Model Theory of Local Real Closed SV-Rings of Finite Rank
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910625338228736 |
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| author | Piepenborn, Ricardo Palomino |
| author_facet | Piepenborn, Ricardo Palomino |
| contents | This note begins the model-theoretic study of local real closed SV-rings of finite rank; to this end, a structure theorem for reduced local SV-rings of finite rank is given and branching ideals in local real closed rings of finite rank are analysed. The class of local real closed SV-rings of rank $ n \in \mathbb{N}^{\geq 2} $ is elementary in the language of rings $ \mathscr{L} := \{ +, -, \cdot, 0, 1 \} $ and its $ \mathscr{L} $-theory $ T_n $ has a model companion $ T_{n,1} $; models of $ T_{n,1} $ are $ n $-fold fibre products $ (((V_1 \times_{\textbf{k}} V_2) \times_{\textbf{k}}V_3) \dots \times_{\textbf{k}} V_{n-1})\times_{\textbf{k}} V_{n} $ of non-trivial real closed valuation rings $ V_i $ with isomorphic residue field $\mathbf{k}$. The $ \mathscr{L} $-theory $ T_{n,1} $ is complete, decidable, and NIP. After enriching $ \mathscr{L} $ with a predicate for the maximal ideal, models of $ T_n $ have prime extensions in models of $ T_{n,1} $, and $ T_{n,1} $ is the model completion of $ T_n $ in this enriched language. A quantifier elimination result for $ T_{n,1} $ is also given. The class of those local real closed SV-rings of rank $ n\in \mathbb{N}^{\geq 2} $ which are $ n $-fold fibre products $ (((V_1 \times_{W} V_2) \times_{W}V_3) \dots \times_{W} V_{n-1})\times_{W} V_{n} $ of non-trivial real closed valuation rings $ V_i $ along surjective morphisms $ V_i \twoheadrightarrow W $ onto a non-trivial domain $ W $ is elementary in the language of rings, and its $ \mathscr{L} $-theory $ T_{n,2} $ is also complete, decidable, and NIP; after enriching $ \mathscr{L} $ with predicates for the maximal ideal and the unique branching ideal, $ T_{n,2} $ is model complete. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_20295 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Model Theory of Local Real Closed SV-Rings of Finite Rank Piepenborn, Ricardo Palomino Logic Commutative Algebra Rings and Algebras 03C10 (Primary), 06F25, 13F99, 13H99 (Secondary) This note begins the model-theoretic study of local real closed SV-rings of finite rank; to this end, a structure theorem for reduced local SV-rings of finite rank is given and branching ideals in local real closed rings of finite rank are analysed. The class of local real closed SV-rings of rank $ n \in \mathbb{N}^{\geq 2} $ is elementary in the language of rings $ \mathscr{L} := \{ +, -, \cdot, 0, 1 \} $ and its $ \mathscr{L} $-theory $ T_n $ has a model companion $ T_{n,1} $; models of $ T_{n,1} $ are $ n $-fold fibre products $ (((V_1 \times_{\textbf{k}} V_2) \times_{\textbf{k}}V_3) \dots \times_{\textbf{k}} V_{n-1})\times_{\textbf{k}} V_{n} $ of non-trivial real closed valuation rings $ V_i $ with isomorphic residue field $\mathbf{k}$. The $ \mathscr{L} $-theory $ T_{n,1} $ is complete, decidable, and NIP. After enriching $ \mathscr{L} $ with a predicate for the maximal ideal, models of $ T_n $ have prime extensions in models of $ T_{n,1} $, and $ T_{n,1} $ is the model completion of $ T_n $ in this enriched language. A quantifier elimination result for $ T_{n,1} $ is also given. The class of those local real closed SV-rings of rank $ n\in \mathbb{N}^{\geq 2} $ which are $ n $-fold fibre products $ (((V_1 \times_{W} V_2) \times_{W}V_3) \dots \times_{W} V_{n-1})\times_{W} V_{n} $ of non-trivial real closed valuation rings $ V_i $ along surjective morphisms $ V_i \twoheadrightarrow W $ onto a non-trivial domain $ W $ is elementary in the language of rings, and its $ \mathscr{L} $-theory $ T_{n,2} $ is also complete, decidable, and NIP; after enriching $ \mathscr{L} $ with predicates for the maximal ideal and the unique branching ideal, $ T_{n,2} $ is model complete. |
| title | Model Theory of Local Real Closed SV-Rings of Finite Rank |
| topic | Logic Commutative Algebra Rings and Algebras 03C10 (Primary), 06F25, 13F99, 13H99 (Secondary) |
| url | https://arxiv.org/abs/2409.20295 |