The Stacks project

Lemma 49.6.5. Let $A \to B$ be a quasi-finite homomorphism of Noetherian rings.

  1. If $A \to A'$ is a flat map of Noetherian rings, then

    \[ \xymatrix{ \omega _{B/A} \times J \ar[r] \ar[d] & B \ar[d] \\ \omega _{B'/A'} \times J' \ar[r] & B' } \]

    is commutative where notation as in Lemma 49.6.2 and horizontal arrows are given by (

  2. If $B = B_1 \times B_2$, then

    \[ \xymatrix{ \omega _{B/A} \times J \ar[r] \ar[d] & B \ar[d] \\ \omega _{B_ i/A} \times J_ i \ar[r] & B_ i } \]

    is commutative for $i = 1, 2$ where notation as in Lemma 49.6.1 and horizontal arrows are given by (

Proof. Because of the construction of the pairing in Remark 49.6.4 both (1) and (2) reduce to the case where $A \to B$ is finite. Then (1) follows from the fact that the contraction map $\mathop{\mathrm{Hom}}\nolimits _ A(M, A) \otimes _ A M \otimes _ A M \to M$, $\lambda \otimes m \otimes m' \mapsto \lambda (m)m'$ commuted with base change. To see (2) use that $J = J_1 \times J_2$ is contained in the summands $B_1 \otimes _ A B_1$ and $B_2 \otimes _ A B_2$ of $B \otimes _ A B$. $\square$

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