# stacks/stacks-project

Blow up formula

 @@ -12668,6 +12668,66 @@ \section{Gysin maps for local complete intersection morphisms} $(f')^! = (i')^! \circ (g')^*$, and $(g')^* = res(g^*)$ we conclude. \end{proof} \begin{lemma}[Blow up formula] \label{lemma-blow-up-formula} Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $i : Z \to X$ be a regular closed immersion of schemes locally of finite type over $S$. Let $b : X' \to X$ be the blowing up with center $Z$. Picture $$\xymatrix{ E \ar[r]_j \ar[d]_\pi & X' \ar[d]^b \\ Z \ar[r]^i & X }$$ Assume that the gysin map exists for $b$. Then we have $$res(b^!) = c_{top}(\mathcal{F}^\vee) \circ \pi^*$$ in $A^*(E \to Z)$ where $\mathcal{F}$ is the kernel of the canonical map $\pi^*\mathcal{C}_{Z/X} \to \mathcal{C}_{E/X'}$. \end{lemma} \begin{proof} Observe that the morphism $b$ is a local complete intersection morphism by More on Algebra, Lemma \ref{more-algebra-lemma-blowup-regular-sequence} and hence the statement makes sense. Since $Z \to X$ is a regular immersion (and hence a fortiori quasi-regular) we see that $\mathcal{C}_{Z/X}$ is finite locally free and the map $\text{Sym}^*(\mathcal{C}_{Z/X}) \to \mathcal{C}_{Z/X, *}$ is an isomorphism, see Divisors, Lemma \ref{divisors-lemma-quasi-regular-immersion}. Since $E = \text{Proj}(\mathcal{C}_{Z/X, *})$ we conclude that $E = \mathbf{P}(\mathcal{C}_{Z/X})$ is a projective space bundle over $Z$. Thus $E \to Z$ is smooth and certainly a local complete intersection morphism. Thus Lemma \ref{lemma-compare-gysin-base-change} applies and we see that $$res(b^!) = c_{top}(\mathcal{C}^\vee) \circ \pi^!$$ with $\mathcal{C}$ as in the statement there. Of course $\pi^* = \pi^!$ by Lemma \ref{lemma-lci-gysin-flat}. It remains to show that $\mathcal{F}$ is equal to the kernel $\mathcal{C}$ of the map $H^{-1}(j^*\NL_{X'/X}) \to H^{-1}(\NL_{E/Z})$. \medskip\noindent Since $E \to Z$ is smooth we have $H^{-1}(\NL_{E/Z}) = 0$, see More on Morphisms, Lemma \ref{more-morphisms-lemma-NL-smooth}. Hence it suffices to show that $\mathcal{F}$ can be identified with $H^{-1}(j^*\NL_{X'/X})$. By More on Morphisms, Lemmas \ref{more-morphisms-lemma-get-triangle-NL} and \ref{more-morphisms-lemma-NL-immersion} we have an exact sequence $$0 \to H^{-1}(j^*\NL_{X'/X}) \to H^{-1}(\NL_{E/X}) \to \mathcal{C}_{E/X'} \to \ldots$$ By the same lemmas applied to $E \to Z \to X$ we obtain an isomorphism $\pi^*\mathcal{C}_{Z/X} = H^{-1}(\pi^*\NL_{Z/X}) \to H^{-1}(\NL_{E/X})$. Thus we conclude. \end{proof} \begin{lemma} \label{lemma-lci-gysin-product-regular} Let $(S, \delta)$ be as in Situation \ref{situation-setup}.