All Questions
Tagged with model-theory ag.algebraic-geometry
42
questions
3
votes
1
answer
128
views
Can we see quantifier elimination by comparing semirings?
This question came up while reading the paper Hales, What is motivic measure?. Broadly speaking, I'm interested in which ideas from motivic measure make sense in arbitrary first-order theories (or ...
- 20.7k
3
votes
1
answer
299
views
Chevalley's theorem on valuation spectra
In the paper On Valuation Spectra (Section 2, Page 176), Huber and Knebusch asserted that: if the ring map $A\to B$ is finitely presented then the associated map of valuation spectra $\mathrm{Spv}(B)\...
- 255
1
vote
0
answers
206
views
Interpretation of model theory in algebraic geometry
I found a paper Some applications of a model theoretic fact to (semi-) algebraic geometry by Lou van den Dries. In this paper, the author uses model theoretical methods to prove the completeness of ...
- 227
5
votes
1
answer
479
views
General algebraic result obtained from consideration on $\mathbb{Q}_p$
There are results in field theory which are obtained from, let's say, the complex numbers and then generalized to all algebraically closed fields.
For instance, the fact that a polynomial $P$ admits a ...
- 231
6
votes
1
answer
363
views
Interpreting group-theoretic sentences as statements about algebraic groups
Suppose we are given a sentence in the language of groups, e.g. $\phi=\forall x\forall y(x\cdot y=y\cdot x)$, and suppose that we are also given the data defining an algebraic group $G/k$. One can ...
- 333
2
votes
1
answer
208
views
Minor Lefschetz principle
I once read (I think) the following equivalent formulation of the Minor Lefschetz principle:
If an elementary sentence holds for one algebraically closed field,
then it holds for every algebraically ...
- 4,503
9
votes
1
answer
734
views
Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?
This is a cross-post! For the original post on SE (9 upvotes, no answer) see:
https://math.stackexchange.com/questions/4475853/is-a-complex-algebraic-set-with-a-zariski-dense-subset-of-algebraic-...
- 145
5
votes
1
answer
382
views
Does the main theorem of elimination theory with $\mathbb{Z}$-coefficients imply that projective varieties are complete?
I have a question concerning the completeness of projective varieties.
Let $k$ be an algebraically closed field. By the "main theorem of elimination theory" I mean the following result:
Let $...
- 145
7
votes
1
answer
488
views
Is there any theory of "étale cohomology" with algebraic coefficients?
For simplicity, I will restrict attention to untwisted coefficients.
Let $k$ be a finite field of characteristic $p$, and $\ell\ne p$ prime. Can one define a cohomology theory with $\mathbb{Q}_\ell^{\...
- 341
2
votes
1
answer
237
views
Example of a bounded imperfect PAC field that is not separably closed
How do you construct a bounded (meaning there are only finitely many separable finite extensions of any given degree) imperfect pseudo-algebraically closed field that is not separably closed? I assume ...
- 311
4
votes
2
answers
646
views
Tarski's original proof of quantifier elimination in algebraically closed fields
I am currently helping teach a course about foundations of mathematics, which has thus far focused mostly on propositional and first-order logic. As part of the course, the students are each required ...
- 741
4
votes
1
answer
213
views
Comparing the first-order theories of different kinds of local rings of a complex variety
Let $X$ be a complex variety containing some point $x$. Then $X$ is naturally a complex-analytic space, and we have an inclusion of rings $\mathbb{C}[X]_x\hookrightarrow\mathbb{C}\{X\}_x\...
- 1,837
4
votes
0
answers
186
views
Issue with "definition" of pseudo algebraically closed fields
I'm having an issue with a sentence in Chapter 11 of Fried & Jarden's Field Arithmetic. As a "motto" for pseudo algebraically closed (PAC) fields, they say they are fields $K$ such that &...
- 477
6
votes
1
answer
275
views
Algebraic geometry additionally equipped with field automorphism operation
I am looking for some facts on theory, which is essentially algebraic geometry but with field automorphisms added as 'basic' operations. (Precisely, I mean universal algebraic geometry for (universal) ...
- 63
4
votes
1
answer
247
views
What is the definable functor associated to an algebraic scheme (model theory of valued fields)
I have a very basic question regarding algebraic model theory. I am trying to read Espaces de Berkovich, polytopes, squelettes et théorie des modèles (MSN) by Antoine Ducros. The relevant section is ...
- 1,133