K-theories and Free Inductive Graded Rings in Abstract Quadratic Forms Theories
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917633996095488 |
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| author | Roberto, Kaique Matias de Andrade mariano, Hugo Luiz |
| author_facet | Roberto, Kaique Matias de Andrade mariano, Hugo Luiz |
| contents | We build on previous work on multirings (\cite{roberto2021quadratic}) that provides generalizations of the available abstract quadratic forms theories (special groups and real semigroups) to the context of multirings (\cite{marshall2006real}, \cite{ribeiro2016functorial}). Here we raise one step in this generalization, introducing the concept of pre-special hyperfields and expand a fundamental tool in quadratic forms theory to the more general multivalued setting: the K-theory. We introduce and develop the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor's K-theory (\cite{milnor1970algebraick}) and Special Groups K-theory, developed by Dickmann-Miraglia (\cite{dickmann2006algebraic}). We develop some properties of this generalized K-theory, that can be seen as a free inductive graded ring, a concept introduced in \cite{dickmann1998quadratic} in order to provide a solution of Marshall's Signature Conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05750 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | K-theories and Free Inductive Graded Rings in Abstract Quadratic Forms Theories Roberto, Kaique Matias de Andrade mariano, Hugo Luiz K-Theory and Homology Commutative Algebra We build on previous work on multirings (\cite{roberto2021quadratic}) that provides generalizations of the available abstract quadratic forms theories (special groups and real semigroups) to the context of multirings (\cite{marshall2006real}, \cite{ribeiro2016functorial}). Here we raise one step in this generalization, introducing the concept of pre-special hyperfields and expand a fundamental tool in quadratic forms theory to the more general multivalued setting: the K-theory. We introduce and develop the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor's K-theory (\cite{milnor1970algebraick}) and Special Groups K-theory, developed by Dickmann-Miraglia (\cite{dickmann2006algebraic}). We develop some properties of this generalized K-theory, that can be seen as a free inductive graded ring, a concept introduced in \cite{dickmann1998quadratic} in order to provide a solution of Marshall's Signature Conjecture. |
| title | K-theories and Free Inductive Graded Rings in Abstract Quadratic Forms Theories |
| topic | K-Theory and Homology Commutative Algebra |
| url | https://arxiv.org/abs/2404.05750 |