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Pré-Publication, Document De Travail Année : 2021

Recovering the Picard group of quadratic algebras from Wood's binary quadratic forms

Résumé

The first part of this article aims at classifying quadratic algebras over an arbitrary scheme : this goal is achieved by using two invariants, the Wood-discriminant and the parity. Next, we obtain a parametrization of the Picard group of a given quadratic algebra by twisted quadratic forms, provided $2$ is not a zero divisor in our base scheme. For that purpose, we start from Wood's set-theoretical bijection (2011) generalizing the classical result over $\mathbb{Z}$. To remedy the fact that some Picard classes are identified in this bijection, we extend the notion of orientation of quadratic algebras to the non-free case, and we use our classification to conclude. All along the paper, we illustrate various notions and obstructions with a wide range of examples.
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Dates et versions

hal-03450812 , version 1 (26-11-2021)
hal-03450812 , version 2 (30-09-2022)
hal-03450812 , version 3 (22-06-2023)
hal-03450812 , version 4 (28-04-2024)

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  • HAL Id : hal-03450812 , version 1

Citer

William Dallaporta. Recovering the Picard group of quadratic algebras from Wood's binary quadratic forms. 2021. ⟨hal-03450812v1⟩
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