The Stacks project

38.44 Blowing up complexes, III

In this section we give an “algebra version” of the version of Macpherson's graph construction given in [Section 18.1, F].

Let $X$ be a scheme. Let $E$ be a perfect object of $D(\mathcal{O}_ X)$. Let $U \subset X$ be the maximal open subscheme such that $E|_ U$ has locally free cohomology sheaves.

Consider the commutative diagram

\[ \xymatrix{ \mathbf{A}^1_ X \ar[r] \ar[rd] & \mathbf{P}^1_ X \ar[d]^ p & (\mathbf{P}^1_ X)_\infty \ar[l] \ar[ld] \\ & X \ar@/_1em/[ur]_\infty } \]

Here we recall that $\mathbf{A}^1 = D_+(T_0)$ is the first standard affine open of $\mathbf{P}^1$ and that $\infty = V_+(T_0)$ is the complementary effective Cartier divisor and the diagram above is the pullback of these schemes to $X$. Observe that $\infty : X \to (\mathbf{P}^1_ X)_\infty $ is an isomorphism. Then

\[ (\mathbf{P}^1_ X, (\mathbf{P}^1_ X)_\infty , Lp^*E) \]

is a triple as in Situation 38.43.1 in Section 38.43. Let

\[ b : W \longrightarrow \mathbf{P}^1_ X\quad \text{and}\quad W_\infty = b^{-1}((\mathbf{P}^1_ X)_\infty ) \]

be the blowing up and effective Cartier divisor constructed starting with this triple in Lemma 38.43.6. We also denote

\[ Q = L\eta _{\mathcal{I}} Lb^*M = L\eta _\mathcal {I} L(p \circ b)^*E \]

the perfect object of $D(\mathcal{O}_ W)$ considered in Lemma 38.43.7. Here $\mathcal{I} \subset \mathcal{O}_ W$ is the ideal sheaf of $W_\infty $.

Lemma 38.44.1. The construction above has the following properties:

  1. $b$ is an isomorphism over $\mathbf{P}^1_ U \cup \mathbf{A}^1_ X$,

  2. the restriction of $Q$ to $\mathbf{A}^1_ X$ is equal to the pullback of $E$,

  3. there exists a closed immersion $i : T \to W_\infty $ of finite presentation such that $(W_\infty \to X)^{-1}U \subset T$ scheme theoretically and such that $Li^*Q$ has locally free cohomology sheaves,

  4. for $t \in T$ with image $u \in U$ we have that the rank $H^ i(Li^*Q)_ t$ over $\mathcal{O}_{T, t}$ is equal to the rank of $H^ i(M)_ u$ over $\mathcal{O}_{U, u}$,

  5. if $E$ can be represented by a locally bounded complex of finite locally free $\mathcal{O}_ X$-modules, then $Q$ can be represented by a locally bounded complex of finite locally free $\mathcal{O}_ W$-modules.

Proof. This follows immediately from the results in Section 38.43; for a statement collecting everything needed, see Lemma 38.43.14. $\square$

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