Abstract
We discuss when an order in a Dedekind domain \(R\) is equal to \(R\) (is the maximal order in \(R\)). Every order in \(R\) is a subring of \(R\). This fact implies the existence of natural homomorphisms between objects related to orders such that the group of Cartier divisors, the Picard group, the group of Weil divisors, the Chow group and the Witt ring of an order. We examine the maximality of an order in \(R\) in the context of such natural homomorphisms.
In [8], we discuss when an order \(\mathcal{O}\) in \(R\) is equal to \(R\) on the assumption that either the Picard group of \(R\) or the Picard group of \(\mathcal{O}\) is a torsion group. In this paper, we abandon this assumption. We formulate equivalent conditions for the maximality of \(\mathcal{O}\) for any Dedekind domain \(R\) and any order \(\mathcal{O}\) in \(R\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Z. Borevich and I. Shafarevich, Number Theory, Nauka (Moscow, 1985)
Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag. Berlin, Heidelberg, New York (1999)
Geroldinger, A., Halter-Koch, F.: Non-unique Factorizations, Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, Chapman & Hall/CRC. London, New York, Boca Raton (2006)
M. Knebusch, Grothendieck-und Wittringe von nichtausgearteten symmetrischen Bilinearformen, S.-B. Heidelberger Akad. Wiss. Math-Natur. Kl. (1969/1970), 93–157
Lombardi, H., Quitté, C.: Comparison of Picard groups in dimension \(1\). MLQ Math. Log. Q. 54, 247–252 (2008)
Milnor, J., Husemoller, D.: Symmetric Bilinear Forms, Springer-Verlag. Berlin, Heidelberg, New York (1973)
M. Rosen, Number Theory in Function Fields, Graduate Texts in Math., Springer-Verlag (New York, Berlin, Heidelberg, 2002)
B. Rothkegel, Maximality of orders in Dedekind domains, J. Algebra Appl., 19 (2020), Article no. 2050125
Rothkegel, B.: Witt functor of a quadratic order. Math. Slovaca 68, 1339–1342 (2018)
Scharlau, W.: Quadratic and Hermitian Forms, Springer-Verlag. Berlin, Heidelberg, New York (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rothkegel, B. Maximality of orders in Dedekind domains. II. Acta Math. Hungar. 164, 428–438 (2021). https://doi.org/10.1007/s10474-021-01161-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-021-01161-7