Stable domination and independence in algebraically closed valued fields. (English) Zbl 1149.03027
Lecture Notes in Logic 30. Cambridge: Cambridge University Press (ISBN 978-0-521-88981-0/hbk). xi, 182 p. (2008).
Contents: Preface. 1. Introduction. Part 1. Stable domination: 2. Some background on stability theory; 3. Definition and basic properties of \(St_C\); 4. Invariant types and change of base; 5. A combinatorial lemma; 6. Strong codes for germs. Part 2. Independence in ACVF: 7. Some background on algebraically valued fields; 8. Sequential independence; 9. Growth of the stable part; 10. Types orthogonal to \(\Gamma\); 11. Opacity and resolutions; 12. Maximally complete fields and domination; 13. Invariant types; 14. A maximum modulus principle; 15. Canonical bases and independence given by modules; 16. Other henselian fields. References. Index.
Over the last twenty-five years, ideas of stability theory have constantly penetrated applied model theory and are currently a mainstream driving force in the subject. In particular, these ideas have migrated successfully to non-stable contexts. One of the most striking examples is perhaps the concept of o-minimality. In the book under review, the authors aim in the same direction to lay some fundamentals for applications in valued fields.
The work grew out of the fundamental results of the authors on the elimination of imaginaries in algebraically closed valued fields [“Definable sets in algebraically closed valued fields: elimination of imaginaries”, J. Reine Angew. Math. 597, 175–236 (2006; Zbl 1127.12006)], which developed over several years. The methods and basic results of that paper are at the heart of this book. Just as an indication, the paper is cited on average almost once every three pages.
The authors work in algebraically valued fields but with the view to encompass other (henselian) valued fields. For that purpose, they adopt Weil’s outlook and see a valued field primarily as embedded in its algebraic closure or in a universal algebraically closed valued field: there are general results about algebraically closed valued fields, and one should then go down from there to reach other valued fields. This contrasts with the usual treatment in the model-theoretic literature on valued fields, which rarely exploits this point of view. It has already been applied by Hrushovski-Martin to obtain elimination of imaginaries for the \(p\)-adic fields (see arXiv, math.LO/0701011).
The book is divided into two parts. In the first part they set the basics on stable domination, and in the second part they use it to study algebraically closed valued fields.
A valued field leaves the universe of conventional stability theory because of the presence of an order in the value group. A fundamental tool in stability theory is independence of types and of substructures. But an idiosyncratic idea at the heart of the study of valued fields is that they are somewhat an extension of the residue field by the value group. This is strictly true at the level of the multiplicative groups of the fields. It is reflected in model theory by classical theorems of Ax-Kochen-Ershov type: in good henselian cases, the first-order theory of the valued field is determined by the theory of the residue field and the theory of the value group. It is also present at the level of types via relative quantifier-elimination results. Only the residue field can take us back to stability, e.g., if it is algebraically closed. The idea of stable domination is that, in an appropiate way, (p. 2) “a type can be controlled by a very small part, lying in the stable part; by analogy, (but it is more than an analogy) a power series is controlled, with respect to the question of invertibility for instance, by its constant coefficient.” The fundamental notion is that of stably dominated type. Let \(\mathcal U\) be a large model and \(C\) a parameter set from \(\mathcal U\). Define \(St_C\) as the many-sorted structure whose sorts are the \(C\)-definable stably embedded stable subsets of \(\mathcal U\). The basic relations of \(St_C\) are given by the \(C\)-definable relations of \(\mathcal U\). It turns out that \(St_C\) is stable. Let \(A\) be a parameter set. Define the stable part of \(A\), \(St_C(A)\), to be the definable closure of \(A\) in \(St_C\). Write \(A \overset{\mid}{\smile}^d_C B\) if \(St_C(A)\overset{\mid}{\smile}_C St_C(B)\) in the stable structure \(St_C\) (i.e. \(St_C(A),St_C(B)\) are independent) and \(tp(B/CSt_C(A)) \vdash tp(B/CA)\). Then \(tp(A/C)\) is said to be stably dominated if, for all \(B\), whenever \(St_C(A) \overset{\mid}{\smile}_C St_C(B)\), we have \(A \overset{\mid}{\smile}^d_C B.\) According to the authors, the usual calculus of (in-) dependence of types becomes essentially available for stably dominated types. This opens the way to applications.
