Imaginaries in pseudo- p -adically closed fields Joint with Samaria - - PowerPoint PPT Presentation

Imaginaries in pseudo-p-adically closed fields

Joint with Samaria Montenegro Silvain Rideau

UC Berkeley

July  

 / 

Bounded pseudo-p-adically closed fields

Definition

L A valuation is p-adic if the residue field is Fp and p has minimal

positive valutation.

L A field extension K B L is totally p-adic if every p-adic valuation of K

can be extended to a p-adic valuation of L.

L A field K is pseudo-p-adically closed if it is existentially closed (as a

ring) in every regular totally p-adic extension.

L A field is bounded if it has finitely many extensions of any given

degree.

Proposition (Montenegro)

Let K be a bounded pseudo-p-adically closed field. There are finitely many p-adic valuations on K and they are definable in the ring language.

 / 

Bounded pseudo-p-adically closed fields

Definition

L A valuation is p-adic if the residue field is Fp and p has minimal

positive valutation.

L A field extension K B L is totally p-adic if every p-adic valuation of K

can be extended to a p-adic valuation of L.

L A field K is pseudo-p-adically closed if it is existentially closed (as a

ring) in every regular totally p-adic extension.

L A field is bounded if it has finitely many extensions of any given

degree.

Proposition (Montenegro)

Let K be a bounded pseudo-p-adically closed field. There are finitely many p-adic valuations on K and they are definable in the ring language.

 / 

Bounded pseudo-p-adically closed fields

Definition

L A valuation is p-adic if the residue field is Fp and p has minimal

positive valutation.

L A field extension K B L is totally p-adic if every p-adic valuation of K

can be extended to a p-adic valuation of L.

L A field K is pseudo-p-adically closed if it is existentially closed (as a

ring) in every regular totally p-adic extension.

L A field is bounded if it has finitely many extensions of any given

degree.

Proposition (Montenegro)

Let K be a bounded pseudo-p-adically closed field. There are finitely many p-adic valuations on K and they are definable in the ring language.

 / 

Bounded pseudo-p-adically closed fields

Definition

L A valuation is p-adic if the residue field is Fp and p has minimal

positive valutation.

L A field extension K B L is totally p-adic if every p-adic valuation of K

can be extended to a p-adic valuation of L.

L A field K is pseudo-p-adically closed if it is existentially closed (as a

ring) in every regular totally p-adic extension.

L A field is bounded if it has finitely many extensions of any given

degree.

Proposition (Montenegro)

Let K be a bounded pseudo-p-adically closed field. There are finitely many p-adic valuations on K and they are definable in the ring language.

 / 

Bounded pseudo-p-adically closed fields

Definition

L A valuation is p-adic if the residue field is Fp and p has minimal

positive valutation.

L A field extension K B L is totally p-adic if every p-adic valuation of K

can be extended to a p-adic valuation of L.

L A field K is pseudo-p-adically closed if it is existentially closed (as a

ring) in every regular totally p-adic extension.

L A field is bounded if it has finitely many extensions of any given

degree.

Proposition (Montenegro)

Let K be a bounded pseudo-p-adically closed field. There are finitely many p-adic valuations on K and they are definable in the ring language.

 / 

The geometric language

Let K,v be a valued field.

Definition (Geometric sorts)

L We define Sn GLnK~GLnO and Tn GLnK~GLn nO. L The geometric language LG has sorts F, Sn and Tn for all n C . It also

contains the ring language on F, the canonical projections sn GLnF Sn and tn GLnF Tn.

Remark

S and s can be identified with the valuation.

Theorem

L Algebraically closed valued fields eliminate imaginaries in LG

(HHM).

L p eliminates imaginaries in LG (HMR).

 / 

The geometric language

Let K,v be a valued field.

Definition (Geometric sorts)

L We define Sn GLnK~GLnO and Tn GLnK~GLn,nO. L The geometric language LG has sorts F, Sn and Tn for all n C . It also

contains the ring language on F, the canonical projections sn GLnF Sn and tn GLnF Tn.

Remark

S and s can be identified with the valuation.

Theorem

L Algebraically closed valued fields eliminate imaginaries in LG

(HHM).

L p eliminates imaginaries in LG (HMR).

 / 

The geometric language

Let K,v be a valued field.

Definition (Geometric sorts)

L We define Sn GLnK~GLnO and Tn GLnK~GLn,nO. L The geometric language LG has sorts F, Sn and Tn for all n C . It also

contains the ring language on F, the canonical projections sn GLnF Sn and tn GLnF Tn.

