This is not a direct answer to the original question, but is what I am interested in.

I found the following in 12.14(iii) of Brylinski and Deligne's paper "Central Extensions of Reductive Groups by K_2". I'll quote the relevant paragraph and comment afterwards.

Suppose that V is henselian and essentially of finite type over a field. For j (resp i) the inclusion of G (resp G_s) in G_V, Quillen resolution gives a short exact sequence of sheaves on G_V.

0 --> K₂ --> jK₂ --> iK₁(D) --> 0.

The K's are sheafified K-theory on the big Zariski site. G is the generic fibre of a smooth group scheme G_V, with special fibre G_s.

What I don't know is what "essentially of finite type over a field" means, nor how this exact sequence arises.

This is not a direct answer to the original question, but is what I am interested in.

I found the following in 12.14(iii) of Brylinski and Deligne's paper "Central Extensions of Reductive Groups by K_2". I'll quote the relevant paragraph and comment afterwards.

Suppose that V is henselian and essentially of finite type over a field. For j (resp i) the inclusion of G (resp G_s) in G_V, Quillen resolution gives a short exact sequence of sheaves on G_V. $$0 \to K_2 \to j_\*K_2 \to i_\*K_1(D) \to 0$$

The K's are sheafified K-theory on the big Zariski site. G is the generic fibre of a smooth group scheme G_V, with special fibre G_s.

What I don't know is what "essentially of finite type over a field" means, nor how this exact sequence arises.