Degree three unramified cohomology groups

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• 作者：Akinari Hoshi^a ; ^{hoshi@math.sc.niigata-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Ming-chang Kang^b ; ^{kang@math.ntu.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; Aiichi Yamasaki^c ; ^{aiichi.yamasaki@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14E08 ; 12G05
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：458
• 期：Complete
• 页码：120-133
• 全文大小：363 K

文摘

Let k be any field, G be a finite group. Let G   act on the rational function field class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si1.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=44dd4d33d7e4b738aaa48ae805bebffd" title="Click to view the MathML source">k(x_g:g∈G)class="mathContainer hidden">class="mathCode">$k(x_g:g\in G)$ by k  -automorphisms defined by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si2.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=49b6e0503570300e255d3fe39628212b" title="Click to view the MathML source">h⋅x_g=x_hgclass="mathContainer hidden">class="mathCode">$h\cdot x_g=x_{hg}$ for any class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si3.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=049eecc58341c64c200b2686ef8a587d" title="Click to view the MathML source">g,h∈Gclass="mathContainer hidden">class="mathCode">$g,h\in G$. Denote by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si4.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=f7e48e7d1ddebdf6298a3d46a1db8626" title="Click to view the MathML source">k(G)=k(x_g:g∈G)^Gclass="mathContainer hidden">class="mathCode">$k(G)=k{(x_g:g\in G)}^G$, the fixed subfield. Noether's problem asks whether class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si5.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=ed398905d47c240cc00214137b37dcb2" title="Click to view the MathML source">k(G)class="mathContainer hidden">class="mathCode">$k(G)$ is rational (= purely transcendental) over k  . The unramified Brauer group class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si26.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=3bf2def1df8bd7b2591bf7b24ed055fa" title="Click to view the MathML source">Br_nr(C(G))class="mathContainer hidden">class="mathCode">${\mathrm{Br}}_{\mathrm{nr}}(\mathbb{C}(G))$ and the unramified cohomology class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si49.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=6559afd9f228ba54dcb37f81c74c9320">class="imgLazyJSB inlineImage" height="18" width="115" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316300047-si49.gif">class="mathContainer hidden">class="mathCode">$H_{\mathrm{nr}}^3(\mathbb{C}(G),\mathbb{Q}/\mathbb{Z})$ are obstructions to the rationality of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si12.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=3a5bdb40b7b973f788f97806a88445eb" title="Click to view the MathML source">C(G)class="mathContainer hidden">class="mathCode">$\mathbb{C}(G)$ (see [14] and [5]). Peyre proves that, if p is an odd prime number, then there is a group G   such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si9.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=aee9bcfa599f8b9f5989be8c50298d17" title="Click to view the MathML source">|G|=p¹²class="mathContainer hidden">class="mathCode">$|G|=p^{12}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si10.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=e80699c86344fb24c08286f1bff84167" title="Click to view the MathML source">Br_nr(C(G))={0}class="mathContainer hidden">class="mathCode">${\mathrm{Br}}_{\mathrm{nr}}(\mathbb{C}(G))=\{0\}$, but class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si11.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=eb3791e9a21e11898abc39fbbdcc760a">class="imgLazyJSB inlineImage" height="18" width="162" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316300047-si11.gif">class="mathContainer hidden">class="mathCode">$H_{\mathrm{nr}}^3(\mathbb{C}(G),\mathbb{Q}/\mathbb{Z})\ne \{0\}$; thus class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si12.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=3a5bdb40b7b973f788f97806a88445eb" title="Click to view the MathML source">C(G)class="mathContainer hidden">class="mathCode">$\mathbb{C}(G)$ is not stably class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si13.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=8b9a790d2e5b00d5162ee1fea49b5baa" title="Click to view the MathML source">Cclass="mathContainer hidden">class="mathCode">$\mathbb{C}$-rational [12]. Using Peyre's method, we are able to find groups G   with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si14.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=05a22d45a04e08285406521286cc7a70" title="Click to view the MathML source">|G|=p⁹class="mathContainer hidden">class="mathCode">$|G|=p^9$ where p   is an odd prime number such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si10.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=e80699c86344fb24c08286f1bff84167" title="Click to view the MathML source">Br_nr(C(G))={0}class="mathContainer hidden">class="mathCode">${\mathrm{Br}}_{\mathrm{nr}}(\mathbb{C}(G))=\{0\}$, class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300047&_mathId=si11.gif&_user=111111111&_pii=S0021869316300047&_rdoc=1&_issn=00218693&md5=eb3791e9a21e11898abc39fbbdcc760a">class="imgLazyJSB inlineImage" height="18" width="162" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316300047-si11.gif">class="mathContainer hidden">class="mathCode">$H_{\mathrm{nr}}^3(\mathbb{C}(G),\mathbb{Q}/\mathbb{Z})\ne \{0\}$.