On factorizations of generically étale double coverings
type: Item
instance of: scholarly article
title: On factorizations of generically étale double coverings
zbMATH Open document ID: 1476.14062
author: Guido Reißner
author: Uwe Storch
published in: Beiträge zur Algebra und Geometrie
publication date: 2021-05-31T00:00:00Z
review text: Let \(X\) be a locally Noetherian scheme, \(B\) a quadratic \({\mathcal O}_X\)-algebra, \(Y =\mathrm{Spec}\ B\) and \(p: Y \rightarrow X\) a double covering. The authors define a genus of \(p\) as the class of an etale double covering \(y: Z \rightarrow Y\) with the property locally, \(q\) can already be defined over the base scheme \(X\), and that the canonical involution of the double covering \(p\) can be lifted to an involution of \(Z\). The genera of \(p\) form a group Gen\((Y|X)\). When specialized to the base scheme \(X = {\mathbb Z}\), the authors recover the genera of quadratic number fields (in the strict sense). They also give an interpretation of Hilbert's theory of genera of quadratic extensions of \({\mathbb Q}(i)\) in terms of their abstract genus theory.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 11S15
zbMATH DE Number: 7352567
zbMATH Keywords: genus theory
zbMATH Keywords: quadratic algebra
zbMATH Keywords: double covering
zbMATH Keywords: quadratic number fields
MaRDI profile type: Publication
full work available at URL: s13366-020-00498-4
OpenAlex ID: W3023319376
Wikidata QID: Q114219941
cites work: Q4177654
cites work: Quadratic algebras, Clifford algebras, and arithmetic Witt groups
cites work: Zur Geschlechtertheorie in quadratischen Zahlkörpern
cites work: Number theory
cites work: The Ambiguous Class Number Formula Revisited
cites work: Q5628266
cites work: Q5187813
cites work: Unramified quadratic extensions of a quadratic field
DOI: 10.1007/S13366-020-00498-4
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description: scientific article; zbMATH DE number 7352567