Inductive graded rings, hyperfields and quadratic forms

Document Type : Research Paper

Authors

Institute of Mathematics and Statistics, University of São Paulo, Brazil.

Abstract

In [6] we developed a k-theory for the category of hyperbolic hyperfields (a category that contains a copy of the category of (pre)special groups): this construction extends, simultaneously, Milnor's k-theory ([20]) and Dickmann-Miraglia's k-theory ([13]). An abstract environment that encapsulate all them, and of course, provide an axiomatic approach to guide new extensions of the concept of K-theory in the context of the algebraic and abstract theories of quadratic forms is given by the concept of inductive graded rings a concept introduced in [9] in order to provide a solution of Marshall's signature conjecture in realm the algebraic theory of quadratic forms for Pythagorean fields. The goal of this work is twofold: (i) to provide a detailed analysis of some categories of inductive graded ring - a concept introduced in [9] in order to provide a solution of Marshall's signature conjecture in the algebraic theory of quadratic forms; (ii) apply this analysis to deepen the connections between the category of special hyperfields ([6]) - equivalent to the category of special groups ([10]) and the categories of inductive graded rings.

Keywords

Main Subjects


[1] Ad´amek, J., Adamek, J., Rosick´y, J., et al., “Locally Presentable and Accessible Categories” London Mathematical Society, Lecture Note Series 189, Cambridge University Press, 1994.
[2] Ameri, R., Eyvazi, M., and Hoskova-Mayerova, S., Superring of Polynomials over a Hyperring, Mathematics 7(10) (2019), 902, https://doi.org/10.3390/math7100902.
[3] Borceux, F. “Handbook of Categorical Algebra: Vol. 1, Basic Category Theory”, Cambridge University Press, 1994.
[4] Roberto, K.M.de A., Ribeiro, H.R.de O., and Mariano, H.L., “Quadratic Extensions of Special Hyperfields and the general Arason-Pfister Hauptsatz”, Preliminary version in https://arxiv.org/abs/2210.03784. Submitted, 2022. 
[5] Roberto, K.M.de A., Ribeiro, H.R.de O., and Mariano, H.L., Quadratic structures associated to (multi) rings, Categ. Gen. Algebr. Struct. Appl. 16(1) (2022), 105-141.
[6] Roberto, K.M.de A. and Mariano, H.L., K-theories and free inductive graded rings in abstract quadratic forms theories, Categ. Gen. Algebr. Struct. Appl. 17(1) (2022), 1-46.
[7] Roberto, K.M.de A. and Mariano, H.L., Galois groups of pre special hyperfields, I, in preparation, 2023.
[8] Ribeiro, H.R.de O., Roberto, K.M.de A., and Mariano, H.L., Functorial relationship between multirings and the  various abstract theories of quadratic forms, S˜ao Paulo J. Math. Sci. 16 (2022), 5-42, https://doi.org/10.1007/s40863-020-00185-1.
[9] Dickmann, M. and Miraglia, F., On quadratic forms whose total signature is zero mod 2n: Solution to a problem of M. Marshall, Invent. Math. 133(2) (1998), 243-278.
[10] Dickmann, M. and Miraglia, F., “Special Groups: Boolean-Theoretic Methods in the Theory of Quadratic Forms”, Mem. Amer. Math. Soc. 689, American Mathematical Society, 2000.
[11] Dickmann, M. and Miraglia, F., Elementary properties of the Boolean hull and reduced quotient functors, J. Symb. Log. 68(3) (2003), 946-971.
[12] Dickmann, M. and Miraglia, F., Lam’s Conjecture, Algebra Colloq. 10(2) (2003), 149-176.
[13] Dickmann, M. and Miraglia, F., Algebraic K-theory of special groups, J. Pure Appl. Algebra 204(1) (2006), 195-234.
[14] Ellerman, D.P., Sheaves of structures and generalized ultraproducts, Ann. Math. Logic 7(2) (1974), 163-195.
[15] Gladki, P. and Worytkiewicz, K., Witt rings of quadratically presentable fields, Categ. Gen. Algebr. Struct. Appl. 12(1) (2020), 1-23.
[16] Jun, J., Algebraic Geometry over Hyperrings, Adv. Math. 323 (2018), 142-192.
[17] Lam, T.Y., “Orderings, Valuations and Quadratic Forms”, CBMS-NSF Regional Conference Series in Applied Mathematics 52, American Mathematical Society, 1983.
[18] Mac Lane, S., “Categories for the Working Mathematician”, Graduate Texts in Mathematics 5, Springr, 2013.
[19] Marshall, M., Real reduced multirings and multifields, J. Pure Appl. Algebra 205(2) (2006), 452-468.
[20] Milnor, J., Algebraic K-theory and quadratic forms, Invent. Math. 9(4) (1970), 318-344.
[21] Min´aˇc, J. and Spira, M., Witt rings and Galois groups, Ann. Math. 144 (1996), 35-60.
[22] Pelea, C. and Purdea, I., Multialgebras, universal algebras and identities, J. Austral. Math. Soc. 81(1) (2006), 121-140.
[23] Viro, O. “Hyperfields for Tropical Geometry I. Hyperfields and dequantization”, arXiv preprint arXiv:1006.3034, 2010.