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Witt sheaves and the \(\eta\)-inverted sphere spectrum. (English) Zbl 1378.14021

Summary: The first author [Ann. K-Theory 2, No. 4, 517–560 (2017; Zbl 1401.14117)] has recently computed the stable operations and cooperations of rational Witt theory. These computations enable us to show a motivic analog of Serre’s finiteness result.{ } Theorem. Let \(k\) be a field of characteristic different from two. Then \(\pi_n^{\mathbb{A}^1}(\mathbb{S}_k^-)_\ast\) is torsion for \(n>0\).
As an application, we define a category of Witt motives and show that rationally this category is equivalent to the minus part of \(\mathcal{SH}(k)_{\mathbb{Q}}\).

MSC:

14F42 Motivic cohomology; motivic homotopy theory
19G38 Hermitian \(K\)-theory, relations with \(K\)-theory of rings
55P42 Stable homotopy theory, spectra
11E81 Algebraic theory of quadratic forms; Witt groups and rings

Citations:

Zbl 1401.14117
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