A note on Gersten’s conjecture in the mixed characteristic case. (English) Zbl 0598.13007

Applications of algebraic K-theory to algebraic geometry and number theory, Proc. AMS-IMS-SIAM joint Summer Res. Conf., Boulder/Colo. 1983, Part I, Contemp. Math. 55, 75-78 (1986).
[For the entire collection see Zbl 0588.00014.]
Suppose A is a local ring essentially of finite type and smooth over a discrete valuation ring R. The author proves that the Gersten resolution \(0\to K_ 2(A)\to \coprod_{cod 0}K_ 2(k(x))\to \coprod_{cod 1}k(x)^*\to \coprod_{cod 0}{\mathbb{Z}}\to 0\quad is\) exact, where the direct sums are taken over all cycles of the indicated codimension.
Reviewer: M.R.Stein


13D15 Grothendieck groups, \(K\)-theory and commutative rings
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
13H99 Local rings and semilocal rings


Zbl 0588.00014