Fibrant resolutions for motivic Thom spectra
Fibrant resolutions for motivic Thom spectra
复制标题
Motivic Thom 光谱的 Fibrant 分辨率
作者：
G. Garkusha;A. Neshitov
本文利用Voevodsky[24]发展的框架对应理论和[6]中引入和发展的框架动机机制，构造了动机Thom谱的各种显式Fibrant分解。结果表明，该双谱具有较高的误码率
$$M_E^{mathbb G}(X)=(M_{E}(X)，M_{E}(X)(1)，M_{E}(X)(2)，ldots)，其中每一项都是$X$的扭曲的$E$-框架动机，本文引入的$X_+楔形E$是双谱范畴中的$X_+楔形E$。作为拓扑应用，证明了当基域$k$是特征零的代数闭域且嵌入$epsilon：khookright tarrowmathbb C$时，点$pt=Spec(K)$的有限系数$M_E(Pt)/N$，$N>；0$的$E$-框架动机是拓扑$S^2$-谱$Re(E)/N$的拟Fibrant模型.此外，还用[15]意义下的$Omega$-对应来计算代数余边谱$MGL$。还证明了$MGL$由一个双谱表示，它的每一项都是单纯光滑拟投射簇的序列极限。
Using the theory of framed correspondences developed by Voevodsky [24] and the machinery of framed motives introduced and developed in [6], various explicit fibrant resolutions for a motivic Thom spectrum $E$ are constructed in this paper. It is shown that the bispectrum
$$M_E^{mathbb G}(X)=(M_{E}(X),M_{E}(X)(1),M_{E}(X)(2),ldots),$$ each term of which is a twisted $E$-framed motive of $X$, introduced in the paper, represents $X_+wedge E$ in the category of bispectra. As a topological application, it is proved that the $E$-framed motive with finite coefficients $M_E(pt)(pt)/N$, $N>0$, of the point $pt=Spec (k)$ evaluated at $pt$ is a quasi-fibrant model of the topological $S^2$-spectrum $Re^epsilon(E)/N$ whenever the base field $k$ is algebraically closed of characteristic zero with an embedding $epsilon:khookrightarrowmathbb C$. Furthermore, the algebraic cobordism spectrum $MGL$ is computed in terms of $Omega$-correspondences in the sense of [15]. It is also proved that $MGL$ is represented by a bispectrum each term of which is a sequential colimit of simplicial smooth quasi-projective varieties.