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Tangent space to Milnor $$K$$-groups of rings. (English. Russian original) Zbl 1334.19003
Proc. Steklov Inst. Math. 290, 26-34 (2015); translation from Tr. Mat. Inst. Steklova 290, 34-42 (2015).
The main result of this elegant paper is the existence of a natural isomorphism from the tangent space to the $$(n+1)$$th Milnor $$K$$-group, $$TK_{n+1}^M(R)$$, of a commutative ring $$R$$ to the $$n$$th group of Kähler differentials, $$\Omega^n_R$$, provided $$1/2\in R$$ and $$R$$ is weakly $$5$$-fold stable. This proves the result for a larger class of rings than was previously known, even when $$n=1$$. Furthermore, the method of proof involves only explicit calculations with symbols and relations among them.
The tangent space, $$TF$$, to a functor $$F$$ from the category of commutative rings to the category of abelian groups is defined to be the kernel of the natural map $$T(R[\epsilon])\to T(R)$$ ($$\epsilon\mapsto 0$$), where $$\mathbb R[\epsilon]$$ is the ring of dual numbers. The condition that a ring $$R$$ be weakly $$k$$-fold stable, due to M. Morrow [J. Algebra 399, 190–204 (2014; Zbl 1308.19003)], is the following: For any collection of elements $$r_1,\dots, r_{k-1}$$ of elements of $$R$$ there is a unit $$r\in R^*$$ such that all of the elements $$r_1+s,\dots,r_{k-1}+s$$ are units in $$R$$.
A ring that is $$k$$-fold stable, in the sense of W. van der Kallen [Ann. Sci. Éc. Norm. Supér. (4) 10, 473–515 (1977; Zbl 0393.18012)], is weakly $$k$$-fold stable. W. van der Kallen proved the theorem in the case $$n=1$$ for rings that are $$5$$-fold stable ([C. R. Acad. Sci., Paris, Sér. A 273, 1204–1207 (1971; Zbl 0225.13006)] and [loc. cit]). However, as the authors point out, for a commutative ring $$A$$ which is not a field, the ring of Laurent series $$A((t))$$ is not $$k$$-fold stable for any $$k\geq 1$$, but is weakly $$k$$-fold stable for any $$k\geq 2$$. This is of particular significance since the authors’ immediate application of their theorem is to derive an explicit formula for the higher-dimensional Contou-Carrére symbols [S. O. Gorchinskiy and D. V. Osipov, Sb. Math. 206, No. 9, 1191–1259 (2015; Zbl 1337.19004); translation from Mat. Sb. 206, No. 9, 21–98 (2015)], for which they must apply the theorem in the case where $$R$$ is an iterated Laurent series ring.

##### MSC:
 19D45 Higher symbols, Milnor $$K$$-theory
##### Keywords:
Milnor K-theory; symbols; Kahler differentials
Full Text:
##### References:
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