Recall the following theorem of S. Bloch from his paper ($K_2$ of Artinian $\mathbb Q$-algebras, with applications to algebraic cycles, 1975): For any local $\mathbb Q$-algebra $B$ and an augmented $B$-algebra $A$ with $J$ nilpotent (here $J$ is the kernel of the augmentation $A \to B$), one has an isomorphism $$\tau: \Omega^1_{A,J}/{dJ} \xrightarrow{\sim} K_2(A,J).$$ Here $\Omega^1_{A,J}$ is the kernel of map $\Omega^1_A \to \Omega^1_B$ of absolute Kahler differentials and $K_2(A,J)$ is the kernel of the (induced by augmentation) map $K_2(A) \to K_2(B)$.

This theorem motivated Bloch's famous conjecture on zero-cycles.

The proof proceeds by constructing $\tau$ and also its inverse. The construction of the inverse of $\tau$ is not too hard to understand; however, the construction of $\tau$ is quite involved with many steps, each involving many intricate calculations with symbols in $K_2$ of local rings. So the proof of the theorem is very hard to follow.

Question: Is there a better/conceptual proof of this result now (33 years later)? Has this been generalized to higher $K$-groups?

Any pointers to the literature would be highly appreciated: MathScinet searches did not bring up anything relevant to the question.


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