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 $M$ is a perfect object of $D(\mathcal{O}_ X)$.

Let $(X, D, M)$ be a triple as in Situation 38.43.1. Consider an affine open $U = \mathop{\mathrm{Spec}}(A) \subset X$ such that

  1. $D \cap U = V(f)$ for some nonzerodivisor $f \in A$, and

  2. there exists a bounded complex $M^\bullet $ of finite free $A$-modules representing $M|_ U$ (via the equivalence of Derived Categories of Schemes, Lemma 36.3.5).

We will say that $(U, A, f, M^\bullet )$ is an affine chart for $(X, D, M)$. Consider the ideals $I_ i(M^\bullet , f) \subset A$ defined in More on Algebra, Section 15.96. Let us say $(X, S, M)$ is a good triple if for every $x \in D$ there exists an affine chart $(U, A, f, M^\bullet )$ with $x \in U$ and $I_ i(M^\bullet , f)$ principal ideals for all $i \in \mathbf{Z}$.

Lemma 38.43.2. In Situation 38.43.1 let $h : Y \to X$ be a morphism of schemes such that the pullback $E = h^{-1}D$ of $D$ is defined (Divisors, Definition 31.13.12). Let $(U, A, f, M^\bullet )$ is an affine chart for $(X, D, M)$. Let $V = \mathop{\mathrm{Spec}}(B) \subset Y$ is an affine open with $h(V) \subset U$. Denote $g \in B$ the image of $f \in A$. Then

  1. $(V, B, g, M^\bullet \otimes _ A B)$ is an affine chart for $(Y, E, Lh^*M)$,

  2. $I_ i(M^\bullet , f)B = I_ i(M^\bullet \otimes _ A B, g)$ in $B$, and

  3. if $(X, D, M)$ is a good triple, then $(Y, E, Lh^*M)$ is a good triple.

Proof. The first statement follows from the folowing observations: $g$ is a nonzerodivisor in $B$ which defines $E \cap V \subset V$ and $M^\bullet \otimes _ A B$ represents $M^\bullet \otimes _ A^\mathbf {L} B$ and hence represents the pullback of $M$ to $V$ by Derived Categories of Schemes, Lemma 36.3.8. Part (2) follows from part (1) and More on Algebra, Lemma 15.96.3. Combined with More on Algebra, Lemma 15.96.3 we conclude that the second statement of the lemma holds. $\square$

Lemma 38.43.3. Let $X, D, \mathcal{I}, M$ be as in Situation 38.43.1. If $(X, D, M)$ is a good triple, then $L\eta _\mathcal {I}M$ is a perfect object of $D(\mathcal{O}_ X)$.

Proof. Translation of More on Algebra, Lemma 15.96.5. To do the translation use Lemma 38.42.2. $\square$

Lemma 38.43.4. Let $X, D, \mathcal{I}, M$ be as in Situation 38.43.1. Assume $(X, D, M)$ is a good triple. If there exists a locally bounded complex $\mathcal{M}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules representing $M$, then there exists a locally bounded complex $\mathcal{Q}^\bullet $ of finite locally free $\mathcal{O}_{X'}$-modules representing $L\eta _\mathcal {I}M$.

Proof. By Cohomology, Lemma 20.53.7 the complex $\mathcal{Q}^\bullet = \eta _\mathcal {I}\mathcal{M}^\bullet $ represents $L\eta _\mathcal {I}M$. To check that this complex is locally bounded and consists of finite locally free, we may work affine locally. Then the boundedness is clear. Choose an affine chart $(U, A, f, M^\bullet )$ for $(X, D, M)$ such that the ideals $I_ i(M^\bullet , f)$ are principal and such that $\mathcal{M}^ i|_ U$ is finite free for each $i$. By our assumption that $(X, D, M)$ is a good triple we can do this. Writing $N^ i = \Gamma (U, \mathcal{M}^ i|_ U)$ we get a bounded complex $N^\bullet $ of finite free $A$-modules representing the same object in $D(A)$ as the complex $M^\bullet $ (by Derived Categories of Schemes, Lemma 36.3.5). Then $I_ i(N^\bullet , f)$ is a principal ideal for all $i$ by More on Algebra, Lemma 15.96.1. Hence the complex $\eta _ fN^\bullet $ is a bounded complex of finite locally free $A$-modules. Since $\mathcal{Q}^ i|_ U$ is the quasi-coherent $\mathcal{O}_ U$-module corresponding to $\eta _ fN^ i$ by Lemma 38.42.1 we conclude. $\square$

