The Stacks project

Lemma 45.14.13. Assume given data (D0), (D1), and (D2') satisfying axioms (A1) – (A7). In order to show that $\gamma $ commutes with pushforward it suffices to show that $i_*(1) = \gamma ([Z])$ if $i : Z \to X$ is a closed immersion of nonempty smooth projective equidimensional schemes over $k$ such that the class of $\mathcal{C}_{Z/X}$ in $K_0(Z)$ is the pullback of a class in $K_0(X)$.

Proof. By Lemma 45.14.11 it suffices to show that $i_*(1) = \gamma ([Z])$ if $i : Z \to X$ is a closed immersion of nonempty smooth projective equidimensional schemes over $k$. Say $Z$ has codimension $r$ in $X$. Let $\mathcal{L}$ be a sufficiently ample invertible module on $X$. Choose $n > 0$ and a surjection

\[ \mathcal{O}_ Z^{\oplus n} \to \mathcal{C}_{Z/X} \otimes \mathcal{L}|_ Z \]

This gives a morphism $g : Z \to \mathbf{G}(n - r, n)$ to the Grassmanian over $k$, see Constructions, Section 27.22. Consider the composition

\[ Z \to X \times \mathbf{G}(n - r, n) \to X \]

Pushforward along the second morphism is compatible with classes of cycles by Lemma 45.14.12. The conormal sheaf $\mathcal{C}$ of the closed immersion $Z \to X \times \mathbf{G}(n - r, n)$ sits in a short exact sequence

\[ 0 \to \mathcal{C}_{Z/X} \to \mathcal{C} \to g^*\Omega _{\mathbf{G}(n - r, n)} \to 0 \]

See More on Morphisms, Lemma 37.11.13. Since $\mathcal{C}_{Z/X} \otimes \mathcal{L}|_ Z$ is the pull back of a finite locally free sheaf on $\mathbf{G}(n - r, n)$ we conclude that the class of $\mathcal{C}$ in $K_0(Z)$ is the pullback of a class in $K_0(X \times \mathbf{G}(n - r, n))$. Hence we have the property for $Z \to X \times \mathbf{G}(n - r, n)$ and we conclude. $\square$


Comments (1)


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Your email address will not be published.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