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:
b is an isomorphism over \mathbf{P}^1_ U \cup \mathbf{A}^1_ X,
the restriction of Q to \mathbf{A}^1_ X is equal to the pullback of E,
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,
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},
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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