Lemma 38.43.5. 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. If $(X, D, M)$ is a good triple, then

\[ Lh^*(L\eta _\mathcal {I}M) = L\eta _\mathcal {J}(Lh^*M) \]

in $D(\mathcal{O}_ Y)$ where $\mathcal{J}$ is the ideal sheaf of $E$.

Proof. Translation of More on Algebra, Lemma 15.96.6. Use Lemmas 38.42.1 and 38.42.2 to do the translation. $\square$

Lemma 38.43.6. In Situation 38.43.1 there is a unique morphism $b : X' \to X$ such that

  1. the pullback $D' = b^{-1}D$ is defined and $(X', D', M')$ is a good triple where $M' = Lb^*M$, and

  2. for any morphism of schemes $h : Y \to X$ such that the pullback $E = h^{-1}D$ is defined and $(Y, E, Lh^*M)$ is a good triple, there is a unique factorization of $h$ through $b$.

Moreover, for any affine chart $(U, A, f, M^\bullet )$ the restriction $b^{-1}(U) \to U$ is the blowing up in the product of the ideals $I_ i(M^\bullet , f)$ and for any quasi-compact open $W \subset X$ the restriction $b|_{b^{-1}(W)} : b^{-1}(W) \to W$ is a $W \setminus D$-admissible blowing up.

Proof. The proof is just that we will locally blow up $X$ in the product ideals $I_ i(M^\bullet , f)$ for any affine chart $(U, A, f, M^\bullet )$. The first few lemmas in More on Algebra, Section 15.96 show that this is well defined. The universal property (2) then follows from the universal property of blowing up. The details can be found below.

Let $U, A, f, M^\bullet $ be an affine chart for $(X, D, M)$. All but a finite number of the ideals $I_ i(M^\bullet , f)$ are equal to $A$ hence it makes sense to look at

\[ I = \prod \nolimits _ i I_ i(M^\bullet , f) \]

and this is a finitely generated ideal of $A$. Denote

\[ b_ U : U' \to U \]

the blowing up of $U$ in $I$. Then $b_ U^{-1}(U \cap D)$ is defined by Divisors, Lemma 31.32.11. Recall that $f^{r_ i} \in I_ i(M^\bullet , f)$ and hence $b_ U$ is a $(U \setminus D)$-admissible blowing up. By Divisors, Lemma 31.32.12 for each $i$ the morphism $b_ U$ factors as $U' \to U'_ i \to U$ where $U'_ i \to U$ is the blowing up in $I_ i(M^\bullet , f)$ and $U' \to U'_ i$ is another blowing up. It follows that the pullback $I_ i(M^\bullet , f)\mathcal{O}_{U'}$ of $I_ i(M^\bullet , f)$ to $U'$ is an invertible ideal sheaf, see Divisors, Lemmas 31.32.11 and 31.32.4. It follows that $(U', b^{-1}D, Lb^*M|_ U)$ is a good triple, see Lemma 38.43.2 for the behaviour of the ideals $I_ i(-,-)$ under pullback. Finally, we claim that $b_ U : U' \to U$ has the universal property mentioned in part (2) of the statement of the lemma. Namely, suppose $h : Y \to U$ is a morphism of schemes such that the pullback $E = h^{-1}(D \cap U)$ is defined and $(Y, E, Lh^*M)$ is a good triple. Then $Y$ is covered by affine charts $(V, B, g, N^\bullet )$ such that $I_ i(N^\bullet , g)$ is an invertible ideal for each $i$. Then $g$ and the image of $f$ in $B$ differ by a unit as they both cut out the effective Cartier divisor $E \cap V$. Hence we may assume $g$ is the image of $f$ by More on Algebra, Lemma 15.96.2. Then $I_ i(N^\bullet , g)$ is isomorphic to $I_ i(M^\bullet \otimes _ A B, g)$ as a $B$-module by More on Algebra, Lemma 15.96.1. Thus $I_ i(M^\bullet \otimes _ A B, g) = I_ i(M^\bullet , f)B$ (Lemma 38.43.2) is an invertible $B$-module. Hence the ideal $IB$ is invertible. It follows that $I\mathcal{O}_ Y$ is invertible. Hence we obtain a unique factorization of $h$ through $b_ U$ by Divisors, Lemma 31.32.5.

