Homotopy invariant presheaves with framed transfers

IF 1.8 2区 数学 Q1 MATHEMATICS Cambridge Journal of Mathematics Pub Date : 2015-04-03 DOI:10.4310/cjm.2020.v8.n1.a1
G. Garkusha, I. Panin
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引用次数: 39

Abstract

The category of framed correspondences $Fr_*(k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [12]. Based on the theory, framed motives are introduced and studied in [7]. The main aim of this paper is to prove that for any $\mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $\mathcal F$, the associated Nisnevich sheaf $\mathcal F_{nis}$ is $\mathbb A^1$-invariant whenever the base field $k$ is infinite of characteristic different from 2. Moreover, if the base field $k$ is infinite perfect of characteristic different from 2, then every $\mathbb A^1$-invariant quasi-stable Nisnevich framed sheaf of Abelian groups is strictly $\mathbb A^1$-invariant and quasi-stable. Furthermore, the same statements are true in characteristic 2 if we also assume that the $\mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $\mathcal F$ is a presheaf of $\mathbb Z[1/2]$-modules. This result and the paper are inspired by Voevodsky's paper [13].
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带框架转移的同伦不变预轴
框架通信$Fr_*(k)$、框架预捆和框架捆的范畴是由Voevodsky在他未发表的笔记[12]中发明的。在此基础上,对b[7]中的框架动机进行了介绍和研究。本文的主要目的是证明对于任意Abelian群$\mathcal F$的$\mathbb A^1$-不变拟稳定共轭框架预集,当基域$k$为无限且特征不等于2时,所关联的$\mathbb A^1$-不变的Nisnevich预集$\mathcal F_{nis}$。此外,如果基域$k$是特征异于2的无限完美,则每一个$\mathbb A^1$不变拟稳定的阿贝群Nisnevich框架丛都是严格的$\mathbb A^1$不变拟稳定的。更进一步,如果我们也假设阿贝尔群的$\mathbb A^1$-不变拟稳定的共轭框架预表$\mathbb F$是$\mathbb Z[1/2]$-模块的预表$ $ $,同样的陈述在特征2中是成立的。这一结果和这篇论文的灵感来自Voevodsky的论文[13]。
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3.10
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7
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