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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
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References:
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