Heinrich Heine Universit¨ at D¨ usseldorf 2017 Triangulation of p -adic semi-algebraic sets Luck Darni` ere Thursday, November 2 nd Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 1 / 34
Introduction 1 Semi-algebraic sets p -adically closed fields Quantifiers elimination Which triangulation? Simplicial complexes 2 Main result and applications 3
1.1 - Semi-algebraic sets K is any field. � � P N := { y N / y ∈ K } P × N := P N \ { 0 } . A ⊆ K m is semi-algebraic if it is a finite union of sets defined by: f 1 = · · · = f r = 0 and g 1 ∈ P × N 1 and · · · and g s ∈ P × N s . with f i , g i ∈ K [ X 1 , . . . , X m ]. Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 3 / 34
1.1 - Semi-algebraic sets K is any field. � � P N := { y N / y ∈ K } P × N := P N \ { 0 } . A ⊆ K m is semi-algebraic if it is a finite union of sets defined by: f 1 = · · · = f r = 0 and g 1 ∈ P × N 1 and · · · and g s ∈ P × N s . with f i , g i ∈ K [ X 1 , . . . , X m ]. Remarks If K is algebraically closed, g i ∈ P × N ⇐ ⇒ g i � = 0. Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 3 / 34
1.1 - Semi-algebraic sets K is any field. � � P N := { y N / y ∈ K } P × N := P N \ { 0 } . A ⊆ K m is semi-algebraic if it is a finite union of sets defined by: f 1 = · · · = f r = 0 and g 1 ∈ P × N 1 and · · · and g s ∈ P × N s . with f i , g i ∈ K [ X 1 , . . . , X m ]. Remarks If K is algebraically closed, g i ∈ P × N ⇐ ⇒ g i � = 0. If K is real closed: g i ∈ P × 2 n ⇐ ⇒ g i > 0 . g i ∈ P × 2 n +1 ⇐ ⇒ g i � = 0 , Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 3 / 34
1.2 - p -adically closed fields Examples Every finite extension K 0 of Q p . Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 4 / 34
1.2 - p -adically closed fields Examples Every finite extension K 0 of Q p . The relative algebraic closure of Q inside K 0 (not complete). Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 4 / 34
1.2 - p -adically closed fields Examples Every finite extension K 0 of Q p . The relative algebraic closure of Q inside K 0 (not complete). The completion w.r.t. the t -adic valuation the field � n ≥ 1 K 0 (( t 1 / n )) of Puiseux series over K 0 (value group Z × Q ). Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 4 / 34
1.2 - p -adically closed fields Examples Every finite extension K 0 of Q p . The relative algebraic closure of Q inside K 0 (not complete). The completion w.r.t. the t -adic valuation the field � n ≥ 1 K 0 (( t 1 / n )) of Puiseux series over K 0 (value group Z × Q ). K is p -adically closed if Q ⊆ K and there is a valuation v on K such that: 1 ( K , v ) is Henselian. 2 The residue field of ( K , v ) is finite, with characteristic p . 3 The value group Z = v ( K × ) is a Z-group : i) Z has a smallest element > 0 ; ii) Z / n Z ≃ Z / n Z for every n ≥ 1. Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 4 / 34
1.3 - Quantifiers elimination Theorem (Chevalley (19??), Tarski (1948), Macintyre (1976)) If K is algebraically closed, real closed or p-adically closed, then the projection on K m of any semi-algebraic set A ⊆ K m +1 is also semi-algebraic. Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 5 / 34
1.3 - Quantifiers elimination Theorem (Chevalley (19??), Tarski (1948), Macintyre (1976)) If K is algebraically closed, real closed or p-adically closed, then the projection on K m of any semi-algebraic set A ⊆ K m +1 is also semi-algebraic. This means that for every such field K : By stabilizing algebraic sets (defined by f = 0 with f pol.) projections and boolean combinations we obtain exactly the semi-algebraic sets. A ⊆ K m is semi-algebraic ⇐ ⇒ A is definable (in the language of rings). Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 5 / 34
1.4 - Which triangulation? A semi-algebraic map ϕ : A ⊆ K m → K n is a map whose graph is semi-algebraic. Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 6 / 34
1.4 - Which triangulation? A semi-algebraic map ϕ : A ⊆ K m → K n is a map whose graph is semi-algebraic. Theorem (Triangulation of real semi-algebraic sets) Let K be a real closed field. Every semi-algebraic set A ⊆ K m is semi-algebraically homeomorphic to the union of a simplicial complex. Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 6 / 34