Let $\mathcal{B}$ be the set of affine opens $U \subset X$ such that there exists an affine chart $(U, A, f, M^\bullet )$ for $(X, D, M)$. Then $\mathcal{B}$ is a basis for the topology on $X$; details omitted. For $U \in \mathcal{B}$ we have the morphism $b_ U : U' \to U$ constructed above which satisfies the universal property over $U$. If $U_1 \subset U_2 \subset X$ are both in $\mathcal{B}$, then $b_{U_1} : U'_1 \to U_1$ is canonically isomorphic to

\[ b_{U_2}|_{b_{U_2}^{-1}(U_1)} : b_{U_2}^{-1}(U_1) \longrightarrow U_1 \]

by the universal propery. In other words, we get an isomorphism $U'_1 \to b_{U_2}^{-1}(U_1)$ over $U_1$. These isomorphisms satisfy the cocycle condition (again by the universal property) and hence by Constructions, Lemma 27.2.1 we get a morphism $b : X' \to X$ whose restriction to each $U$ in $\mathcal{B}$ is isomorphic to $U' \to U$. Then the morphism $b : X' \to X$ satisfies properties (1) and (2) of the statement of the lemma as these properties may be checked locally (details omitted).

We still have to prove the final assertion of the lemma. Let $W \subset X$ be a quasi-compact open. Choose a finite covering $W = U_1 \cup \ldots \cup U_ T$ such that for each $1 \leq t \leq T$ there exists an affine chart $(U_ t, A_ t, f_ t, M_ t^\bullet )$. We will use below that for any affine open $V = \mathop{\mathrm{Spec}}(B) \subset U_ t \cap U_{t'}$ we have (a) the images of $f_ t$ and $f_{t'}$ in $B$ differ by a unit, and (b) the complexes $M_ t^\bullet \otimes _ A B$ and $M_{t'} \otimes _ A B$ define isomorphic objects of $D(B)$. For $i \in \mathbf{Z}$, set

\[ N_ i = \max \nolimits _{t = 1, \ldots , T} \left(\sum \nolimits _{j \geq i} (-1)^{j - i} rk(M_ t^ j)\right) \]

Then $N_ t - \sum \nolimits _{j \geq i} (-1)^{j - i} rk(M_ t^ j) \geq 0$ and we can consider the ideals

\[ I_{t, i} = f_ t^{N_ i - \sum \nolimits _{j \geq i} (-1)^{j - i} rk(M_ t^ j)} I_ i(M_ t^\bullet , f_ t) \subset A_ t \]

It follows from More on Algebra, Lemmas 15.96.2 and 15.96.1 that the ideals $I_{t, i}$ glue to a quasi-coherent, finite type ideal $\mathcal{I}_ i \subset \mathcal{O}_ W$. Moreover, all but a finite number of these ideals are equal to $\mathcal{O}_ W$. Clearly, the morphism $X' \to X$ constructed above restricts to the blowing up of $W$ in the product of the ideals $\mathcal{I}_ i$. This finishes the proof. $\square$

Lemma 38.43.7. In Situation 38.43.1 let $b : X' \to X$ be the morphism of Lemma 38.43.6. Consider the effective Cartier divisor $D' = b^{-1}D$ with ideal sheaf $\mathcal{I}' \subset \mathcal{O}_{X'}$. Then $Q = L\eta _{\mathcal{I}'}Lb^*M$ is a perfect object of $D(\mathcal{O}_{X'})$.

