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Motivic spheres and the image of the Suslin-Hurewicz map. (English) Zbl 1444.19004
The authors use computations in \(\mathbb{A}^1\)-homotopy theory to etablish a new case of a delicate conjecture of Suslin in \(K\)-theory. In [Lect. Notes Math. 1046, 357–375 (1984; Zbl 0528.18007)], A. A. Suslin constructed, for infinite fields \(F\), natural maps \(K_n^Q(F)\to K_n^M(F)\) from the Quillen \(K\)-theory to the Milnor \(K\)-theory of \(F\). He showed that image contains \((n-1)!K_n^M(F)\) and conjectured that this was precisely the image for all \(n\) and all infinite fields \(F\). He showed that when \(n=3\) this conjecture is equivalent to the case \(n=3\) of Milnor’s conjecture on quadratic forms. In the article under review, the authors prove the case \(n=5\) of Suslin’s conjecture for local algebras \(A\) that are essentially smooth over an infinite field \(k\) of characteristic not equal to \(2\) or \(3\).
The main idea is to compare Suslin’s homomorphism above (this is the Suslin-Hurewicz map of the title) to a morphism: \(\psi_n:\mathbf{K}_n^Q\to \mathbf{K}_n^M\) of sheaves occurring in a previous work of the first two authors [J. Topol. 7, No. 3, 894–926 (2014; Zbl 1326.14098)]. Letting \(\mathbf{S}_n\) denote the cokernel of \(\psi_n\), the authors build on the seminal work of F. Morel [\(\mathbb A^1\)-algebraic topology over a field. Berlin: Springer (2012; Zbl 1263.14003)] as well as recent work by M. Schlichting [Adv. Math. 320, 1–81 (2017; Zbl 1387.19002)] to prove that for rings \(A\) as above (but for fields \(k\) of any characteristic) Suslin’s conjecture holds in degree \(n\) if and only if the surjection \(\mathbf{K}_n^M\to \mathbf{S}_n\) induces an isomorphism \(\left(\mathbf{K}^M_n/(n-1)!\right)(A)\to \mathbf{S}_n(A)\). In the last part of the article, the authors use intricate and interesting calculations in the \(\mathbb{A}^1\)-homotopy theory of motivic spheres to arrive at their result in the case \(n=5\).
MSC:
19D50 Computations of higher \(K\)-theory of rings
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