The Stacks project

Proof. The first statement follows from the following 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. $\square$

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

Proof. By Cohomology, Lemma 20.55.7 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. $\square$

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

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

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

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

by the universal property. 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

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

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. $\square$

Proof. Follows from Lemmas 38.43.6 and 38.43.3. $\square$

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

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

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. $\square$

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

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 20.55.10. $\square$

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. $\square$

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. $\square$

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. $\square$

Proof. This statement collects the information obtained in Lemmas 38.43.2, 38.43.3, 38.43.5, 38.43.6, 38.43.7, 38.43.8, 38.43.9, 38.43.10, 38.43.11, and 38.43.13. $\square$