TY - JOUR
T1 - Noether's problem and unramified brauer groups
AU - Hoshi, Akinari
AU - Kang, Ming Chang
AU - Kunyavskii, Boris E.
PY - 2013
Y1 - 2013
N2 - Let k be any field, G be a finite group acting on the rational function field k(xg : g ε G) by h · xg = xhg for any h, g ε G. Define k(G) = k(xg : g ε G)G. Noether's problem asks whether k(G) is rational (= purely transcendental) over k. It is known that, if ℂ(G) is rational over ℂ, then B0(G) = 0 where B0(G) is the unramified Brauer group of ℂ(G) over ℂ. Bogomolov showed that, if G is a p-group of order p5, then B0(G) = 0. This result was disproved by Moravec for p = 3, 5, 7 by computer calculations. We will prove the following theorem. Theorem. Let p be any odd prime number, G be a group of order p5. Then B0(G) ≠ 0 if and only if G belongs to the isoclinism family π10 in R. James's classification of groups of order p5.
AB - Let k be any field, G be a finite group acting on the rational function field k(xg : g ε G) by h · xg = xhg for any h, g ε G. Define k(G) = k(xg : g ε G)G. Noether's problem asks whether k(G) is rational (= purely transcendental) over k. It is known that, if ℂ(G) is rational over ℂ, then B0(G) = 0 where B0(G) is the unramified Brauer group of ℂ(G) over ℂ. Bogomolov showed that, if G is a p-group of order p5, then B0(G) = 0. This result was disproved by Moravec for p = 3, 5, 7 by computer calculations. We will prove the following theorem. Theorem. Let p be any odd prime number, G be a group of order p5. Then B0(G) ≠ 0 if and only if G belongs to the isoclinism family π10 in R. James's classification of groups of order p5.
KW - Bogomolov multipliers
KW - Noether's problem
KW - Rationality
KW - Rationality problem
KW - Retract rationality
KW - Unramified Brauer groups
UR - http://www.scopus.com/inward/record.url?scp=84884952911&partnerID=8YFLogxK
U2 - 10.4310/AJM.2013.v17.n4.a8
DO - 10.4310/AJM.2013.v17.n4.a8
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AN - SCOPUS:84884952911
SN - 1093-6106
VL - 17
SP - 689
EP - 714
JO - Asian Journal of Mathematics
JF - Asian Journal of Mathematics
IS - 4
ER -