The Stacks project

Lemma 42.58.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ and $g : Y \to Z$ be local complete intersection morphisms of schemes locally of finite type over $S$. Assume the gysin map exists for $g \circ f$ and $g$. Then the gysin map exists for $f$ and $(g \circ f)^! = f^! \circ g^!$.

Proof. Observe that $g \circ f$ is a local complete intersection morphism by More on Morphisms, Lemma 37.55.7 and hence the statement of the lemma makes sense. If $X \to P$ is an immersion of $X$ into a scheme $P$ smooth over $Z$ then $X \to P \times _ Z Y$ is an immersion of $X$ into a scheme smooth over $Y$. This prove the first assertion of the lemma. Let $Y \to P'$ be an immersion of $Y$ into a scheme $P'$ smooth over $Z$. Consider the commutative diagram

\[ \xymatrix{ X \ar[r] \ar[d] & P \times _ Z Y \ar[r]_ a \ar[ld]^ p & P \times _ Z P' \ar[ld]^ q \\ Y \ar[r]_ b \ar[d] & P' \ar[ld] \\ Z } \]

Here the horizontal arrows are regular immersions, the south-west arrows are smooth, and the square is cartesian. Whence $a^! \circ q^* = p^* \circ b^!$ as bivariant classes commute with flat pullback. Combining this fact with Lemmas 42.58.1 and 42.14.3 the reader finds the statement of the lemma holds true. Small detail omitted. $\square$


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