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Existentially closed models of the theory of Artinian local rings. (English) Zbl 1060.03056

In recent years an increasing interaction between algebraic geometry and model theory has developed. But, as the author of the paper under review observes in his introduction, a closer dialog requires that the study of definable sets from algebraically closed fields to Artinian local rings be enlarged. Artinianity is not an elementary property. But the author shows that, if one bounds the length of the Artinian rings involved and one considers the class of all Artinian local rings of length at most \(l\) for a given \(l\), then the corresponding theory is \(\forall_2\)-elementary, and even finitely axiomatizable.
Then the author looks at existentially closed models in this class. The main theorem of the paper is devoted to characterizing these models: they are exactly the Artinian local rings \(R\) of length \(\leq l\) satisfying the following conditions: (i) \(R\) is Gorenstein (equivalently \(R\) is quasi-Frobenius, owing to the Artinian local framework); (ii) the residue field \(\kappa\) of \(R\) is algebraically closed; (iii) the length of \(R\) is just \(l\). Model theory is mainly used in showing (i), (ii) and (iii) for an existentially closed \(R\). The proof of the converse implication has a more algebraic flavour, and is concerned with the ideal structure of an Artinian local Gorenstein ring. First the author deals with the equicharacteristic case, when the residue field \(\kappa\) has the same characteristic as \(R\) and is embeddable. In this setting, an equicharacteristic Artinian local Gorenstein ring \(S\) extending \(R\) and having the same length \(l\) is of the form \(S = R \otimes_{\kappa} \lambda\) where \(\lambda\) is the residue field of \(S\). Under this assumption, one sees that any system of polynomial equalities and inequalities over \(R\) having a solution in \(S\) inherits a solution in \(R\). The mixed characteristic case, when \(\kappa\) is not embeddable, is handled by using Witt vectors, to show that the multiplicative part of \(\kappa\) is still embeddable. The approach is as in the former case, but needs a deeper analysis.
Finally, the author observes that the theory of Artinian local Gorenstein rings of length \(l\) having an algebraically closed residue field is model complete; but he produces a counterexample proving that quantifier elimination fails in this setting (actually, a quantifier elimination theorem in a suitable natural enlarged language is announced).

MSC:

03C60 Model-theoretic algebra
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13L05 Applications of logic to commutative algebra
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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References:

[1] Transactions of the American Mathematical Society 59 pp 54–106– (1946)
[2] Cohen-Macaulay rings (1993) · Zbl 0788.13005
[3] Mathematische Zeitschrift 82 pp 8–28– (1963)
[4] Existentially closed models of the theory of Artinian local rings (1999) · Zbl 1060.03056
[5] Building Models by Games (1985) · Zbl 0569.03015
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[8] Model Theory (1993)
[9] Introduction to model theory and to the metamathematics of algebra (1965)
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