Noncommutative motives of separable algebras

详细信息    查看全文

• 作者：Gonç ; alo Tabuada^a ; ^b ; ^c ; ¹ ; ^{tabuada@math.mit.edu" class="auth_mail" title="E-mail the corresponding author} ; Michel Van den Bergh^d ; ² ; ^{michel.vandenbergh@uhasselt.be" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16H05 ; 16K50 ; 14M15 ; 18D20 ; 20C08
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：5 November 2016
• 年：2016
• 卷：303
• 期：Complete
• 页码：1122-1161
• 全文大小：712 K

文摘

In this article we study in detail the category of noncommutative motives of separable algebras class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816310891&_mathId=si1.gif&_user=111111111&_pii=S0001870816310891&_rdoc=1&_issn=00018708&md5=c0088f9e14b120c84b9447f72dec86bc" title="Click to view the MathML source">Sep(k)class="mathContainer hidden">class="mathCode">$\mathrm{Sep}(k)$ over a base field k  . We start by constructing four different models of the full subcategory of commutative separable algebras class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816310891&_mathId=si2.gif&_user=111111111&_pii=S0001870816310891&_rdoc=1&_issn=00018708&md5=facaf4d5f2e00c5671ccd44c9624e971" title="Click to view the MathML source">CSep(k)class="mathContainer hidden">class="mathCode">$\mathrm{CSep}(k)$. Making use of these models, we then explain how the category class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816310891&_mathId=si1.gif&_user=111111111&_pii=S0001870816310891&_rdoc=1&_issn=00018708&md5=c0088f9e14b120c84b9447f72dec86bc" title="Click to view the MathML source">Sep(k)class="mathContainer hidden">class="mathCode">$\mathrm{Sep}(k)$ can be described as a “fibered class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816310891&_mathId=si3.gif&_user=111111111&_pii=S0001870816310891&_rdoc=1&_issn=00018708&md5=6ec748137adec7deb3e7a024314cd8c4" title="Click to view the MathML source">Zclass="mathContainer hidden">class="mathCode">$\mathbb{Z}$-order” over class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816310891&_mathId=si2.gif&_user=111111111&_pii=S0001870816310891&_rdoc=1&_issn=00018708&md5=facaf4d5f2e00c5671ccd44c9624e971" title="Click to view the MathML source">CSep(k)class="mathContainer hidden">class="mathCode">$\mathrm{CSep}(k)$. This viewpoint leads to several computations and structural properties of the category class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816310891&_mathId=si1.gif&_user=111111111&_pii=S0001870816310891&_rdoc=1&_issn=00018708&md5=c0088f9e14b120c84b9447f72dec86bc" title="Click to view the MathML source">Sep(k)class="mathContainer hidden">class="mathCode">$\mathrm{Sep}(k)$. For example, we obtain a complete dictionary between directs sums of noncommutative motives of central simple algebras (= CSA) and sequences of elements in the Brauer group of k. As a first application, we establish two families of motivic relations between CSA which hold for every additive invariant (e.g. algebraic K-theory, cyclic homology, and topological Hochschild homology). As a second application, we compute the additive invariants of twisted flag varieties using solely the Brauer classes of the corresponding CSA. Along the way, we categorify the cyclic sieving phenomenon and compute the (rational) noncommutative motives of purely inseparable field extensions and of dg Azumaya algebras.