On generalized integral representations over Dedekind rings

D. K. Faddeev

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 4, Tome 227 (1995), pp. 113-118

Résumé

The present paper develops the ideas presented in [1]. Let $\mathfrak o$ be a Dedeking ring, and let $\Lambda$ be a finitely generated algebra over $\mathfrak o$. An integral representation in the broad sense of the ring $\Lambda$ over $\mathfrak o$ is a homomorphism of $\Lambda$ to the endomorphism ring of a finitely generated module over $\mathfrak o$. A representation in the restricted sense is a representation by matrices over $\mathfrak o$. Thus, the problem of describing the integral representations over $\mathfrak o$ is subdivided into the following two problems: the description of representations in the broad sense and the selection of them of representations in the restricted sense. It is proved that any representation of $\Lambda$ by matrices over the field $k$ of fractions of the ring $\mathfrak o$ is equivalent over $k$ to an integral representation in the broad sense. This fact simplifies the problem of describing the representations in the broad sense. A representation is equivalent to a representation in the restricted sense if its degree over $k$ and the order of the ideal class group of the ring $\mathfrak o$ are relatively prime, or if it is the direct sum of $h$ copies of one and the same representation over $k$, where $h$ is the exponent of the ideal class group of $\mathfrak o$. Bibliography: 3 titles.