Remark

S and s can be identified with the valuation.

Theorem

L Algebraically closed valued fields eliminate imaginaries in LG

(HHM).

L p eliminates imaginaries in LG (HMR).

 / 

The geometric language

Let K,v be a valued field.

Definition (Geometric sorts)

L We define Sn GLnK~GLnO and Tn GLnK~GLn,nO. L The geometric language LG has sorts F, Sn and Tn for all n C . It also

contains the ring language on F, the canonical projections sn GLnF Sn and tn GLnF Tn.

Remark

S Γ and s can be identified with the valuation.

Theorem

L Algebraically closed valued fields eliminate imaginaries in LG

(HHM).

L p eliminates imaginaries in LG (HMR).

 / 

The geometric language

Let K,v be a valued field.

Definition (Geometric sorts)

L We define Sn GLnK~GLnO and Tn GLnK~GLn,nO. L The geometric language LG has sorts F, Sn and Tn for all n C . It also

contains the ring language on F, the canonical projections sn GLnF Sn and tn GLnF Tn.

Remark

S Γ and s can be identified with the valuation.

Theorem

L Algebraically closed valued fields eliminate imaginaries in LG

(HHM).

L p eliminates imaginaries in LG (HMR).

 / 

The geometric language

Let K,v be a valued field.

Definition (Geometric sorts)

L We define Sn GLnK~GLnO and Tn GLnK~GLn,nO. L The geometric language LG has sorts F, Sn and Tn for all n C . It also

contains the ring language on F, the canonical projections sn GLnF Sn and tn GLnF Tn.

Remark

S Γ and s can be identified with the valuation.

Theorem

L Algebraically closed valued fields eliminate imaginaries in LG

(HHM).

L Qp eliminates imaginaries in LG (HMR).

 / 

An orthogonality result

L Let K be a bounded pseudo-p-adically closed fields with n p-adic

valuations viiBn.

L Let Li denote n copies of LG, with sorts Gi, sharing the sort F. L Let K l K, L i Li 8 K and T ThLK. L Let M T, Mi be the algebraic closure of M with an extension of vi

and Mi be the p-adic closure of M inside Mi

Remark

Let Ui x g be vi-open, then i Ui x g.

Proposition

Let K b A b FM and si ti > Si nM. If i si Mi

LiA ti

then siiBn M

LA tiiBn

 / 

An orthogonality result

L Let K be a bounded pseudo-p-adically closed fields with n p-adic

valuations viiBn.

L Let Li denote n copies of LG, with sorts Gi, sharing the sort F. L Let K l K, L i Li 8 K and T ThLK. L Let M T, Mi be the algebraic closure of M with an extension of vi

and Mi be the p-adic closure of M inside Mi

Remark

Let Ui x g be vi-open, then i Ui x g.

Proposition

Let K b A b FM and si ti > Si nM. If i si Mi

LiA ti

then siiBn M

LA tiiBn

 / 

An orthogonality result

L Let K be a bounded pseudo-p-adically closed fields with n p-adic

valuations viiBn.

L Let Li denote n copies of LG, with sorts Gi, sharing the sort F. L Let K l K, L i Li 8 K and T ThLK. L Let M T, Mi be the algebraic closure of M with an extension of vi

and Mi be the p-adic closure of M inside Mi

Remark

Let Ui x g be vi-open, then i Ui x g.

Proposition

Let K b A b FM and si ti > Si nM. If i si Mi

LiA ti

then siiBn M

LA tiiBn

 / 

An orthogonality result

L Let K be a bounded pseudo-p-adically closed fields with n p-adic

valuations viiBn.

L Let Li denote n copies of LG, with sorts Gi, sharing the sort F. L Let K l K, L i Li 8 K and T ThLK. L Let M T, Mi be the algebraic closure of M with an extension of vi

and Mi be the p-adic closure of M inside Mi

Remark

Let Ui x g be vi-open, then i Ui x g.

Proposition

Let K b A b FM and si ti > Si nM. If i si Mi

LiA ti

then siiBn M

LA tiiBn

 / 

An orthogonality result

L Let K be a bounded pseudo-p-adically closed fields with n p-adic

valuations viiBn.

L Let Li denote n copies of LG, with sorts Gi, sharing the sort F. L Let K l K, L i Li 8 K and T ThLK. L Let M T, Mi be the algebraic closure of M with an extension of vi

and Mi be the p-adic closure of M inside Mi

Remark

Let Ui x g be vi-open, then i Ui x g.