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 $k = h'$). $\square$

Lemma 38.43.9. In Situation 38.43.1 let $W \subset X$ be the maximal open subscheme over which the cohomology sheaves of $M$ are locally free. Then the morphism $b : X' \to X$ of Lemma 38.43.6 is an isomorphism over $W$.

Proof. This is true because for any affine chart $(U, A, f, M^\bullet )$ with $U \subset W$ we have that $I_ i(M^\bullet , f)$ are locally generated by a power of $f$ by More on Algebra, Lemma 15.96.4. Since $f$ is a nonzerodivisor, the blowing up $b^{-1}(U) \to U$ is an isomorphism. $\square$

Lemma 38.43.10. Let $X, D, \mathcal{I}, M$ be as in Situation 38.43.1. If $(X, D, M)$ is a good triple, then there exists a closed immersion

\[ i : T \longrightarrow D \]

of finite presentation with the following properties

  1. $T$ scheme theoretically contains $D \cap W$ where $W \subset X$ is the maximal open over which the cohomology sheaves of $M$ are locally free,

  2. the cohomology sheaves of $Li^*L\eta _\mathcal {I}M$ are locally free, and

  3. for any point $t \in T$ with image $x = i(t) \in W$ the rank of $H^ i(M)_ x$ over $\mathcal{O}_{X, x}$ and the rank of $H^ i(Li^*L\eta _\mathcal {I}M)_ t$ over $\mathcal{O}_{T, t}$ agree.

Proof. Let $(U, A, f, M^\bullet )$ be an affine chart for $(X, D, M)$ such that $I_ i(M^\bullet , f)$ is a principal ideal for all $i \in \mathbf{Z}$. Then we define $T \cap U \subset D \cap U$ as the closed subscheme defined by the ideal

\[ J(M^\bullet , f) = \sum J_ i(M^\bullet , f) \subset A/fA \]

studied in More on Algebra, Lemmas 15.96.8 and 15.96.9; in terms of the second lemma we see that $T \cap U \to D \cap U$ is given by the ring map $A/fA \to C$ studied there. Since $(X, D, M)$ is a good triple we can cover $X$ by affine charts of this form and by the first of the two lemmas, this construction glues. Hence we obtain a closed subscheme $T \subset D$ which on good affine charts as above is given by the ideal $J(M^\bullet , f)$. Then properties (1) and (2) follow from the second lemma. Details omitted. Small observation to help the reader: since $\eta _ fM^\bullet $ is a complex of locally free modules by More on Algebra, Lemma 15.96.5 we see that $Li^*L\eta _\mathcal {I}M|_{T \cap U}$ is represented by the complex $\eta _ fM^\bullet \otimes _ A C$ of $C$-modules. The statement (3) on ranks follows from Cohomology, Lemma 20.53.10. $\square$

Lemma 38.43.11. In Situation 38.43.1. Let $b : X' \to X$ and $D'$ be as in Lemma 38.43.6. Let $Q = L\eta _{\mathcal{I}'}Lb^*M$ be as in Lemma 38.43.7. Let $W \subset X$ be the maximal open where $M$ has locally free cohomology modules. Then there exists a closed immersion $i : T \to D'$ of finite presentation such that

  1. $D' \cap b^{-1}(W) \subset T$ scheme theoretically,

  2. $Li^*Q$ has locally free cohomology sheaves, and

  3. for $t \in T$ mapping to $w \in W$ the rank of $H^ i(Li^*Q)_ t$ over $\mathcal{O}_{T, t}$ is equal to the rank of $H^ i(M)_ x$ over $\mathcal{O}_{X, x}$.

