Gersten's Injectivity for Smooth Algebras over Valuation Rings

Gersten's injectivity conjecture for a functor $F$ of ``motivic type'', predicts that given a semilocal, ``non-singular'', integral domain $R$ with a fraction field $K$, the restriction morphism induces an injection of $F(R)$ inside $F(K)$. We prove two new cases of this conjecture for smooth algebras over valuation rings. Namely, we show that the higher algebraic $K$-groups of a semilocal, integral domain that is an essentially smooth algebra over an equicharacteristic valuation ring inject inside the same of its fraction field. Secondly, we show that Gersten's injectivity is true for smooth algebras over, possibly of mixed-characteristic, valuation rings in the case of torsors under tori and also in the case of the Brauer group.