Quotient divisible groups and torsion-free groups corresponding to finite Abelian groups

E. I. Kompantseva ; A. A. Fomin

Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 221-233.

Résumé

The category of sequences $\mathcal{S}$ has been introduced in [1, 2, 3]. Objects of the category $\mathcal{S}$ are finite sequences of the form $a_{1},\ldots,a_{n}$, where the elements $a_{1},\ldots,a_{n}$ belong to a finitely presented module over the ring of polyadic numbers $\widehat{{Z}}$. The ring of polyadic numbers $\widehat{{Z}}=\prod\limits_{p}{\widehat{Z}}_{p}$ is the product of the rings of $p$-adic integers over all prime numbers $p$. Morphisms of the category $\mathcal{S}$ from the object $a_{1},\ldots,a_{n}$ to an object $b_{1},\ldots,b_{k}$ are all possible pairs $(\varphi, T),$ where $\varphi: \langle a_{1},\ldots,a_{n}\rangle_{\widehat{{Z}}} \rightarrow \langle b_{1},\ldots,b_{k}\rangle_{\widehat{{Z}}}$ is a homomorphism of $\widehat{{Z}}$-modules, generated by given elements, and $T$ is a matrix of dimension $k\times n$ with integer entries such that the following matrix equality takes place $$(\varphi a_{1},\ldots,\varphi a_{n})=(b_{1},\ldots,b_{k})T.$$ It is proved in [2] that the category $\mathcal{S}$ is equivalent to the category $\mathcal{D}$ of mixed quotient divisible abelian groups with marked bases. It is proved in [3] that the category $\mathcal{S}$ is dual to the category $\mathcal{F}$ of torsion-free finite-rank abelian groups with marked bases, a basis means here a maximal linearly independent set of elements. The composition of these equivalence and duality is the duality introduced in [1] and in [4], which can be considered as a version of the duality introduced in [5]. If an object of the category $\mathcal{S}$ consists of one element, then it corresponds to rank-1 groups of the categories $\mathcal{\mathcal{D}}$ and $\mathcal{F}$. This case is considered in [6] and we obtain the following. The duality $\mathcal{S}\leftrightarrow\mathcal{F}$ gives us the classical description by R. Baer [7] of rank-$1$ torsion-free groups. The equivalence $\mathcal{S}\leftrightarrow\mathcal{D}$ coincides with the description by O.I. Davydova [8] of rank-$1$ quotient divisible groups. We consider another marginal case in the present paper. Every torsion abelian group can be considered as a module over the ring of polyadic numbers. Moreover, a torsion group is a finitely presented $\widehat{{Z}}$-module if and only if it is finite. Thus, for every set of generators $g_{1},\ldots,g_{n}$ of every finite abelian group $G$ the sequence $g_{1},\ldots,g_{n}$ is an object of the category $\mathcal{S}$. Such objects determine a complete subcategory of the category $\mathcal{S}$. We show in the present paper that the object $g_{1},\ldots,g_{n}$ of the category $\mathcal{S}$ corresponds to an object of the category $\mathcal{D}$, which is of the form $G\oplus Q^{n}$ with the marked basis $g_{1}+e_{1},\ldots,g_{n}+e_{n}$, where $e_{1},\ldots,e_{n}$ is the standard basis of the vector space $Q^{n}$ over the field of rational numbers $Q$. The same object $g_{1},\ldots,g_{n}$ corresponds to an object of the category $\mathcal{F}$, which is a free group $A$, satisfying the conditions $Z^{n}\subset A\subset Q^{n}$ and $A/Z^{n}\cong G^{\ast}$, where $ G^{\ast}=Hom(G,Q/Z)$ is the dual finite group. We consider also the group homomorphisms corresponding to morphisms of the category $\mathcal{S}$.

Détail

Keywords: abelian groups, modules, dual categories.

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