scholarly journals TWISTED GALOIS STRATIFICATION

2016 ◽  
Vol 222 (1) ◽  
pp. 1-60 ◽  
Author(s):  
IVAN TOMAŠIĆ
Keyword(s):  
Quantifier Elimination ◽  
Difference Schemes ◽  
Direct Image ◽  
Geometric Description ◽  
Elimination Procedure ◽  
Definable Sets ◽  
Galois Covers ◽  
Image Theorem

We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic–geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulas associated with finite Galois covers of difference schemes.

More on definable sets of p-adic numbers

1988 ◽  
Vol 53 (3) ◽  
pp. 912-920 ◽  
Author(s):  
Philip Scowcroft
Keyword(s):  
Cross Section ◽  
Quantifier Elimination ◽  
Order Theory ◽  
Model Completeness ◽  
First Order ◽  
First Order Theory ◽  
Ordered Fields ◽  
Definable Sets ◽  
The Cross ◽  
Primitive Recursive

To eliminate quantifiers in the first-order theory of the p-adic field Qp, Ax and Kochen use a language containing a symbol for a cross-section map n → pn from the value group Z into Qp [1, pp. 48–49]. The primitive-recursive quantifier eliminations given by Cohen [2] and Weispfenning [10] also apply to a language mentioning the cross-section, but none of these authors seems entirely happy with his results. As Cohen says, “all the operations… introduced for our simple functions seem natural, with the possible exception of the map n → pn” [2, p. 146]. So all three authors show that various consequences of quantifier elimination—completeness, decidability, model-completeness—also hold for a theory of Qp not employing the cross-section [1, p. 453; 2, p. 146; 10, §4]. Macintyre directs a more specific complaint against the cross-section [5, p. 605]. Elementary formulae which use it can define infinite discrete subsets of Qp; yet infinite discrete subsets of R are not definable in the language of ordered fields, and so certain analogies between Qp and R suggested by previous model-theoretic work seem to break down.To avoid this problem, Macintyre gives up the cross-section and eliminates quantifiers in a theory of Qp written just in the usual language of fields supplemented by a predicate V for Qp's valuation ring and by predicates Pn for the sets of nth powers in Qp (for all n ≥ 2).

The 1-convex generalization of Grauert's direct image theorem

1971 ◽  
Vol 190 (3) ◽  
pp. 203-214 ◽  
Author(s):  
Yum-Tong Siu
Keyword(s):  
Direct Image ◽  
Image Theorem

A version of p-adic minimality

2012 ◽  
Vol 77 (2) ◽  
pp. 621-630 ◽  
Author(s):  
Raf Cluckers ◽  
Eva Leenknegt
Keyword(s):  
Positive Characteristic ◽  
Quantifier Elimination ◽  
Cell Decomposition ◽  
Real Numbers ◽  
Valued Fields ◽  
The Real ◽  
Definable Sets ◽  
A Cell ◽  
Definable Functions

AbstractWe introduce a very weak language on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language are trivial functions. We also give a definitional expansion of in which K has quantifier elimination, and we obtain a cell decomposition result for -definable sets.Our language can serve as a p-adic analogue of the very weak language (<) on the real numbers, to define a notion of minimality on the field of p-adic numbers and on related valued fields. These fields are not necessarily Henselian and may have positive characteristic.

The direct image theorem in formal and rigid geometry

1995 ◽  
Vol 301 (1) ◽  
pp. 69-104 ◽  
Author(s):  
Peter Ullrich
Keyword(s):  
Direct Image ◽  
Image Theorem
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