1.4 - Which triangulation? A semi-algebraic map ϕ : A ⊆ K m → K n is a map whose graph is semi-algebraic. Theorem (Triangulation of real semi-algebraic sets) Let K be a real closed field. Every semi-algebraic set A ⊆ K m is semi-algebraically homeomorphic to the union of a simplicial complex. Aim Same result for a p -adically closed field. Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 6 / 34
1.4 - Which triangulation? A semi-algebraic map ϕ : A ⊆ K m → K n is a map whose graph is semi-algebraic. Theorem (Triangulation of real semi-algebraic sets) Let K be a real closed field. Every semi-algebraic set A ⊆ K m is semi-algebraically homeomorphic to the union of a simplicial complex. Aim Same result for a p -adically closed field. Tools Cell decomposition. Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 6 / 34
1.4 - Which triangulation? A semi-algebraic map ϕ : A ⊆ K m → K n is a map whose graph is semi-algebraic. Theorem (Triangulation of real semi-algebraic sets) Let K be a real closed field. Every semi-algebraic set A ⊆ K m is semi-algebraically homeomorphic to the union of a simplicial complex. Aim Same result for a p -adically closed field. Tools Cell decomposition. “Good Direction” Lemma. Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 6 / 34
1.4 - Which triangulation? A semi-algebraic map ϕ : A ⊆ K m → K n is a map whose graph is semi-algebraic. Theorem (Triangulation of real semi-algebraic sets) Let K be a real closed field. Every semi-algebraic set A ⊆ K m is semi-algebraically homeomorphic to the union of a simplicial complex. Aim Same result for a p -adically closed field. Tools Cell decomposition. “Good Direction” Lemma. Simplexes (faces, splitting. . . ). Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 6 / 34
Introduction 1 Simplicial complexes 2 The real case Topological complexes The discrete case Division The p -adic case Main result and applications 3
2.1 - The real case A real polytope A is the strict convex hull of a finite set A 0 ⊆ R q (the points of its frontier ∂ A are excluded). It is a simplex if A 0 can be chosen a finite set of affinely independent points. Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 8 / 34
2.1 - The real case A real polytope A is the strict convex hull of a finite set A 0 ⊆ R q (the points of its frontier ∂ A are excluded). It is a simplex if A 0 can be chosen a finite set of affinely independent points. Properties Let A ⊆ R q be a real polytope. 1 A is relatively open and precompact. 2 A can be defined by finitely many inequalities on linear maps. 3 Every face of A is a polytope. 4 The faces of A form a complex and a partition of A. Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 8 / 34
The specialisation order on the subsets of a topological space is defined by B ≤ A ⇐ ⇒ B ⊆ A . The facets of a polytope are its proper faces which are maximal (with respect to the specialization order). Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 9 / 34
The specialisation order on the subsets of a topological space is defined by B ≤ A ⇐ ⇒ B ⊆ A . The facets of a polytope are its proper faces which are maximal (with respect to the specialization order). Proposition Let A ⊆ R q be a real polytope. 1 A has at least ≥ dim( A ) + 1 facets. 2 Equality holds ⇐ ⇒ A is a simplex. Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 9 / 34
2.2 - Topological complexes Let X be a topological space, and A a finite family of subsets of X . A is a complex of subsets of X if: 1 the elements of A are pairwise disjoint; Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 10 / 34
2.2 - Topological complexes Let X be a topological space, and A a finite family of subsets of X . A is a complex of subsets of X if: 1 the elements of A are pairwise disjoint; 2 every A ∈ A is relatively open ( i.e. A \ A is closed) and � � � B ∈ A / B ≤ A A = . Thursday, November 2 nd Luck Darni` ere Triangulation of p -adic semi-algebraic sets 10 / 34
Recommend
More recommend
