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Framed motives of algebraic varieties (after V. Voevodsky)
Journal of the American Mathematical Society ( IF 3.9 ) Pub Date : 2020-12-03 , DOI: 10.1090/jams/958
Grigory Garkusha , Ivan Panin

Using the machinery of framed sheaves developed by Voevodsky, a triangulated category of framed motives $SH_{S^1}^{fr}(k)$ is introduced and studied. To any smooth algebraic variety $X\in Sm/k$, the framed motive $M_{fr}(X)$ is associated in the category $SH_{S^1}^{fr}(k)$. Also, for any smooth scheme $X\in Sm/k$ an explicit quasi-fibrant motivic replacement of its suspension P^1-spectrum is given. Moreover, it is shown that the bispectrum $(M_{fr}(X),M_{fr}(X)(1),M_{fr}(X)(2),\ldots),$ each term of which is a twisted framed motive of X, has motivic homotopy type of the suspension bispectrum of X. We also construct a compactly generated triangulated category of framed bispectra $SH^{fr}(k)$ and show that it reconstructs the Morel-Voevodsky category SH(k). As a topological application, it is proved that the framed motive $M_{fr}(pt)(pt)$ of the point pt=spec k evaluated at $pt$ is a quasi-fibrant model of the classical sphere spectrum whenever the base field k is algebraically closed of characteristic zero. As another application, a reformulation of the motivic Serre Finiteness Conjecture is given.

中文翻译:

代数簇的框架动机(在 V. Voevodsky 之后)

使用 Voevodsky 开发的框架滑轮机械,介绍和研究了框架动机 $SH_{S^1}^{fr}(k)$ 的三角范畴。对于任何平滑代数变量 $X\in Sm/k$,框架动机 $M_{fr}(X)$ 与类别 $SH_{S^1}^{fr}(k)$ 相关联。此外,对于任何平滑方案 $X\in Sm/k$,给出了其悬浮 P^1 谱的显式准纤维动机替换。此外,还表明双谱 $(M_{fr}(X),M_{fr}(X)(1),M_{fr}(X)(2),\ldots),$ 的每一项是X 的扭曲框架动机,具有 X 的悬浮双谱的动机同伦类型。我们还构建了一个紧凑生成的框架双谱 $SH^{fr}(k)$ 三角化范畴,并表明它重建了 Morel-Voevodsky 范畴 SH (k)。作为拓扑应用,证明了在 $pt$ 处评估的点 pt=spec k 的框架动机 $M_{fr}(pt)(pt)$ 是经典球谱的准纤维模型,只要基场 k 代数闭合特征为零。作为另一个应用,给出了动机塞尔有限猜想的重新表述。
更新日期:2020-12-03
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