Let $A$ be a comm. local $k$-algebra with max. ideal $m$. Let $gr(-)$ be the associated graded algebra or module for the $m$-adic filtration. The canonical derivation $d:A \to \Omega^1_A$ satisfies $d(m^{n+1}) \subset m^n \Omega^1_A$ and therefore induces a derivation $gr A \to gr \Omega^1_A$, which has degree -1 for the grading. This derivation factors through the canonical derivation $gr A \to \Omega^1_{gr A}$ and we obtain a $gr A$-module homomorphism $\Omega^1_{grA} \to gr \Omega^1_A$.

Under what conditions is this an isomorphism?


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