reals, or etc. Of course, in many cases our results will be different. So the meaning of the title question hinges on what you mean by "construct cohomology theory;" the "how much" version is more welldefined. 
17h

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Can we construct cohomolgy theory on noetherian separated schemes without Axiom of Choice?
I'm not entirely clear what the question in the OP is. If, however, it is "How much of the cohomology theory of noetherian separated schemes relies on AC?", this seems a real and appropriateforMO question. Or am I missing something? 
1d

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Why integer should have finite many digits?
This question is not appropriate for this site; however, you might be interested in $p$adic numbers, which can be thought of as allowing digits to run infinitely far to the left, but not the right: en.wikipedia.org/wiki/Padic_number. 
1d

revised 
Why integer should have finite many digits?
edited tags 
1d

answered  What year was Hechler forcing created? 
1d

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$\infty$Borel Determinacy?
A silly question: is it obvious that there is an $\infty$Borel set outside $L(\mathbb{R})$ (assuming, say, $ZF+AD+$"for every set of reals $A$, $\mathcal{P}(\mathbb{R})\not\subset L(\mathbb{R}, A)$)"? 
1d

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$\infty$Borel Determinacy?
@Carlo, I think that's exactly Andres' second comment. But thanks for the citation. 
1d

revised 
$\infty$Borel Determinacy?
added 159 characters in body 
1d

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$\infty$Borel Determinacy?
I didn't know that it was still open whether $AD$ proved "all sets of reals are $\infty$Borel," and it didn't occur to me that all sets of reals in $L(\mathbb{R})$ are $\infty$Borel; your two comments (as far as I'm concerned) completely answer question 1. 
1d

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Contraction between basis vectors and basis oneforms
I think this is probably a better fit for mathstackexchange? 
1d

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The (non)absoluteness of secondorder elementary equivalence
I'm still interested in the remaining questions, but this is certainly enough for now! 
1d

accepted  The (non)absoluteness of secondorder elementary equivalence 
1d

asked  $\infty$Borel Determinacy? 
2d

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SHPS and SPHS inequality using monounary algebra
Where does this problem come from? 
Apr 14 
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infinitary logic and partial fixed point logic
Yeah, I deleted it because it used infinitely many variables and I didn't see a way to fix it. This is very nice! 
Apr 13 
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Recurrence relation practice problem that I can't figure out
I killed the relation algebra tag, since that's not what it means. 
Apr 13 
revised 
Recurrence relation practice problem that I can't figure out
edited tags 