The exactness of the Gersten complex for affineregular, equicharacteristic excellent local rings with infinite residue field (or the like) is also known as the Gersten conjecture (see e.g. Kerz 08, p. 4).

References

Original references include

Daniel Quillen, Algebraic K-theory, I: Higher K-theories (Battelle Memorial Inst., Seattle 1972), Proc. Conf., Lecture Notes in Math. 341, 1973, pp. 85–147.

Spencer Bloch, Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986), 267–304.

Review includes

Ishai Dan-Cohen, Bloch’s formula and the Gersten resolution (pdf)

Moritz Kerz, Milnor K-theory of local rings Dissertation, Universität Regensburg (2008) (web)

Thomas Geisser, Motivic cohomology, K-theory and topological cyclic homology, chapter II.1 in Handbook of K-theory, pdf

See also

S. O. Gorchinskii, Adelic resolution for sheaves of K-groups, Russ. Math. Surv. 62, p. 184-186 (2007) pdf

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