\(K_2\) of localisations of local rings
Let \(A\) be a commutative local ring whose residue field has \(>5\) elements. Then it is known that the Quillen \(K\)-group \(K_2(A)\) is isomorphic to the target \(K_2^M(A)\) of the universal Steinberg symbol. This target is also known as the second Milnor \(K\)-group of \(A\). Now assume that \(A\) is also regular. Let \(t_1,\dots,t_n\) be irreducible elements in \(A\) such that the quotient of \(A\) by an ideal generated by any number of the \(t_i\) is still regular. Then the remarkable result is that \(K_2^M(A_{t_1\cdots t_n})\to K_2(A_{t_1\cdots t_n})\) is also an isomorphism.
scientific article; zbMATH DE number 6329088
scientific article