The Stacks project

38.43 Blowing up complexes, II

The material in this section will be used to construct a version of Macpherson's graph construction in Section 38.44.

Situation 38.43.1. Here $X$ is a scheme, $D \subset X$ is an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$, and $\mathcal{E}^\bullet $ is a bounded complex of finite locally free $\mathcal{O}_ X$-modules with differentials $d^ i : \mathcal{E}^ i \to \mathcal{E}^{i + 1}$.

We are going to construct a canonical blowing up of this situation.

Remark 38.43.2. In Situation 38.43.1 for any $i \in \mathbf{Z}$ there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J}_ i \subset \mathcal{O}_ X$ with the following property: for any $U \subset X$ open such that $\mathcal{I}|_ U$, $\mathcal{E}^ i|_ U$, and $\mathcal{E}^{i + 1}|_ U$ are free of ranks $1$, $r_ i$, and $r_{i + 1}$, the ideal $\mathcal{J}_ i$ is generated by the $r_ i \times r_ i$ minors of the map

\[ 1, d^ i : \mathcal{I}\mathcal{E}^ i \longrightarrow \mathcal{E}^ i \oplus \mathcal{I}\mathcal{E}^{i + 1} \]

with notation as in Section 38.42. By convention we set $\mathcal{J}_ i|_ U = \mathcal{O}_ U$ if $r_ i = 0$. Observe that $\mathcal{I}^{r_ i}|_ U \subset \mathcal{J}_ i|_ U$ in other words, the closed subscheme $V(\mathcal{J})$ is set theoretically contained in $D$. Formation of the ideal $\mathcal{J}_ i$ commutes with base change by any morphism $f : Y \to X$ such that the pullback of $D$ by $f$ is defined (Divisors, Definition 31.13.12).

Lemma 38.43.3. In Situation 38.43.1 let $b : X' \to X$ be the blowing up of the product of the ideals $\mathcal{J}_ i$ from Remark 38.43.2. Denote $D' = b^{-1}D$ with ideal sheaf $\mathcal{I}' \subset \mathcal{O}_{X'}$. Then

\[ \mathcal{Q}^\bullet = \eta _{\mathcal{I}'}b^*\mathcal{E}^\bullet \]

is a bounded complex of finite locally free $\mathcal{O}_{X'}$-modules.

Proof. Recall that $D'$ is an effective Cartier divisor (Divisors, Lemma 31.32.11). Observe that $\mathcal{J}_ i$ pulls back to an invertible ideal sheaf on $X'$ as $X'$ dominates the blowing up in $\mathcal{J}_ i$, see Divisors, Lemma 31.32.12. By Remark 38.43.2 we may replace $X$ by $X'$ and assume $\mathcal{J}_ i$ is invertible for all $i$. Via Lemma 38.42.1 we obtain the result from More on Algebra, Lemma 15.95.1. $\square$

Lemma 38.43.4. In Situation 38.43.1 let $f : Y \to X$ be a morphism of schemes such that the inverse image $f^{-1}D$ is an effective Cartier divisor with ideal sheaf $\mathcal{J}$. Assume $\mathcal{J}_ i$ as in Remark 38.43.2 is invertible for all $i$. Then $f^*(\eta _\mathcal {I}\mathcal{E}^\bullet ) = \eta _\mathcal {J}(f^*\mathcal{E}^\bullet )$.

Proof. Follows from More on Algebra, Lemma 15.95.2 via Lemma 38.42.1. $\square$

Lemma 38.43.5. In Situation 38.43.1 let $f : Y \to X$ be a morphism of schemes such that the inverse image $f^{-1}D$ is an effective Cartier divisor. Let $X' \to X$ and $\mathcal{Q}^\bullet $, resp. $Y' \to Y$ and $\mathcal{Q}_{Y'}^\bullet $, be as constructed in Lemma 38.43.3 for $D \subset X$ and $\mathcal{E}^\bullet $, resp. $f^{-1}D \subset Y$ and $f^*\mathcal{E}^\bullet $. Then $Y'$ is the strict transform of $Y$ with respect to $X' \to X$ and $\mathcal{Q}_{Y'}^\bullet = (Y' \to X')^*\mathcal{Q}^\bullet $.

