Classes in Hpmn+1(F) of lower exponent

Adam Chapman; Daniel Krashen; Kelly McKinnie

doi:10.2140/akt.2026.11.37

Classes in H_pmⁿ⁺¹(F) of lower exponent

Adam Chapman
, Daniel Krashen
, Kelly McKinnie

Mathematical Sciences

Academic College of Tel-Aviv - Yaffo
University of Pennsylvania

Research output: Contribution to journal › Article › peer-review

Abstract

Let F be a field of characteristic p > 0. We prove that if a symbol (Formula presented) in (Formula presented) is of exponent dividing p^m−1, then its symbol length in (Formula presented) is at most pⁿ. In the case n = 1, we also prove that if (Formula presented) in (Formula presented) satisfies exp (Formula presented), then the symbol length of A in (Formula presented) is at most p^r + r − 1. We conclude by looking at the case p = 2 and proving that if A is a sum of two symbols in (Formula presented) and exp (Formula presented), then the symbol length of A in (Formula presented) is at most (2n + 1)2ⁿ. Our results use norm conditions in characteristic p in the same manner as Matzri in his 2024 paper “On the symbol length of symbols”.

Original language	English
Pages (from-to)	37-46
Number of pages	10
Journal	Annals of K-Theory
Volume	11
Issue number	1
DOIs	https://doi.org/10.2140/akt.2026.11.37
State	Published - 2026

Keywords

Brauer group
cyclic algebras
fields of positive characteristic
Kato–Milne cohomology
symbol length

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10.2140/akt.2026.11.37

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title = "Classes in Hpmn+1(F) of lower exponent",

abstract = "Let F be a field of characteristic p > 0. We prove that if a symbol (Formula presented) in (Formula presented) is of exponent dividing pm−1, then its symbol length in (Formula presented) is at most pn. In the case n = 1, we also prove that if (Formula presented) in (Formula presented) satisfies exp (Formula presented), then the symbol length of A in (Formula presented) is at most pr + r − 1. We conclude by looking at the case p = 2 and proving that if A is a sum of two symbols in (Formula presented) and exp (Formula presented), then the symbol length of A in (Formula presented) is at most (2n + 1)2n. Our results use norm conditions in characteristic p in the same manner as Matzri in his 2024 paper “On the symbol length of symbols”.",

keywords = "Brauer group, cyclic algebras, fields of positive characteristic, Kato–Milne cohomology, symbol length",

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TY - JOUR

T1 - Classes in Hpmn+1(F) of lower exponent

AU - Chapman, Adam

AU - Krashen, Daniel

AU - McKinnie, Kelly

PY - 2026

Y1 - 2026

N2 - Let F be a field of characteristic p > 0. We prove that if a symbol (Formula presented) in (Formula presented) is of exponent dividing pm−1, then its symbol length in (Formula presented) is at most pn. In the case n = 1, we also prove that if (Formula presented) in (Formula presented) satisfies exp (Formula presented), then the symbol length of A in (Formula presented) is at most pr + r − 1. We conclude by looking at the case p = 2 and proving that if A is a sum of two symbols in (Formula presented) and exp (Formula presented), then the symbol length of A in (Formula presented) is at most (2n + 1)2n. Our results use norm conditions in characteristic p in the same manner as Matzri in his 2024 paper “On the symbol length of symbols”.

AB - Let F be a field of characteristic p > 0. We prove that if a symbol (Formula presented) in (Formula presented) is of exponent dividing pm−1, then its symbol length in (Formula presented) is at most pn. In the case n = 1, we also prove that if (Formula presented) in (Formula presented) satisfies exp (Formula presented), then the symbol length of A in (Formula presented) is at most pr + r − 1. We conclude by looking at the case p = 2 and proving that if A is a sum of two symbols in (Formula presented) and exp (Formula presented), then the symbol length of A in (Formula presented) is at most (2n + 1)2n. Our results use norm conditions in characteristic p in the same manner as Matzri in his 2024 paper “On the symbol length of symbols”.

KW - Brauer group

KW - cyclic algebras

KW - fields of positive characteristic

KW - Kato–Milne cohomology

KW - symbol length

UR - https://www.scopus.com/pages/publications/105033106033

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DO - 10.2140/akt.2026.11.37

M3 - Article

AN - SCOPUS:105033106033

SN - 2379-1683

VL - 11

SP - 37

EP - 46

JO - Annals of K-Theory

JF - Annals of K-Theory

IS - 1

ER -

Classes in H_pmⁿ⁺¹(F) of lower exponent

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Keywords

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