Proposition

Let K b A b FM and si ti > Si nM. If i si Mi

LiA ti

then siiBn M

LA tiiBn

 / 

An orthogonality result

L Let K be a bounded pseudo-p-adically closed fields with n p-adic

valuations viiBn.

L Let Li denote n copies of LG, with sorts Gi, sharing the sort F. L Let K l K, L i Li 8 K and T ThLK. L Let M T, Mi be the algebraic closure of M with an extension of vi

and Mi be the p-adic closure of M inside Mi

Remark

Let Ui x g be vi-open, then i Ui x g.

Proposition

Let K b A b FM and si,ti > Si,nM. If i, si Mi

LiA ti

then siiBn M

LA tiiBn.

 / 

A local density result

Let A b Meq containing i Giacleq

MA.

Proposition

Let c > FM. Then, for all i, there exists an GiA-invariant LiMi-type pi such that tpM

Lc~A 8 i pi is consistent.

Proposition

Let c > FM and d > Giacleq

MAc be some tuples. Assume tpMi Lic~Mi is

GiA-invariant, then so is tpMi

Li d~Mi.

 / 

A local density result

Let A b Meq containing i Giacleq

MA.

Proposition

Let c > FM. Then, for all i, there exists an GiA-invariant LiMi-type pi such that tpM

Lc~A 8 i pi is consistent.

Proposition

Let c > FM and d > Giacleq

MAc be some tuples. Assume tpMi Lic~Mi is

GiA-invariant, then so is tpMi

Li d~Mi.

 / 

A local density result

Let A b Meq containing i Giacleq

MA.

Proposition

Let c > FM. Then, for all i, there exists an GiA-invariant LiMi-type pi such that tpM

Lc~A 8 i pi is consistent.

Proposition

Let c > FM and d > Giacleq

MAc be some tuples. Assume tpMi Lic~Mi is

GiA-invariant, then so is tpMi

Li d~Mi.

 / 

A local density result

Let A b Meq containing i Giacleq

MA.

Proposition

Let c > FM. Then, for all i, there exists an GiA-invariant LiMi-type pi such that tpM

Lc~A 8 i pi is consistent.

Proposition

Let c > FM and d > Giacleq

MAc be some tuples. Assume tpMi Lic~Mi is

GiA-invariant, then so is tpMi

Li d~Mi.

 / 

A local density result

Let A b Meq containing i Giacleq

MA.

Proposition

Let c > FM. Then, for all i, there exists an GiA-invariant LiMi-type pi such that tpM

Lc~A 8 i pi is consistent.

Proposition

Let c > FM and d > Giacleq

MAc be some tuples. Assume tpMi Lic~Mi is

GiA-invariant, then so is tpMi

Li d~Mi.

Corollary

Let c > FM be some tuple. Then, for all i, there exists a GiA-invariant LiMi-type pi such that tpM

Lc~A 8 i pi is consistent.

 / 

A local density result

Let A b Meq containing i Giacleq

MA.

Proposition

Let c > FM. Then, for all i, there exists an GiA-invariant LiMi-type pi such that tpM

Lc~A 8 i pi is consistent.

Proposition

Let c > FM and d > Giacleq

MAc be some tuples. Assume tpMi Lic~Mi is

GiA-invariant, then so is tpMi

Li d~Mi.

Corollary

Let c > FM be some tuple. Then, for all i, there exists a GiA-invariant LiMi-type pi such that tpM

Lc~A 8 i pi is consistent.

 / 

A criterion using amalgamation

L Let Lii>I be languages, with sorts Ri, sharing a dominant sort D,

and let L c i Li.

L Let Ti be Li-theories and T c i Ti be an L-theory. L Let M T and M b Mi Ti be sufficiently saturated and

homogeneous.

L For all C B Meq and all tuples a b > DM, write a C b if there are

RiA-invariant LiMi-types pi with a i piSRiCb.

Proposition

Assume: . for all A acleq

MA b Meq and tuple c > DM, there exists d M LA c

with d A M; . For all A aclMA b M and a b c > DM tuples, if b A a, c A ab, a M

LA b and ac Mi LiRiA bc, for all i, then there exists d such that

db M

LA da M LA ca.

Then T weakly eliminates imaginaries.

 / 

A criterion using amalgamation

L Let Lii>I be languages, with sorts Ri, sharing a dominant sort D,

and let L c i Li.

L Let Ti be Li-theories and T c i Ti be an L-theory. L Let M T and M b Mi Ti be sufficiently saturated and

homogeneous.

