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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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##### References:
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