Elementary duality of modules.

*(English)*Zbl 0815.16002The duality that is the topic of this paper is one which connects the model theories of right and left modules over a given ring. It was defined at the level of formulas by the reviewer [M. Prest, J. Lond. Math. Soc., II. Ser. 38, 403-409 (1988; Zbl 0674.16019)] and, at this level, it can also be discerned, in rather different form, in earlier work of L. Gruson and C. U. Jensen [Lect. Notes Math. 867, 234-294 (1981; Zbl 0505.18005)] (also see [B. Zimmermann- Huisgen and W. Zimmermann, Trans. Am. Math. Soc. 320, 695-713 (1990; Zbl 0699.16019)]).

In Herzog’s paper, this duality is extended considerably and put to many uses. For example, the author shows that the right and left Ziegler spectra over a ring are essentially homeomorphic and that there is a bijection between theories of right and left \(R\)-modules. He shows that a wide class of indecomposable pure-injectives (= points of the Ziegler spectrum) have well-defined duals.

The paper contains interesting and sometimes startling results on a variety of special and general topics. These include modules over Dedekind domains, localisation at a closed subset of the Ziegler spectrum, strongly minimal and \(pp\)-simple indecomposables, local purity, the relation between the dual \(DU\) of an indecomposable pure-injective \(U\) and \(\operatorname{Hom}(U,DU\otimes U)\), duality between flat and absolutely pure modules, totally transcendental modules, Morita duality and elementary duality versus Hom-duality.

The paper is well written and the background material is summarised. An early, key, lemma is a reformulation of the usual criterion for a tensor to be zero: if \(\overline a\) is an \(n\)-tuple from a left \(R\)-module \(M\) and if \(\overline c\) is an \(n\)-tuple from a right \(R\)-module \(N\) then \(\overline a\otimes\overline c= \) (that is, \(\sum a_ i\otimes c_ i = 0\)) iff there is some \(pp\) formula \(\varphi\) for left modules such that \(M\) satisfies \(\varphi(\overline a)\) and \(N\) satisfies \(D\varphi(\overline c)\), where \(D \varphi\) denotes the dual of \(\varphi\). In many parts of the paper the author works in the context of a category which he denotes \((R\text{-Mod})^{\text{eq}}\) – this can be thought of as “positive-\(^{\text{eq}}\)” of the incomplete theory of left \(R\)-modules, with \(pp\)-defined sorts for objects and \(pp\)-definable morphisms between sorts. This category, which is a very natural one for the study of modules using model-theoretic ideas, is, in fact, equivalent to the category (\(R\text{-mod},Ab)^{fp}\) of finitely presented additive functors from the category of finitely presented \(R\)-modules to the category of abelian groups. The ideas and techniques of this paper have already found many uses and are sure to find many more.

In Herzog’s paper, this duality is extended considerably and put to many uses. For example, the author shows that the right and left Ziegler spectra over a ring are essentially homeomorphic and that there is a bijection between theories of right and left \(R\)-modules. He shows that a wide class of indecomposable pure-injectives (= points of the Ziegler spectrum) have well-defined duals.

The paper contains interesting and sometimes startling results on a variety of special and general topics. These include modules over Dedekind domains, localisation at a closed subset of the Ziegler spectrum, strongly minimal and \(pp\)-simple indecomposables, local purity, the relation between the dual \(DU\) of an indecomposable pure-injective \(U\) and \(\operatorname{Hom}(U,DU\otimes U)\), duality between flat and absolutely pure modules, totally transcendental modules, Morita duality and elementary duality versus Hom-duality.

The paper is well written and the background material is summarised. An early, key, lemma is a reformulation of the usual criterion for a tensor to be zero: if \(\overline a\) is an \(n\)-tuple from a left \(R\)-module \(M\) and if \(\overline c\) is an \(n\)-tuple from a right \(R\)-module \(N\) then \(\overline a\otimes\overline c= \) (that is, \(\sum a_ i\otimes c_ i = 0\)) iff there is some \(pp\) formula \(\varphi\) for left modules such that \(M\) satisfies \(\varphi(\overline a)\) and \(N\) satisfies \(D\varphi(\overline c)\), where \(D \varphi\) denotes the dual of \(\varphi\). In many parts of the paper the author works in the context of a category which he denotes \((R\text{-Mod})^{\text{eq}}\) – this can be thought of as “positive-\(^{\text{eq}}\)” of the incomplete theory of left \(R\)-modules, with \(pp\)-defined sorts for objects and \(pp\)-definable morphisms between sorts. This category, which is a very natural one for the study of modules using model-theoretic ideas, is, in fact, equivalent to the category (\(R\text{-mod},Ab)^{fp}\) of finitely presented additive functors from the category of finitely presented \(R\)-modules to the category of abelian groups. The ideas and techniques of this paper have already found many uses and are sure to find many more.

Reviewer: M.Prest (Manchester)

##### MSC:

16B70 | Applications of logic in associative algebras |

16D90 | Module categories in associative algebras |

03C60 | Model-theoretic algebra |

16D40 | Free, projective, and flat modules and ideals in associative algebras |

##### Keywords:

duality; model theories of right and left modules; Ziegler spectra; modules over Dedekind domains; \(pp\)-simple indecomposables; absolutely pure modules; totally transcendental modules; Morita duality; elementary duality; \(pp\) formula; incomplete theory of left modules; \(pp\)-definable morphisms; finitely presented additive functors; categories of finitely presented modules
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\textit{I. Herzog}, Trans. Am. Math. Soc. 340, No. 1, 37--69 (1993; Zbl 0815.16002)

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##### References:

[1] | Paul Eklof and Gabriel Sabbagh, Model-completions and modules, Ann. Math. Logic 2 (1970/1971), no. 3, 251 – 295. · Zbl 0227.02029 |

[2] | Ivo Herzog, Some model theory of modules, Doctoral Dissertation, Univ. Notre Dame, 1989. · Zbl 0898.03014 |

[3] | Carl Faith, Algebra. II, Springer-Verlag, Berlin-New York, 1976. Ring theory; Grundlehren der Mathematischen Wissenschaften, No. 191. · Zbl 0335.16002 |

[4] | Mike Prest, Model theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988. · Zbl 0634.03025 |

[5] | Gabriel Sabbagh and Paul Eklof, Definability problems for modules and rings, J. Symbolic Logic 36 (1971), 623 – 649. · Zbl 0251.02052 |

[6] | Bo StenstrĂ¶m, Rings of quotients, Springer-Verlag, New York-Heidelberg, 1975. Die Grundlehren der Mathematischen Wissenschaften, Band 217; An introduction to methods of ring theory. · Zbl 0296.16001 |

[7] | Martin Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), no. 2, 149 – 213. · Zbl 0593.16019 |

[8] | Birge Zimmermann-Huisgen and Wolfgang Zimmermann, On the sparsity of representations of rings of pure global dimension zero, Trans. Amer. Math. Soc. 320 (1990), no. 2, 695 – 711. · Zbl 0699.16019 |

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