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Period-index in top cohomology over semiglobal fields (2312.03934v1)
Published 6 Dec 2023 in math.NT, math.AG, and math.RA
Abstract: We prove a common slot lemma for symbols in top cohomology classes over semiglobal fields. Furthermore, we prove that period and index agree for general top cohomology classes over such fields. We discuss applications to quadratic forms and related open problems.
- Open problems on central simple algebras. Transform. Groups, 16(1):219–264, 2011.
- Beweis eines Hauptsatzes in der Theorie der Algebren. J. Reine Angew. Math., 167:399–404, 1932.
- J. W. S. Cassels. Local Fields. London Mathematical Society Student Texts. Cambridge University Press, 1986.
- Saurabh Gosavi. Generalized period-index problem with an application to quadratic forms, 2022. arXiv:1910.02473.
- Central simple algebras and Galois cohomology, volume 165 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2nd revised and updated edition edition, 2017.
- Applications of patching to quadratic forms and central simple algebras. Invent. Math., 178(2):231–263, 2009.
- Local-global principles for Galois cohomology. Comment. Math. Helv., 89(1):215–253, 2014.
- Bounding cohomology classes over semiglobal fields, 2023. arXiv:2203.06770.
- Local-global principles for curves over semi-global fields. Bull. Lond. Math. Soc., 53(1):177–193, 2021.
- Daniel Krashen. Period and index, symbol lengths, and generic splittings in galois cohomology. Bulletin of the London Mathematical Society, page bdw060, 2016.
- Joseph Lipman. Introduction to resolution of singularities. In Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), pages 187–230. Amer. Math. Soc., Providence, R.I., 1975.
- Joseph Lipman. Desingularization of two-dimensional schemes. Ann. of Math. (2), 107(1):151–207, 1978.
- A. S. Merkurjev. Simple algebras and quadratic forms. Izv. Akad. Nauk SSSR Ser. Mat., 55(1):218–224, 1991.
- Cohomology of number fields, volume 323 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 2008.
- An exact sequence for K∗M/2subscriptsuperscript𝐾𝑀∗2K^{M}_{\ast}/2italic_K start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT / 2 with applications to quadratic forms. Ann. of Math. (2), 165(1):1–13, 2007.
- Richard S. Pierce. Associative algebras, volume 9 of Studies in the History of Modern Science. Springer-Verlag, New York-Berlin, 1982.
- R. Parimala and V. Suresh. Isotropy of quadratic forms over function fields of p𝑝pitalic_p-adic curves. Inst. Hautes Études Sci. Publ. Math., (88):129–150 (1999), 1998.
- Raman Parimala and V. Suresh. The u𝑢uitalic_u-invariant of the function fields of p𝑝pitalic_p-adic curves. Ann. Math. (2), 172(2):1391–1405, 2010.
- R. Parimala and V. Suresh. Period-index and u𝑢uitalic_u-invariant questions for function fields over complete discretely valued fields. Invent. Math., 197(1):215–235, 2014.
- R. Parimala and V. Suresh. On the u-invariant of function fields of curves over complete discretely valued fields. Advances in Mathematics, 280:729–742, 2015.
- David J. Saltman. Division algebras over p𝑝pitalic_p-adic curves. J. Ramanujan Math. Soc., 12(1):25–47, 1997.
- David J. Saltman. Correction to: “Division algebras over p𝑝pitalic_p-adic curves” [J. Ramanujan Math. Soc. 12 (1997), no. 1, 25–47; MR1462850 (98d:16032)]. J. Ramanujan Math. Soc., 13(2):125–129, 1998.
- David J. Saltman. Finite u𝑢uitalic_u invariant and bounds on cohomology symbol lengths. Doc. Math., pages 577–590, 2015.
- Jean-Pierre Serre. Cohomologie Galoisienne, volume Vol. 5 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1973. Cours au Collège de France, Paris, 1962–1963, Avec des textes inédits de J. Tate et de Jean-Louis Verdier, Quatrième édition.
- Jean-Pierre Serre. Galois Cohomology. Springer Monographs in Mathematics. Springer-Verlag, New York-Berlin, 1997.
- T. A. Springer. Quadratic forms over fields with a discrete valuation. I. Equivalence classes of definite forms. Indag. Math., 17:352–362, 1955. Nederl. Akad. Wetensch. Proc. Ser. A 58.
- Venapally Suresh. Third Galois cohomology group of function fields of curves over number fields. Algebra Number Theory, 14(3):701–729, 2020.
- John Tate. Relations between K2subscript𝐾2K_{2}italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and Galois cohomology. Invent. Math., 36:257–274, 1976.
- Vladimir Voevodsky. Motivic cohomology with 𝐙/2𝐙2{\bf Z}/2bold_Z / 2-coefficients. Publ. Math. Inst. Hautes Études Sci., (98):59–104, 2003.