In the Introduction (Chapter 1), the authors explain very well their goal and the scope of their results.
Part 1, on stable domination, covers about a third of the book with Chapters 2 to 6. Chapter 2 rounds up an appropriate background on stability theory (saturation, universal domain, imaginaries, invariant types, conditions equivalent to stability, independence and forking, totally transcendental theories and Morley rank, prime models, indiscernibles, Morley sequences, stably embedded sets). The target reader is one (p. 13) “who is familiar with o-minimality or some model theory of valued fields, but has not worked with notions from stability”. The rest of Part 1 gives basic results about stable domination. It is perhaps worthwhile to say something about Chapter 6, on strong codes for germs. This plays an important role in the paper on elimination of imaginaries. Suppose \(D\) is a definable set defined by the formula \(\varphi(x,a)\) where \(a\) is a parameter. There is an equivalence relation \(E_\varphi(y_1,y_2)\), where \(E_\varphi(y_1,y_2)\) holds iff \(\forall x (\varphi(x,y_1)\leftrightarrow \varphi(x,y_2))\). The equivalence class \(a/E_\varphi\) is referred to as a code for \(D\). Let \(p\) be a \(C\)-definable type over \(\mathcal U\), with \({\mathcal U}, C\) as before. Let \(\varphi(x,y,b)\) be a formula defining a function \(f_b\) whose domain contains all realizations of \(p\). The germ of \(f_b\) on \(p\) is the equivalence class of \(b\) under the equivalence relation \(\sim\), where \(b\sim b'\) if the formula \(f_b(x)=f_{b'}(x)\) lies in \(p\). Now the germ of \(f_b\) on \(p\) is a definable object, and a code \(e\) for the germ \(b/\!\!\sim\) is said to be strong if for any \(a\models p|_{Cb}\), we have that \(f_b(a)\) lies in the definable closure of \(C\cup \{e,a\}\). The main result of Chapter 6 is the existence of strong codes for functions on stably dominated types.
Part 2 of the book on independence in algebraically closed valued fields (henceforth ACVF) covers the rest of the book with Chapters 7 to 16. Chapter 7 gives background on algebraically closed valued fields, with emphasis on the main results and methods from the imaginaries paper useful for the present material (background on valued fields; some model theory of valued fields; basics of ACVF; imaginaries and the ACVF sorts; sorts internal to the residue field; unary sets, 1-torsors and generic 1-types; 1-types orthogonal to the value group; generic bases of lattices). We note the typo that, contrary to what is asserted, the generalized power series with rational exponents do not coincide with the Puiseux series. The remaining chapters are roughly described in the table of contents cited above. We indicate here only one key result:
Let \(T\) be a multisorted theory with universal domain \(\mathcal U\), and a stably embedded sort \(\Gamma\) with a definable (without parameters) linear ordering. Then \(T\) is said to be metastable over \(\Gamma\) if for any product \(D\) of sorts and any small \(C\leq {\mathcal U}\), if \(C\) is algebraically closed, then for any \(a\in D({\mathcal U}), tp(a/C)\) extends to an Aut\(({\mathcal U}/C)\)-invariant type, and for some small \(B\) with \(C\subset B\leq {\mathcal U}\), for any \(a\in D({\mathcal U}), tp(a/B,\Gamma(Ba))\) is stably dominated.
The authors show in Theorem 12.18 that algebraically closed valued fields are metastable over the value group, and then also (in Theorems 16.1 and 16.7) the theory of the field of Laurent series over the complex numbers, as well as the model completion of the theory of valued differential fields. As mentioned earlier, this opens the way to a calculus of (in-) dependence of types.