L For all C B Meq and all tuples a b > DM, write a C b if there are

RiA-invariant LiMi-types pi with a i piSRiCb.

Proposition

Assume: . for all A acleq

MA b Meq and tuple c > DM, there exists d M LA c

with d A M; . For all A aclMA b M and a b c > DM tuples, if b A a, c A ab, a M

LA b and ac Mi LiRiA bc, for all i, then there exists d such that

db M

LA da M LA ca.

Then T weakly eliminates imaginaries.

 / 

A criterion using amalgamation

L Let Lii>I be languages, with sorts Ri, sharing a dominant sort D,

and let L c i Li.

L Let Ti be Li-theories and T c i Ti, be an L-theory. L Let M T and M b Mi Ti be sufficiently saturated and

homogeneous.

L For all C B Meq and all tuples a b > DM, write a C b if there are

RiA-invariant LiMi-types pi with a i piSRiCb.

Proposition

Assume: . for all A acleq

MA b Meq and tuple c > DM, there exists d M LA c

with d A M; . For all A aclMA b M and a b c > DM tuples, if b A a, c A ab, a M

LA b and ac Mi LiRiA bc, for all i, then there exists d such that

db M

LA da M LA ca.

Then T weakly eliminates imaginaries.

 / 

A criterion using amalgamation

L Let Lii>I be languages, with sorts Ri, sharing a dominant sort D,

and let L c i Li.

L Let Ti be Li-theories and T c i Ti, be an L-theory. L Let M T and M b Mi Ti be sufficiently saturated and

homogeneous.

L For all C B Meq and all tuples a b > DM, write a C b if there are

RiA-invariant LiMi-types pi with a i piSRiCb.

Proposition

Assume: . for all A acleq

MA b Meq and tuple c > DM, there exists d M LA c

with d A M; . For all A aclMA b M and a b c > DM tuples, if b A a, c A ab, a M

LA b and ac Mi LiRiA bc, for all i, then there exists d such that

db M

LA da M LA ca.

Then T weakly eliminates imaginaries.

 / 

A criterion using amalgamation

L Let Lii>I be languages, with sorts Ri, sharing a dominant sort D,

and let L c i Li.

L Let Ti be Li-theories and T c i Ti, be an L-theory. L Let M T and M b Mi Ti be sufficiently saturated and

homogeneous.

L For all C B Meq and all tuples a,b > DM, write a C b if there are

RiA-invariant LiMi-types pi with a i piSRiCb.

Proposition

Assume: . for all A acleq

MA b Meq and tuple c > DM, there exists d M LA c

with d A M; . For all A aclMA b M and a b c > DM tuples, if b A a, c A ab, a M

LA b and ac Mi LiRiA bc, for all i, then there exists d such that

db M

LA da M LA ca.

Then T weakly eliminates imaginaries.

 / 

A criterion using amalgamation

L Let Lii>I be languages, with sorts Ri, sharing a dominant sort D,

and let L c i Li.

L Let Ti be Li-theories and T c i Ti, be an L-theory. L Let M T and M b Mi Ti be sufficiently saturated and

homogeneous.

L For all C B Meq and all tuples a,b > DM, write a C b if there are

RiA-invariant LiMi-types pi with a i piSRiCb.

Proposition

Assume: . for all A acleq

MA b Meq and tuple c > DM, there exists d M LA c

with d A M; . For all A aclMA b M and a b c > DM tuples, if b A a, c A ab, a M

LA b and ac Mi LiRiA bc, for all i, then there exists d such that

db M

LA da M LA ca.

Then T weakly eliminates imaginaries.

 / 

A criterion using amalgamation

L Let Lii>I be languages, with sorts Ri, sharing a dominant sort D,

and let L c i Li.

L Let Ti be Li-theories and T c i Ti, be an L-theory. L Let M T and M b Mi Ti be sufficiently saturated and

homogeneous.

L For all C B Meq and all tuples a,b > DM, write a C b if there are

RiA-invariant LiMi-types pi with a i piSRiCb.

Proposition

Assume: . for all A acleq

MA b Meq and tuple c > DM, there exists d M LA c

with d A M; . For all A aclMA b M and a b c > DM tuples, if b A a, c A ab, a M

LA b and ac Mi LiRiA bc, for all i, then there exists d such that

db M

LA da M LA ca.

Then T weakly eliminates imaginaries.

 / 

A criterion using amalgamation

L Let Lii>I be languages, with sorts Ri, sharing a dominant sort D,

and let L c i Li.

