Abstract:The category of framed correspondences $\operatorname {Fr}_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [Notes on framed correspondences, 2001, https://www.math.ias.edu/vladimir/publications]. Based on the theory, framed motives are introduced and studied in Garkusha and Panin [J. Amer. Math. Soc. 34 (2021), pp. 261–313]. These are Nisnivich sheaves of $S^1$-spectra and the major computational tool of Garkusha and Panin. The aim of this paper is to show the following result which is essential in proving the main theorem of Garkusha and Panin: given an infinite perfect base field $k$, any $k$-smooth scheme $X$ and any $n\geqslant 1$, the map of simplicial pointed Nisnevich sheaves $(-,\mathbb {A}^1//\mathbb G_m)^{\wedge n}_+\to T^n$ induces a Nisnevich local level weak equivalence of $S^1$-spectra \begin{equation*} M_{fr}(X\times (\mathbb {A}^1// \mathbb G_m)^{\wedge n})\to M_{fr}(X\times T^n). \end{equation*} Moreover, it is proven that the sequence of $S^1$-spectra \begin{equation*} M_{fr}(X \times T^n \times \mathbb G_m) \to M_{fr}(X \times T^n \times \mathbb A^1) \to M_{fr}(X \times T^{n+1}) \end{equation*} is locally a homotopy cofiber sequence in the Nisnevich topology. Another important result of this paper shows that homology of framed motives is computed as linear framed motives in the sense of Garkusha and Panin. This computation is crucial for the whole machinery of framed motives.

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中文翻译：

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相关动机领域的框架动机

摘要：框对应$ \ operatorname的分类{}神父_ *（K）$，嫁祸滑轮被沃沃斯基在他未发表的笔记[发明说明上的对应框架, 2001, https://www.math.ias.edu/vladimir/publications]。基于该理论，加库沙和帕宁引入并研究了框架动机[J. 阿米尔。数学。社会。34 (2021)，第 261-313 页]。这些是 $S^1$-spectra 的 Nisnivich 层和 Garkusha 和 Panin 的主要计算工具。本文的目的是展示以下对证明 Garkusha 和 Panin 的主要定理至关重要的结果：给定一个无限完美基场 $k$，任何 $k$-smooth 方案 $X$ 和任何 $n\geqslant 1$，单纯指向 Nisnevich 层的映射 $(-,\mathbb {A}^1//\mathbb G_m)^{\wedge n}_+\to T^n$ 诱导 $ 的 Nisnevich 局部弱等价S^1$-spectra \begin{equation*} M_{fr}(X\times (\mathbb {A}^1// \mathbb G_m)^{\wedge n})\to M_{fr}(X\次 T^n)。\end{equation*} 此外，证明$S^1$-spectra的序列 \begin{equation*} M_{fr}(X \times T^n \times \mathbb G_m) \to M_{fr}(X \times T^n \times \mathbb A^1) \to M_{fr}(X \times T^{n+1}) \end{equation*} 是 Nisnevich 拓扑中的局部同伦共纤序列。本文的另一个重要结果表明，框架动机的同源性被计算为 Garkusha 和 Panin 意义上的线性框架动机。这种计算对于框架动机的整个机制至关重要。

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