One should be warned about the Index: page references should all be translated downward by four (4).
This book deserves the study of anyone looking for new tools to understand valued fields.
Over the last twenty-five years, ideas of stability theory have constantly penetrated applied model theory and are currently a mainstream driving force in the subject. In particular, these ideas have migrated successfully to non-stable contexts. One of the most striking examples is perhaps the concept of o-minimality. In the book under review, the authors aim in the same direction to lay some fundamentals for applications in valued fields.
The work grew out of the fundamental results of the authors on the elimination of imaginaries in algebraically closed valued fields [“Definable sets in algebraically closed valued fields: elimination of imaginaries”, J. Reine Angew. Math. 597, 175–236 (2006; Zbl 1127.12006)], which developed over several years. The methods and basic results of that paper are at the heart of this book. Just as an indication, the paper is cited on average almost once every three pages.
The authors work in algebraically valued fields but with the view to encompass other (henselian) valued fields. For that purpose, they adopt Weil’s outlook and see a valued field primarily as embedded in its algebraic closure or in a universal algebraically closed valued field: there are general results about algebraically closed valued fields, and one should then go down from there to reach other valued fields. This contrasts with the usual treatment in the model-theoretic literature on valued fields, which rarely exploits this point of view. It has already been applied by Hrushovski-Martin to obtain elimination of imaginaries for the \(p\)-adic fields (see arXiv, math.LO/0701011).
The book is divided into two parts. In the first part they set the basics on stable domination, and in the second part they use it to study algebraically closed valued fields.
A valued field leaves the universe of conventional stability theory because of the presence of an order in the value group. A fundamental tool in stability theory is independence of types and of substructures. But an idiosyncratic idea at the heart of the study of valued fields is that they are somewhat an extension of the residue field by the value group. This is strictly true at the level of the multiplicative groups of the fields. It is reflected in model theory by classical theorems of Ax-Kochen-Ershov type: in good henselian cases, the first-order theory of the valued field is determined by the theory of the residue field and the theory of the value group. It is also present at the level of types via relative quantifier-elimination results. Only the residue field can take us back to stability, e.g., if it is algebraically closed. The idea of stable domination is that, in an appropiate way, (p. 2) “a type can be controlled by a very small part, lying in the stable part; by analogy, (but it is more than an analogy) a power series is controlled, with respect to the question of invertibility for instance, by its constant coefficient.” The fundamental notion is that of stably dominated type. Let \(\mathcal U\) be a large model and \(C\) a parameter set from \(\mathcal U\). Define \(St_C\) as the many-sorted structure whose sorts are the \(C\)-definable stably embedded stable subsets of \(\mathcal U\). The basic relations of \(St_C\) are given by the \(C\)-definable relations of \(\mathcal U\). It turns out that \(St_C\) is stable. Let \(A\) be a parameter set. Define the stable part of \(A\), \(St_C(A)\), to be the definable closure of \(A\) in \(St_C\). Write \(A \overset{\mid}{\smile}^d_C B\) if \(St_C(A)\overset{\mid}{\smile}_C St_C(B)\) in the stable structure \(St_C\) (i.e. \(St_C(A),St_C(B)\) are independent) and \(tp(B/CSt_C(A)) \vdash tp(B/CA)\). Then \(tp(A/C)\) is said to be stably dominated if, for all \(B\), whenever \(St_C(A) \overset{\mid}{\smile}_C St_C(B)\), we have \(A \overset{\mid}{\smile}^d_C B.\) According to the authors, the usual calculus of (in-) dependence of types becomes essentially available for stably dominated types. This opens the way to applications.
In the Introduction (Chapter 1), the authors explain very well their goal and the scope of their results.
