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The cohomological dimension of the quotient field of the two dimensional complete local domain. (English) Zbl 0731.13010
Let A be a complete Noetherian local domain with separable closed residue field k and with quotient field K. If \(p\neq char(K)\) is prime, let \(cd_ p(K)\) be the p-cohomological dimension of the absolute Galois group \(Gal(K^{sep}/K)\), where \(K^{sep}\) denotes the separable closure of K. In the case when \(\dim (A)=2\), \(char(k)=p\) and \(char(K)=0\), the author proves a conjecture of Artin, namely, that \(cd_ p(K)=\dim (A)+\dim_ k(\Omega_ k^ 1)\), where \(\Omega_ k^ 1=\Omega^ 1_{k/{\mathbb{Z}}}\) is the absolute differential module.

MSC:
13D05 Homological dimension and commutative rings
13G05 Integral domains
13H99 Local rings and semilocal rings
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