Graded Witt kernels of the compositum of multiquadratic extensions with the function fields of Pfister forms

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• 作者：Roberto Aravire^a ; ^{raravire@unap.cl" class="auth_mail" title="E-mail the corresponding author} ; Ahmed Laghribi^b ; ^{ahmed.laghribi@univ-artois.fr" class="auth_mail" title="E-mail the corresponding author} ; Manuel O'Ryan^c ; ^{moryan@inst-mat.utalca.cl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11E04 ; 11E81
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：449
• 期：Complete
• 页码：635-659
• 全文大小：500 K

文摘

Let F   be a field of characteristic 2 and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si1.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=1c06753d08eca65fd0abcc6dbca385f3" title="Click to view the MathML source">W_q(F)class="mathContainer hidden">class="mathCode">$W_q(F)$ be the Witt group of nonsingular quadratic forms over F. Let φ be a bilinear Pfister form over F and L be a multiquadratic extension of F   of separability degree less than of equal to 2. In this paper we compute the kernel of the natural homomorphism class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si2.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=4fbc3b5c5286c3519827c91a6878a2e5">class="imgLazyJSB inlineImage" height="19" width="194" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si2.gif">class="mathContainer hidden">class="mathCode">$H_2^{m+1}(F)⟶H_2^{m+1}(L(\phi ))$, where class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si3.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=55782a01236639ec12e8a75827dee106">class="imgLazyJSB inlineImage" height="19" width="69" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si3.gif">class="mathContainer hidden">class="mathCode">$H_2^{m+1}(F)$ is the cokernel of the Artin–Schreier operator class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si4.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=7cea30e3acbd5b13fc70e4866e5a73c3">class="imgLazyJSB inlineImage" height="20" width="164" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si4.gif">class="mathContainer hidden">class="mathCode">$\wp :{\mathrm{\Omega }}_F^m⟶{\mathrm{\Omega }}_F^m/d{\mathrm{\Omega }}_F^{m-1}$ given by class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si5.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=bc113f745cdf1548a52b9390cf8cde52">class="imgLazyJSB inlineImage" height="22" width="307" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si5.gif">class="mathContainer hidden">class="mathCode">$x\frac{d{x}_1}{x_1}\wedge \cdots \wedge \frac{d{x}_m}{x_m}\mapsto (x^2-x)\frac{d{x}_1}{x_1}\wedge \cdots \wedge \frac{d{x}_m}{x_m}$, where class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si6.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=7d10ea0a7171092f26ce0eab1b26e00a">class="imgLazyJSB inlineImage" height="17" width="25" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si6.gif">class="mathContainer hidden">class="mathCode">${\mathrm{\Omega }}_F^m$ is the space of m-differential forms over F  , and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si7.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=a19f9329fe6027ebf015ddd9d7e247ec" title="Click to view the MathML source">F(φ)class="mathContainer hidden">class="mathCode">$F(\phi )$ is the function field of the affine quadric given by the diagonal quadratic form associated to the bilinear form φ  . As a consequence, we deduce the kernel of the natural homomorphisms class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si8.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=9942664ed6e23ffb05df71c052523dcc">class="imgLazyJSB inlineImage" height="21" width="181" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si8.gif">class="mathContainer hidden">class="mathCode">$\overline{I_q^{m+1}}(F)⟶\overline{I_q^{m+1}}(L(\phi ))$ and class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si9.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=5b7286aa8ae151128f5c0d68d843e20d">class="imgLazyJSB inlineImage" height="20" width="181" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si9.gif">class="mathContainer hidden">class="mathCode">$I_q^{m+1}(F)⟶I_q^{m+1}(L(\phi ))$, where class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si10.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=8c3ba1e66bccfa94a07b36692a329ee7">class="imgLazyJSB inlineImage" height="21" width="63" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si10.gif">class="mathContainer hidden">class="mathCode">$\overline{I_q^{m+1}}(F)$ denotes the quotient class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si11.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=a8f9cab7a383c74b7c976400347b0c01">class="imgLazyJSB inlineImage" height="20" width="133" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si11.gif">class="mathContainer hidden">class="mathCode">$I_q^{m+1}(F)/I_q^{m+2}(F)$ such that class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si12.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=f9bf7bd6cc5e9685edfc2cb16d6bd558">class="imgLazyJSB inlineImage" height="20" width="185" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005918-si12.gif">class="mathContainer hidden">class="mathCode">$I_q^{m+1}(F)=I^{m}F\otimes W_q(F)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005918&_mathId=si13.gif&_user=111111111&_pii=S0021869315005918&_rdoc=1&_issn=00218693&md5=6eb393cd89d76094edd672daf0527e3c" title="Click to view the MathML source">I^mFclass="mathContainer hidden">class="mathCode">$I^{m}F$ is the m-th power of the fundamental ideal IF of the Witt ring of F-bilinear forms. We also include some results concerning the case where φ is replaced by a bilinear Pfister neighbor or a quadratic Pfister form.