The Stacks project

Lemma 38.43.8. In Situation 38.43.1 let $h : Y \to X$ be a morphism of schemes such that the pullback $E = h^{-1}D$ is defined. Let $b : X' \to X$, resp. $c : Y' \to Y$ be as constructed in Lemma 38.43.6 for $D \subset X$ and $M$, resp. $E \subset Y$ and $Lh^*M$. Then $Y'$ is the strict transform of $Y$ with respect to $b : X' \to X$ (see proof for a precise formulation of this) and

\[ L\eta _{\mathcal{J}'}L(h \circ c)^*M = L(Y' \to X')^*Q \]

where $Q = L\eta _{\mathcal{I}'}Lb^*M$ as in Lemma 38.43.7. In particular, if $(Y, E, Lh^*M)$ is a good triple and $k : Y \to X'$ is the unique morphism such that $h = b \circ k$, then $L\eta _\mathcal {J}Lh^*M = Lk^*Q$.

Proof. Denote $E' = c^{-1}E$. Then $(Y', E', L(h \circ c)^*M)$ is a good triple. Hence by the universal property of Lemma 38.43.6 there is a unique morphism

\[ h' : Y' \longrightarrow X' \]

such that $b \circ h' = h \circ c$. In particular, there is a morphism $(h', c) : Y' \to X' \times _ X Y$. We claim that given $W \subset X$ quasi-compact open, such that $b^{-1}(W) \to W$ is a blowing up, this morphism identifies $Y'|_ W$ with the strict transform of $Y_ W$ with respect to $b^{-1}(W) \to W$. In turn, to see this is true is a local question on $W$, and we may therefore prove the statement over an affine chart. We do this in the next paragraph.

Let $(U, A, f, M^\bullet )$ be an affine chart for $(X, D, M)$. Recall from the proof of Lemma 38.43.7 that the restriction of $b : X' \to X$ to $U$ is the blowing up of $U = \mathop{\mathrm{Spec}}(A)$ in the product of the ideals $I_ i(M^\bullet , f)$. Now if $V = \mathop{\mathrm{Spec}}(B) \subset Y$ is any affine open with $h(V) \subset U$, then $(V, B, g, M^\bullet \otimes _ A B)$ is an affine chart for $(Y, E, Lh^*M)$ where $g \in B$ is the image of $f$, see Lemma 38.43.2. Hence the restriction of $c : Y' \to Y$ to $V$ is the blowing up in the product of the ideals $I_ i(M^\bullet , f)B$, i.e., the morphism $c : Y' \to Y$ over $h^{-1}(U)$ is the blowing up of $h^{-1}(U)$ in the ideal $\prod I_ i(M^\bullet , f) \mathcal{O}_{h^{-1}(U)}$. Since this is also true for the strict transform, we see that our claim on strict transforms is true.

Having said this the equality $L\eta _{\mathcal{J}'}L(h \circ c)^*M = L(Y' \to X')^*Q$ follows from Lemma 38.43.5. The final statement is a special case of this (namely, the case where $c = \text{id}_ Y$ and $k = h'$). $\square$

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