Witt, GW, K-theory of quasi-projective schemes

详细信息    查看全文

• 作者：Satya Mandal¹ ; ^{mandal@ku.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：13D09 ; 14F05 ; 18E30 ; 19E08
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：221
• 期：2
• 页码：286-303
• 全文大小：481 K

文摘

In this article, we prove some results on Witt, Grothendieck–Witt (GW) and K-theory of noetherian quasi-projective schemes X  , over affine schemes class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300809&_mathId=si1.gif&_user=111111111&_pii=S0022404916300809&_rdoc=1&_issn=00224049&md5=b8724c1830d7ff5256703d2379c359dc" title="Click to view the MathML source">Spec(A)class="mathContainer hidden">class="mathCode">$\mathrm{Spec}\left(A\right)$. For integers class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300809&_mathId=si145.gif&_user=111111111&_pii=S0022404916300809&_rdoc=1&_issn=00224049&md5=dc9256b1ad1e838259c0ad1e48063cd4" title="Click to view the MathML source">k≥0class="mathContainer hidden">class="mathCode">$k\ge 0$, let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300809&_mathId=si3.gif&_user=111111111&_pii=S0022404916300809&_rdoc=1&_issn=00224049&md5=0de56e18b4866c69bdffde86af8f1db8" title="Click to view the MathML source">CM^k(X)class="mathContainer hidden">class="mathCode">$C{\mathbb{M}}^k(X)$ denote the category of coherent class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300809&_mathId=si191.gif&_user=111111111&_pii=S0022404916300809&_rdoc=1&_issn=00224049&md5=e77c3d14aec4a8d95c09d5e44ca1642e" title="Click to view the MathML source">O_Xclass="mathContainer hidden">class="mathCode">${\mathcal{O}}_X$-modules class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300809&_mathId=si5.gif&_user=111111111&_pii=S0022404916300809&_rdoc=1&_issn=00224049&md5=808c1a3a6988bd8a733ed347278d2ad1" title="Click to view the MathML source">Fclass="mathContainer hidden">class="mathCode">$\mathcal{F}$, with locally free dimension class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300809&_mathId=si6.gif&_user=111111111&_pii=S0022404916300809&_rdoc=1&_issn=00224049&md5=9394c4086aab2f57b65af3721121f315" title="Click to view the MathML source">dim_V(X)⁡(F)=k=grade(F)class="mathContainer hidden">class="mathCode">${\dim }_{\mathcal{V}(X)}(\mathcal{F})=k=\text{<italic>grade</italic>}(\mathcal{F})$. We prove that there is an equivalence class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300809&_mathId=si7.gif&_user=111111111&_pii=S0022404916300809&_rdoc=1&_issn=00224049&md5=8a3de55e32d219b4cbed0fb27459c5aa" title="Click to view the MathML source">D^b(CM^k(X))→D^k(V(X))class="mathContainer hidden">class="mathCode">${\mathcal{D}}^b(C{\mathbb{M}}^k(X))\to {\mathcal{D}}^k(\mathcal{V}(X))$ of the derived categories. It follows that there is a sequence of zig-zag maps class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300809&_mathId=si8.gif&_user=111111111&_pii=S0022404916300809&_rdoc=1&_issn=00224049&md5=35af3c1e6e46eb79e93a3c682663a646" title="Click to view the MathML source">K(CM^k+1(X))⟶K(CM^k(X))⟶∐_x∈X^(k)K(CM^k(X_x))class="mathContainer hidden">class="mathCode">$\mathbb{K}(C{\mathbb{M}}^{k+1}(X))⟶\mathbb{K}(C{\mathbb{M}}^k(X))⟶{\coprod }_{x\in X^{(k)}}\mathbb{K}(C{\mathbb{M}}^k(X_x))$ of the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300809&_mathId=si164.gif&_user=111111111&_pii=S0022404916300809&_rdoc=1&_issn=00224049&md5=4ee3955783295d5c1a3acd5d74ae94de" title="Click to view the MathML source">Kclass="mathContainer hidden">class="mathCode">$\mathbb{K}$-theory spectra that is a homotopy fibration. In fact, this is analogous to the homotopy fiber sequence of the G-theory spaces of Quillen (see proof of [16, Theorem 5.4]). We also establish similar homotopy fibrations of class="boldFont">GW-spectra and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300809&_mathId=si10.gif&_user=111111111&_pii=S0022404916300809&_rdoc=1&_issn=00224049&md5=5574770436b31cacc6dba46db0a61ff2" title="Click to view the MathML source">GWclass="mathContainer hidden">class="mathCode">$\mathbb{G}W$-bispectra, by application of the same equivalence theorem.