The Equicharacteristic Case of the Gersten Conjecture

One of the well-known problems in the algebraic $K$-theory is the Gersten conjecture. The geometric case of this conjecture was proved by D. Quillen. The equicharacteristic case of the conjecture is proved in this paper. This covers the result of Quillen. Actually we use the result of Quillen and certain results of D. Popescu and A. Grothendieck.