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

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• 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 Spec(A)$\mathrm{Spec}\left(A\right)$. For integers k≥0$k\ge 0$, let e56e18b4866c69bdffde86af8f1db8" title="Click to view the MathML source">CM^k(X)$C{\mathbb{M}}^k(X)$ denote the category of coherent O_X${\mathcal{O}}_X$-modules a6988bd8a733ed347278d2ad1" title="Click to view the MathML source">F$\mathcal{F}$, with locally free dimension dim_V(X)⁡(F)=k=grade(F)${\dim }_{\mathcal{V}(X)}(\mathcal{F})=k=\text{<italic>grade</italic>}(\mathcal{F})$. We prove that there is an equivalence e55e32d219b4cbed0fb27459c5aa" title="Click to view the MathML source">D^b(CM^k(X))→D^k(V(X))${\mathcal{D}}^b(C{\mathbb{M}}^k(X))\to {\mathcal{D}}^k(\mathcal{V}(X))$ of the derived categories. It follows that there is a sequence of zig-zag maps e93a3c682663a646" title="Click to view the MathML source">K(CM^k+1(X))⟶K(CM^k(X))⟶∐_x∈X^(k)K(CM^k(X_x))$\mathbb{K}(C{\mathbb{M}}^{k+1}(X))⟶\mathbb{K}(C{\mathbb{M}}^k(X))⟶{\coprod }_{x\in X^{(k)}}\mathbb{K}(C{\mathbb{M}}^k(X_x))$ of the e94de" title="Click to view the MathML source">K$\mathbb{K}$-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 GW-spectra and ba46db0a61ff2" title="Click to view the MathML source">GW$\mathbb{G}W$-bispectra, by application of the same equivalence theorem.