Proof. In Remark 38.43.2 we have seen that $\mathcal{J}_ i$ pulls back to the corresponding ideal on $Y$. Hence $Y'$ is the strict transform of $Y$ by Divisors, Lemma 31.33.2. The final statement follows from Lemma 38.43.4 applied to $Y' \to X'$. $\square$

Lemma 38.43.6. In Situation 38.43.1 let $U \subset X$ be the maximal open subscheme over which the cohomology sheaves of $\mathcal{E}^\bullet $ are locally free. Then the blowing up $b : X' \to X$ of Lemma 38.43.3 is an isomorphism over $U$.

Proof. Over $U$ all of the modules $\mathop{\mathrm{Im}}(d^ i)$ and $\mathop{\mathrm{Ker}}(d^ i)$ are finite locally free, see for example the discussion in Remark 38.40.4. Let $x \in U$. Choose an open neighbourhood $x \in V \subset U$ such that $\mathcal{I}|_ V$, $\mathcal{E}^ i$, $\mathop{\mathrm{Ker}}(d^ i)$, $\mathop{\mathrm{Im}}(d^ i)$, and $H^ i(\mathcal{E}^\bullet )$ are free and choose splittings for the short exact sequences

\[ 0 \to \mathop{\mathrm{Im}}(d^ i) \to \mathop{\mathrm{Ker}}(d^{i + 1}) \to H^{i + 1}(\mathcal{E}^\bullet ) \to 0 \quad \text{and}\quad 0 \to \mathop{\mathrm{Ker}}(d^ i) \to \mathcal{E}^ i \to \mathop{\mathrm{Im}}(d^ i) \to 0 \]

Then we see that our complex looks like

\[ \ldots \to \mathcal{O}_ V^{\oplus n_{i - 1} + m_ i + n_ i} \to \mathcal{O}_ V^{\oplus n_ i + m_{i + 1} + n_{i + 1}} \to \ldots \]

where the map identifies the last $n_ i$ summands with the first $n_ i$ summands. Thus $\mathcal{J}_ i|V$ is the ideal generated by $f^{n_{i - 1} + m_ i}$ where $f \in \mathcal{O}_ X(V)$ is a generator for $\mathcal{I}|_ V$. Thus over $V$ we are blowing up an invertible ideal, which produces the identity morphism (Divisors, Lemma 31.32.7). $\square$

Lemma 38.43.7. In Situation 38.43.1. Let $b : X' \to X$, $D' \subset X'$, and $\mathcal{Q}^\bullet $ be as in Lemma 38.43.3. Let $U \subset X$ be as in Lemma 38.43.6. Then there exists a closed immersion $T \to D'$ of finite presentation with $D' \cap b^{-1}(U) \subset T$ scheme theoretically such that $\mathcal{Q}^\bullet |_ T$ has finite locally free cohomology sheaves.

Proof. Arguing exactly as in the proof of Lemma 38.43.3 we may replace $X$ by $X'$ and $U$ by $b^{-1}(U)$ and assume that the ideals $\mathcal{J}_ i$ are invertible for all $i$.