Part 1, on stable domination, covers about a third of the book with Chapters 2 to 6. Chapter 2 rounds up an appropriate background on stability theory (saturation, universal domain, imaginaries, invariant types, conditions equivalent to stability, independence and forking, totally transcendental theories and Morley rank, prime models, indiscernibles, Morley sequences, stably embedded sets). The target reader is one (p. 13) “who is familiar with o-minimality or some model theory of valued fields, but has not worked with notions from stability”. The rest of Part 1 gives basic results about stable domination. It is perhaps worthwhile to say something about Chapter 6, on strong codes for germs. This plays an important role in the paper on elimination of imaginaries. Suppose \(D\) is a definable set defined by the formula \(\varphi(x,a)\) where \(a\) is a parameter. There is an equivalence relation \(E_\varphi(y_1,y_2)\), where \(E_\varphi(y_1,y_2)\) holds iff \(\forall x (\varphi(x,y_1)\leftrightarrow \varphi(x,y_2))\). The equivalence class \(a/E_\varphi\) is referred to as a code for \(D\). Let \(p\) be a \(C\)-definable type over \(\mathcal U\), with \({\mathcal U}, C\) as before. Let \(\varphi(x,y,b)\) be a formula defining a function \(f_b\) whose domain contains all realizations of \(p\). The germ of \(f_b\) on \(p\) is the equivalence class of \(b\) under the equivalence relation \(\sim\), where \(b\sim b'\) if the formula \(f_b(x)=f_{b'}(x)\) lies in \(p\). Now the germ of \(f_b\) on \(p\) is a definable object, and a code \(e\) for the germ \(b/\!\!\sim\) is said to be strong if for any \(a\models p|_{Cb}\), we have that \(f_b(a)\) lies in the definable closure of \(C\cup \{e,a\}\). The main result of Chapter 6 is the existence of strong codes for functions on stably dominated types.
Part 2 of the book on independence in algebraically closed valued fields (henceforth ACVF) covers the rest of the book with Chapters 7 to 16. Chapter 7 gives background on algebraically closed valued fields, with emphasis on the main results and methods from the imaginaries paper useful for the present material (background on valued fields; some model theory of valued fields; basics of ACVF; imaginaries and the ACVF sorts; sorts internal to the residue field; unary sets, 1-torsors and generic 1-types; 1-types orthogonal to the value group; generic bases of lattices). We note the typo that, contrary to what is asserted, the generalized power series with rational exponents do not coincide with the Puiseux series. The remaining chapters are roughly described in the table of contents cited above. We indicate here only one key result:
Let \(T\) be a multisorted theory with universal domain \(\mathcal U\), and a stably embedded sort \(\Gamma\) with a definable (without parameters) linear ordering. Then \(T\) is said to be metastable over \(\Gamma\) if for any product \(D\) of sorts and any small \(C\leq {\mathcal U}\), if \(C\) is algebraically closed, then for any \(a\in D({\mathcal U}), tp(a/C)\) extends to an Aut\(({\mathcal U}/C)\)-invariant type, and for some small \(B\) with \(C\subset B\leq {\mathcal U}\), for any \(a\in D({\mathcal U}), tp(a/B,\Gamma(Ba))\) is stably dominated.
The authors show in Theorem 12.18 that algebraically closed valued fields are metastable over the value group, and then also (in Theorems 16.1 and 16.7) the theory of the field of Laurent series over the complex numbers, as well as the model completion of the theory of valued differential fields. As mentioned earlier, this opens the way to a calculus of (in-) dependence of types.
One should be warned about the Index: page references should all be translated downward by four (4).
This book deserves the study of anyone looking for new tools to understand valued fields.
Reviewer: Luc Bélair (Montréal)
MSC:
| 03C98 | Applications of model theory |
| 03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |
| 12-02 | Research exposition (monographs, survey articles) pertaining to field theory |
| 03C45 | Classification theory, stability, and related concepts in model theory |
| 03C60 | Model-theoretic algebra |
| 12J10 | Valued fields |
| 12L12 | Model theory of fields |
