ON KERVAIRE AND MURTHY'S CONJECTURE

Let p be a semi-regular prime, let ${C_{{p^n}}}$ be a cyclic group of order pn and let ζn be a primitive pn+1th root of unity. There is a short exact sequence $0 \to V_n^ + \oplus V_n^ - \to Pic\,Z{C_{{p^{n + 1}}} \to C1\,Q\left( {{\zeta _n}} \right) + Pic\,Z{C_{{p^n}}} \to 0$ In 1977 Kervaire and M...

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Bibliographic Details
Published in:The Rocky Mountain journal of mathematics 2002-12, Vol.32 (4), p.1467-1483
Main Authors: HELENIUS, OLA, STOLIN, ALEXANDER
Format: Article
Language:English
Subjects:
11R21
11R65
19A31
Algebra
Exact sciences and technology
Integers
integral group rings
K-theory
Mathematical rings
Mathematical sequences
Mathematical theorems
Mathematics
Number theory
Picard groups
Quotients
Sciences and techniques of general use
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Summary:Let p be a semi-regular prime, let ${C_{{p^n}}}$ be a cyclic group of order pn and let ζn be a primitive pn+1th root of unity. There is a short exact sequence $0 \to V_n^ + \oplus V_n^ - \to Pic\,Z{C_{{p^{n + 1}}} \to C1\,Q\left( {{\zeta _n}} \right) + Pic\,Z{C_{{p^n}}} \to 0$ In 1977 Kervaire and Murthy established an exact structure for $V_n^ - $, proved that Char $\left( {V_n^ + } \right)$ ⊆ Char $\left( {V_n^ + } \right)$ (Q(ζn-1)), where Vn is a canonical quotient of Vn and conjectured that Char $\left( {V_n^ + } \right)$ ≅ (Z/pnZ)r where r is the index of irregularity of p. We prove that, under a certain extra condition on p, Vn ≅ C1(p)(Q(ζn-1)) ≅(Z/pnZ)r and ${V_n} \cong \oplus _{i = 1}^r\left( {Z/{p^{n - {\delta _i}}}Z} \right)$, where δi is 0 or 1.
ISSN:0035-7596
1945-3795
DOI:10.1216/rmjm/1181070034