Assume $\mathcal{J}_ i$ invertible for all $i$ so that $b = \text{id}_ X$ and $\mathcal{Q}^\bullet = \eta _{\mathcal{I}}\mathcal{E}^\bullet $. Let $x \in D \cap U$ and choose a generator $f \in \mathcal{I}_ x$. Since $H^ i(\mathcal{E}^\bullet )_ x$ is a finite free $\mathcal{O}_{X, x}$-module for all $i$ (by choice of $U$), we see that

\[ \mathop{\mathrm{Ker}}(d^ i : \mathcal{E}^ i_ x/f^2\mathcal{E}^ i_ x \to \mathcal{E}^{i + 1}_ x/f^2\mathcal{E}^{i + 1}_ x) \to \mathop{\mathrm{Ker}}(d^ i : \mathcal{E}^ i_ x/f\mathcal{E}^ i_ x \to \mathcal{E}^{i + 1}_ x/f\mathcal{E}^{i + 1}_ x) \]

is surjective, see More on Algebra, Lemma 15.95.3. This means that if $X = \mathop{\mathrm{Spec}}(A)$ is affine, then via Lemma 38.42.1 we may apply More on Algebra, Lemma 15.95.4 to get a closed subscheme $T \subset D$ with all the desired properties (some details omitted).

To glue this affine local construction, we remark that in the proof of More on Algebra, Lemma 15.95.6 the ideal cutting out $T$ is constructed with a certain universal property. Namely, the result of More on Algebra, Lemma 15.95.1 tells us that the canonical maps

\[ c^ i : \mathcal{Q}^ i \to \mathcal{I}^ i\mathcal{E}^ i \oplus \mathcal{I}^{i + 1}\mathcal{E}^{i + 1} \]

are locally split. The closed subscheme $T \subset D$ is characterized by the property that a morphism of schemes $g : W \to D$ factors through $T$ if and only if $g^*\mathcal{Q}^ i$ is a direct sum of sheaves compatible with the map $g^*c^ i$ for all $i$. Hence it is clear that the affine locally constructed closed subschemes glue. $\square$

Lemma 38.43.8. In Situation 38.43.1. Let $b : X' \to X$, $D' \subset X'$, and $\mathcal{Q}^\bullet $ be as in Lemma 38.43.3. Given integers $\rho _ i \geq 0$ almost all zero, let $U' \subset X$ be the maximal open subscheme where $H^ i(\mathcal{E}^\bullet )$ is finite locally free of rank $\rho _ i$ for all $i$. Let $T \subset D'$ be as in Lemma 38.43.7. Then there exists an open and closed subscheme $T' \subset T$ containing $D' \cap b^{-1}(U')$ scheme theoretically such that $\mathcal{Q}^\bullet |_{T'}$ has finite locally free cohomology sheaves $H^ i(\mathcal{Q}^\bullet |_{T'})$ of rank $\rho _ i$.

Proof. This is obvious. $\square$

Lemma 38.43.9. Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor. Let $E \in D(\mathcal{O}_ X)$ be a perfect object. Let $U \subset X$ be the maximal open over which the cohomology sheaves $H^ i(E)$ are locally free. There exists a proper morphism $b : X' \longrightarrow X$ and an object $Q \in D(\mathcal{O}_{X'})$ with the following properties

  1. $D' = b^{-1}D$ is an effective Cartier divisor,

  2. $Q = L\eta _{\mathcal{I}'}Lb^*E$ where $\mathcal{I}'$ is the ideal sheaf of $D'$,

  3. $Q$ is a perfect object of $D(\mathcal{O}_{X'})$,

  4. there exists a closed immersion $T \to D'$ of finite presentation with $D' \cap b^{-1}(U) \subset T$ scheme theoretically such that $Q|_ T$ has finite locally free cohomology sheaves,

  5. for any open subscheme $V \subset X$ such that $E|_ V$ can be represented by a bounded complex $\mathcal{E}^\bullet $ of finite locally free $\mathcal{O}_ V$-modules, the base changes of $X' \to X$, $Q$, $D'$, and $T$ to $V$ are given by the constructions of Lemmas 38.43.3 and 38.43.7.

