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We propose a definition of improved Milnor K-groups of local rings with finite residue fields, such that the improved Milnor Ksheaf in the Zariski topology is a universal extension of the naive Milnor K-sheaf with a certain transfer map for etale extensions of local rings. The main theorem states that the improved Milnor K-ring is generated by elements of degree one.

Milnor 𝐾-theory of local rings with finite residue fields