Proof. Lemma 38.43.9 tells us that $b$ is an isomorphism over $W$. Hence $b^{-1}(W) \subset X'$ is contained in the maximal open $W' \subset X'$ where $Lb^*M$ has locally free cohomology sheaves. Then the actual statements in the lemma are an immediate consequence of Lemma 38.43.10 applied to $(X', D', Lb^*M)$ and the other lemmas mentioned in the statement. $\square$

Lemma 38.43.12. In Situation 38.43.1. Let $b : X' \to X$, $D' \subset X'$, and $Q$ be as in Lemma 38.43.7. Let $\rho = (\rho _ i)_{i \in \mathbf{Z}}$ be integers. Let $W(\rho ) \subset X$ be the maximal open subscheme where $H^ i(M)$ is locally free of rank $\rho _ i$ for all $i$. Let $i : T \to D'$ be as in Lemma 38.43.11. Then there exists an open and closed subscheme $T(\rho ) \subset T$ containing $D' \cap b^{-1}(W(\rho ))$ scheme theoretically such that $H^ i(Li^*Q|_{T(\rho )})$ is locally free of rank $\rho _ i$ for all $i$.

Proof. Let $T(\rho ) \subset T$ be the open and closed subscheme where $H^ i(Li^*Q)$ has rank $\rho _ i$ for all $i$. Then the statement is immediate from the assertion in Lemma 38.43.11 on ranks of the cohomology modules. $\square$

Lemma 38.43.13. In Situation 38.43.1. Let $b : X' \to X$, $D' \subset X'$, and $Q$ be as in Lemma 38.43.7. If there exists a locally bounded complex $\mathcal{M}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules representing $M$, then there exists a locally bounded complex $\mathcal{Q}^\bullet $ of finite locally free $\mathcal{O}_{X'}$-modules representing $Q$.

Proof. Recall that $Q = L\eta _{\mathcal{I}'}Lb^*M$ where $\mathcal{I}'$ is the ideal sheaf of the effective Cartier divisor $D'$. The locally bounded complex $(\mathcal{M}')^\bullet = b^*\mathcal{M}^\bullet $ of finite locally free $\mathcal{O}_{X'}$-modules represents $Lb^*M$. Thus the lemma follows from Lemma 38.43.4. $\square$

Lemma 38.43.14. Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor. Let $M \in D(\mathcal{O}_ X)$ be a perfect object. Let $W \subset X$ be the maximal open over which the cohomology sheaves $H^ i(M)$ 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. $b : X' \to X$ is an isomorphism over $X \setminus D$,

  2. $b : X' \to X$ is an isomorphism over $W$,

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

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

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

  6. there exists a closed immersion $i : T \to D'$ of finite presentation such that

    1. $D' \cap b^{-1}(W) \subset T$ scheme theoretically,

    2. $Li^*Q$ has finite locally free cohomology sheaves,

    3. for $t \in T$ with image $w \in W$ the rank of $H^ i(Li^*Q)_ t$ over $\mathcal{O}_{T, t}$ is equal to the rank of $H^ i(M)_ x$ over $\mathcal{O}_{X, x}$,

  7. for any affine chart $(U, A, f, M^\bullet )$ for $(X, D, M)$ the restriction of $b$ to $U$ is the blowing up of $U = \mathop{\mathrm{Spec}}(A)$ in the ideal $I = \prod I_ i(M^\bullet , f)$, and

  8. for any affine chart $(V, B, g, N^\bullet )$ for $(X', D', Lb^*N)$ such that $I_ i(N^\bullet , g)$ is principal, we have

    1. $Q|_ V$ corresponds to $\eta _ gN^\bullet $,

    2. $T \subset V \cap D'$ corresponds to the ideal $J(N^\bullet , g) = \sum J_ i(N^\bullet , g) \subset B/gB$ studied in More on Algebra, Lemma 15.96.9.

  9. If $M$ can be represented by a locally bounded complex of finite locally free $\mathcal{O}_ X$-modules, then $Q$ can be represented by a bounded complex of finite locally free $\mathcal{O}_{X'}$-modules.


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