L Let Ti be Li-theories and T c i Ti, be an L-theory. L Let M T and M b Mi Ti be sufficiently saturated and

homogeneous.

L For all C B Meq and all tuples a,b > DM, write a C b if there are

RiA-invariant LiMi-types pi with a i piSRiCb.

Proposition

Assume: . for all A acleq

MA b Meq and tuple c > DM, there exists d M LA c

with d A M; . For all A aclMA b M and a,b,c > DM tuples, if b A a, c A ab, a M

LA b and ac Mi LiRiA bc, for all i, then there exists d such that

db M

LA da M LA ca.

Then T weakly eliminates imaginaries.

 / 

Amalgamation

In pseudo-p-adically closed fields, Condition  follows from a more general result:

Proposition

Let M T, A b M and a a c c c > FM be tuples. Assume FA

a 9 M b A, FAa a 9 FAa a FA a, c A aa, c M LA c,

c Mi

LiAa c and c LiAa c, for all i. Then

tpM

Lc~Aa 8 tpM Lc~Aa 8 i

tpMi

L c~Aaa

is satisfiable.

Remark

L If A b F, this is an earlier result of Montenegro. L The general result follows from the older version and the description

• f the structure on the geometric sorts given by the orthogonality

result.

 / 

Amalgamation

In pseudo-p-adically closed fields, Condition  follows from a more general result:

Proposition

Let M T, A b M and a,a,c,c,c > FM be tuples. Assume FA

a 9 M b A, FAa a 9 FAa a FA a, c A aa, c M LA c,

c Mi

LiAa c and c LiAa c, for all i. Then

tpM

Lc~Aa 8 tpM Lc~Aa 8 i

tpMi

L c~Aaa

is satisfiable.

Remark

L If A b F, this is an earlier result of Montenegro. L The general result follows from the older version and the description

• f the structure on the geometric sorts given by the orthogonality

result.

 / 

Amalgamation

In pseudo-p-adically closed fields, Condition  follows from a more general result:

Proposition

Let M T, A b M and a,a,c,c,c > FM be tuples. Assume FA

a 9 M b A, FAa a 9 FAa a FA a, c A aa, c M LA c,

c Mi

LiAa c and c LiAa c, for all i. Then

tpM

Lc~Aa 8 tpM Lc~Aa 8 i

tpMi

L c~Aaa

is satisfiable.

Remark

L If A b F, this is an earlier result of Montenegro. L The general result follows from the older version and the description

• f the structure on the geometric sorts given by the orthogonality

result.

 / 

Amalgamation

In pseudo-p-adically closed fields, Condition  follows from a more general result:

Proposition

Let M T, A b M and a,a,c,c,c > FM be tuples. Assume FA

a 9 M b A, FAa a 9 FAa a FA a, c A aa, c M LA c,

c Mi

LiAa c and c LiAa c, for all i. Then

tpM

Lc~Aa 8 tpM Lc~Aa 8 i

tpMi

L c~Aaa

is satisfiable.

Remark

L If A b F, this is an earlier result of Montenegro. L The general result follows from the older version and the description

• f the structure on the geometric sorts given by the orthogonality

result.

 / 

Elimination of imaginaries

Theorem

The theory T eliminates imaginaries.

Remark

L Coding finite sets is not completely obvious. L Since the valuations vi are discrete, Ti n is coded in Si n. L Let O i Oi. We have a bijection

M

i

Si n M

i

GLnF~GLnOi GLnF~GLnO

 / 

Elimination of imaginaries

Theorem

The theory T eliminates imaginaries.

Remark

L Coding finite sets is not completely obvious. L Since the valuations vi are discrete, Ti n is coded in Si n. L Let O i Oi. We have a bijection

M

i

Si n M

i

GLnF~GLnOi GLnF~GLnO

 / 

Elimination of imaginaries

Theorem

The theory T eliminates imaginaries.

Remark

L Coding finite sets is not completely obvious. L Since the valuations vi are discrete, Ti,n is coded in Si,n. L Let O i Oi. We have a bijection

M

i

Si n M

i

GLnF~GLnOi GLnF~GLnO

 / 

Elimination of imaginaries

Theorem

The theory T eliminates imaginaries.

Remark

L Coding finite sets is not completely obvious. L Since the valuations vi are discrete, Ti,n is coded in Si,n. L Let O i Oi. We have a bijection

M

i

Si,n M

i

GLnF~GLnOi GLnF~GLnO.

 / 