## 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

### MSC:

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

### Keywords:

local ring; Gersten resolution

Zbl 0588.00014