The Stacks project

Lemma 42.59.7. Let $(S, \delta )$ be as in Situation 42.7.1. Consider a commutative diagram

\[ \xymatrix{ X'' \ar[d] \ar[r] & X' \ar[d] \ar[r] & X \ar[d]^ f \\ Y'' \ar[r] & Y' \ar[r] & Y } \]

of schemes locally of finite type over $S$ with both square cartesian. Assume $f : X \to Y$ is a local complete intersection morphism such that the gysin map exists for $f$. Let $c \in A^*(Y'' \to Y')$. Denote $res(f^!) \in A^*(X' \to Y')$ the restriction of $f^!$ to $Y'$ (Remark 42.33.5). Then $c$ and $res(f^!)$ commute (Remark 42.33.6).

Proof. Choose a factorization $f = g \circ i$ with $g$ smooth and $i$ an immersion. Since $f^! = i^! \circ g^!$ it suffices to prove the lemma for $g^!$ (which is given by flat pullback) and for $i^!$. The result for flat pullback is part of the definition of a bivariant class. The case of $i^!$ follows immediately from Lemma 42.54.8. $\square$

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