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 (49.6.4.1).

  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 (49.6.4.1).

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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