The Stacks project

Lemma 42.59.6. 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 37.62.7 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 42.59.1 and 42.14.3 the reader finds the statement of the lemma holds true. Small detail omitted. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FF5. Beware of the difference between the letter 'O' and the digit '0'.