Proof. We first construct the morphism $b : X' \to X$ by glueing the blowings up constructed over opens $V \subset X$ as in (5). By Constructions, Lemma 27.2.1 to do this it suffices to show that given $V \subset X$ open and two bounded complexes $\mathcal{E}^\bullet $ and $(\mathcal{E}')^\bullet $ of finite locally free $\mathcal{O}_ V$-modules representing $E|_ V$ the resulting blowing ups are canonically isomorphic. To do this, it suffices, by the universal property of blowing up of Divisors, Lemma 31.32.5, to show that the ideals $\mathcal{J}_ i$ and $\mathcal{J}'_ i$ from Remark 38.43.2 constructed using $\mathcal{E}^\bullet $ and $(\mathcal{E}')^\bullet $ locally differ by multiplication by an invertible ideal. We will in fact show that they differ locally by a power of the ideal sheaf $\mathcal{I}$ of $D$. By More on Algebra, Lemma 15.74.7 working locally it suffices to prove the relationship when

\[ (\mathcal{E}')^\bullet = \mathcal{E}^\bullet \oplus ( \ldots \to 0 \to \mathcal{O}_ V \xrightarrow {1} \mathcal{O}_ V \to 0 \to \ldots ) \]

with the two summands $\mathcal{O}_ V$ placed in degrees $i$ and $i + 1$ say. Computing minors explicitly one finds that $\mathcal{J}'_{i + 1} = \mathcal{I}\mathcal{J}_{i + 1}$ and all other ideals stay the same.

Thus we have the morphism $b : X' \to X$ agreeing locally with the blowing ups in (5). Of course this immediately gives us the effective Cartier divisor $D' = b^{-1}D$, its invertible ideal sheaf $\mathcal{I}'$ and the object $Q = L\eta _{\mathcal{I}'}Lb^*E$. See Remark 38.42.4 for the construction of $L\eta _{\mathcal{I}'}$. Since the construction commutes with restricting to opens we find that $Q|_{V'}$ is represented by the complex $\mathcal{Q}^\bullet $ over the open $V' = b^{-1}(V)$ constructed using $\mathcal{E}^\bullet $ over $V$.

To finish the proof it suffices to show that the closed subschemes $T_ V \subset V'$ constructed in Lemma 38.43.7 glue. Again by relative glueing, it suffices to show that the construction of $T$ does not depend on the choice of the complex $\mathcal{E}^\bullet $ representing $E|_ V$. Again we reduce to the case where

\[ (\mathcal{E}')^\bullet = \mathcal{E}^\bullet \oplus ( \ldots \to 0 \to \mathcal{O}_ V \xrightarrow {1} \mathcal{O}_ V \to 0 \to \ldots ) \]

with the two summands $\mathcal{O}_ V$ placed in degrees $i$ and $i + 1$ say. Note that in this case $(\mathcal{Q}')^\bullet $ and $\mathcal{Q}^\bullet $ differ as follows

\[ (\mathcal{Q}')^\bullet = \mathcal{Q}^\bullet \oplus ( \ldots \to 0 \to (\mathcal{I}')^{i + 1}|_{V'} \xrightarrow {1} (\mathcal{I}')^{i + 1}|_{V'} \to 0 \to \ldots ) \]

In the proof of Lemma 38.43.7 we defined $T \subset D'$ as the largest closed subscheme of $D'$ such that $\mathcal{Q}^ i|_ T$ is a direct sum of two parts compatible with the restriction to $T$ of the canonical split injective maps

\[ c^ i : \mathcal{Q}^ i \longrightarrow (\mathcal{I}')^ ib^*\mathcal{E}^ i \oplus (\mathcal{I}')^{i + 1}b^*\mathcal{E}^{i + 1} \]

for all $i$. The direct sum decomposition for $(\mathcal{Q}')^\bullet $ in terms of $\mathcal{Q}^\bullet $ and the explicit complex $(\mathcal{I}')^{i + 1}|_{V'} \to (\mathcal{I}')^{i + 1}|_{V'}$ implies in a straightforward manner that $T$ plays the same role for $(\mathcal{Q}')^\bullet $ and the proof is complete. $\square$


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