The Stacks project

Lemma 45.14.3. Let $b : X' \to X$ be the blowing up of a smooth projective scheme over $k$ in a smooth closed subscheme $Z \subset X$. Picture

\[ \xymatrix{ E \ar[r]_ j \ar[d]_\pi & X' \ar[d]^ b \\ Z \ar[r]^ i & X } \]

Assume there exists an element of $K_0(X)$ whose restriction to $Z$ is equal to the class of $\mathcal{C}_{Z/X}$ in $K_0(Z)$. Assume every irreducible component of $Z$ has codimension $r$ in $X$. Then there exists a cycle $\theta \in \mathop{\mathrm{CH}}\nolimits ^{r - 1}(X')$ such that $b^![Z] = [E] \cdot \theta $ in $\mathop{\mathrm{CH}}\nolimits ^ r(X')$ and $\pi _*j^!(\theta ) = [Z]$ in $\mathop{\mathrm{CH}}\nolimits ^ r(Z)$.

Proof. The scheme $X$ is smooth and projective over $k$ and hence we have $K_0(X) = K_0(\textit{Vect}(X))$. See Derived Categories of Schemes, Lemmas 36.33.2 and 36.35.5. Let $\alpha \in K_0(\text{Vect}(X))$ be an element whose restriction to $Z$ is $[\mathcal{F}]$. By Chow Homology, Lemma 42.55.3 there exists an element $\alpha ^\vee $ which restricts to $\mathcal{C}_{Z/X}^\vee $. By the blow up formula (Chow Homology, Lemma 42.58.11) we have

\[ b^![Z] = b^!i_*[Z] = j_* res(b^!)([Z]) = j_*(c_{r - 1}(\mathcal{F}^\vee ) \cap \pi ^*[Z]) = j_*(c_{r - 1}(\mathcal{F}^\vee ) \cap [E]) \]

where $\mathcal{F}$ is the kernel of the surjection $\pi ^*\mathcal{C}_{Z/X} \to \mathcal{C}_{E/X'}$. Observe that $b^*\alpha ^\vee - [\mathcal{O}_{X'}(E)]$ is an element of $K_0(\text{Vect}(X'))$ which restricts to $[\pi ^*\mathcal{C}_{Z/X}^\vee ] - [\mathcal{C}_{E/X'}^\vee ] = [\mathcal{F}^\vee ]$ on $E$. Since capping with chern classes commutes with $j_*$ we conclude that the above is equal to

\[ c_{r - 1}(b^*\alpha ^\vee - [\mathcal{O}_{X'}(E)]) \cap [E] \]

in the chow group of $X'$. Hence we see that setting

\[ \theta = c_{r - 1}(b^*\alpha ^\vee - [\mathcal{O}_{X'}(E)]) \cap [X'] \]

we get the first relation $\theta \cdot [E] = b^![Z]$ for example by Chow Homology, Lemma 42.61.2. For the second relation observe that

\[ j^!\theta = j^!(c_{r - 1}(b^*\alpha ^\vee - [\mathcal{O}_{X'}(E)]) \cap [X']) = c_{r - 1}(\mathcal{F}^\vee ) \cap j^![X'] = c_{r - 1}(\mathcal{F}^\vee ) \cap [E] \]

in the chow groups of $E$. To prove that $\pi _*$ of this is equal to $[Z]$ it suffices to prove that the degree of the codimension $r - 1$ cycle $(-1)^{r - 1}c_{r - 1}(\mathcal{F}) \cap [E]$ on the fibres of $\pi $ is $1$. This is a computation we omit. $\square$


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