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 positive valutation. can be extended to a p -adic valuation of L . ring) in every regular totally p -adic extension. 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.  /  L A valuation is p-adic if the residue field is F p and p has minimal L A field extension K B L is totally p-adic if every p -adic valuation of K L A field K is pseudo-p-adically closed if it is existentially closed (as a L A field is bounded if it has finitely many extensions of any given

The geometric language Definition (Geometric sorts) contains the ring language on F , the canonical projections Remark Theorem (HHM).  /  Let � K , v � be a valued field. L We define S n � � GL n � K �~ GL n � O � and T n � � GL n � K �~ GL n , n � O � . L The geometric language L G has sorts F , S n and T n for all n C . It also s n � GL n � F � � S n and t n � GL n � F � � T n . S  � Γ and s  can be identified with the valuation. L Algebraically closed valued fields eliminate imaginaries in L G L Q p eliminates imaginaries in L G (HMR).

An orthogonality result Remark then Proposition  /  L Let K be a bounded pseudo- p -adically closed fields with n p -adic valuations � v i � i B n . L Let L i denote n copies of L G , with sorts G i , sharing the sort F . L Let K  l K , L � � i L i 8 K  and T � Th L � K � . L Let M � T , M i be the algebraic closure of M with an extension of v i and M i be the p -adic closure of M inside M i Let U i x g be v i -open, then � i U i x g . Let K  b A b F � M � and s i , t i > S i , n � M � . If � i , s i � M i L i � A � t i � s i � i B n � M L � A � � t i � i B n .

A local density result such that tp M Corollary Proposition  /  Proposition Let A b M eq containing � i G i � acl eq M � A �� . Let c > F � M � . Then, for all i , there exists an G i � A � -invariant L i � M i � -type p i L � c ~ A � 8 � i p i is consistent. Let c > F � M � and d > G i � acl eq M � Ac �� be some tuples. Assume tp M i L i � c ~ M i � is G i � A � -invariant, then so is tp M i L i � d ~ M i � . Let c > F � M � be some tuple. Then, for all i , there exists a G i � A � -invariant L � c ~ A � 8 � i p i is consistent. L i � M i � -type p i such that tp M

A criterion using amalgamation Proposition Then T weakly eliminates imaginaries. Assume:  /  homogeneous. L Let � L i � i > I be languages, with sorts R i , sharing a dominant sort D , and let L c � i L i . L Let T i be L i -theories and T c � i T i , � be an L -theory. L Let M � T and M b M i � T i be sufficiently saturated and L For all C B M eq and all tuples a , b > D � M � , write a � C b if there are R i � A � -invariant L i � M i � -types p i with a � � i p i S R i � C � b . M � A � b M eq and tuple c > D � M � , there exists d � M . for all A � acl eq L � A � c with d � A M ; . For all A � acl M � A � b M and a , b , c > D � M � tuples, if b � A a , c � A ab , a � M L � A � b and ac � M i L i � R i � A �� bc , for all i , then there exists d such that db � M L � A � da � M L � A � ca .

Amalgamation In pseudo- p -adically closed fields, Condition  follows from a more result. of the structure on the geometric sorts given by the orthogonality Remark is satisfiable. tp M i i tp M  /  general result: Proposition Let M � T , A b M and a  , a  , c  , c  , c > F � M � be tuples. Assume a 9 M b A , F � A �� a  � a 9 F � A �� a  � a � F � A � F � A � a , c � A a  a  , c  � M L � A � c  , c � M i L i � Aa  � c  and c � L i � Aa  � c  , for all i . Then L � c  ~ Aa  � 8 � L � c  ~ Aa  � 8 tp M L � c ~ Aa  a  � L If A b F , this is an earlier result of Montenegro. L The general result follows from the older version and the description

Elimination of imaginaries Theorem The theory T eliminates imaginaries. Remark i i  /  L Coding finite sets is not completely obvious. L Since the valuations v i are discrete, T i , n is coded in S i , n �  . L Let O � � i O i . We have a bijection S i , n � M M GL n � F �~ GL n � O i � � GL n � F �~ GL n � O � .