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. $\square$

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Your email address will not be published.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 0FF6. Beware of the difference between the letter 'O' and the digit '0'.