Lemma 15.34.2. Let $M/L/K$ be field extensions. Then the Jacobi-Zariski sequence

\[ 0 \to H_1(L_{L/K}) \otimes _ L M \to H_1(L_{M/K}) \to H_1(L_{M/L}) \to \Omega _{L/K} \otimes _ L M \to \Omega _{M/K} \to \Omega _{M/L} \to 0 \]

is exact.

Lemma 15.34.2. Let $M/L/K$ be field extensions. Then the Jacobi-Zariski sequence

\[ 0 \to H_1(L_{L/K}) \otimes _ L M \to H_1(L_{M/K}) \to H_1(L_{M/L}) \to \Omega _{L/K} \otimes _ L M \to \Omega _{M/K} \to \Omega _{M/L} \to 0 \]

is exact.

**Proof.**
Combine Lemma 15.33.7 with Algebra, Lemma 10.158.11.
$\square$

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