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In the seminal paper $K_2$ and algebraic cycles, Bloch make the following conjecture :

Suppose $A$ is a local Noetherian integral domain with quotient field $F$

  • $K_2(A)$ → $K_2(F)$ is injective

  • Assume in addition $A$ is normal, $K_2(A)$ = $∩_pK_2(A_p)$ where $p$ runs through all height 1 prime ideals in $A$.

What is the current status of this conjecture?

I only know that the first statement is true for discrete valuation ring by a theorem of Dennis and Stein. Can we prove it for a local algebra over a field?

Moreover, the second claim in this conjecture is a Hartogs like statement, so we want it still to hold without the local assumption, could this be true? For example,can we prove it for Dedekind domain or more specifically coordinate ring of a smooth affine curve over a (finite) field?

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up vote 9 down vote accepted

The second statement is false (even if we modify it by replacing $K_2(A)$ by its image in $K_2(F)$). A counterexample is $A=k[x,y,z]_(x,y,z)/(z^2-xy)$. See J. Reine Angew. Math. 381 (1987), 37–50.

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