We establish a kind of ‘degree $0$ Freudenthal ${\mathbb {G}_m}$ -suspension theorem’ in motivic homotopy theory. From this we deduce results about the conservativity of the $\mathbb P^1$ -stabilization functor.

In order to establish these results, we show how to compute certain pullbacks in the cohomology of a strictly homotopy-invariant sheaf in terms of the Rost–Schmid complex. This establishes the main conjecture of [2], which easily implies the aforementioned results.

中文翻译：

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动空间的零稳定同伦层

我们在动机同伦理论中建立了一种“度 $0$ Freudenthal ${\mathbb {G}_m}$ - 悬置定理”。由此我们推导出关于 $\mathbb P^1$ -稳定函子的保守性的结果。

为了建立这些结果，我们展示了如何根据 Rost-Schmid 复形计算严格同伦不变层的上同调中的某些回调。这就建立了[2]的主要猜想，很容易推导出上述结果。