Lemma 54.3.5. Let (A, \mathfrak m) be a regular local ring of dimension 2. Let f : X \to S = \mathop{\mathrm{Spec}}(A) be the blowing up of A in \mathfrak m. Let \mathfrak m^ n \subset I \subset \mathfrak m be an ideal. Let d \geq 0 be the largest integer such that
I \mathcal{O}_ X \subset \mathcal{O}_ X(-dE)
where E is the exceptional divisor. Set \mathcal{I}' = I\mathcal{O}_ X(dE) \subset \mathcal{O}_ X. Then d > 0, the sheaf \mathcal{O}_ X/\mathcal{I}' is supported in finitely many closed points x_1, \ldots , x_ r of X, and
\begin{align*} \text{length}_ A(A/I) & > \text{length}_ A \Gamma (X, \mathcal{O}_ X/\mathcal{I}') \\ & \geq \sum \nolimits _{i = 1, \ldots , r} \text{length}_{\mathcal{O}_{X, x_ i}} (\mathcal{O}_{X, x_ i}/\mathcal{I}'_{x_ i}) \end{align*}
Proof.
Since I \subset \mathfrak m we see that every element of I vanishes on E. Thus we see that d \geq 1. On the other hand, since \mathfrak m^ n \subset I we see that d \leq n. Consider the short exact sequence
0 \to I\mathcal{O}_ X \to \mathcal{O}_ X \to \mathcal{O}_ X/I\mathcal{O}_ X \to 0
Since I\mathcal{O}_ X is globally generated, we see that H^1(X, I\mathcal{O}_ X) = 0 by Lemma 54.3.4. Hence we obtain a surjection A/I \to \Gamma (X, \mathcal{O}_ X/I\mathcal{O}_ X). Consider the short exact sequence
0 \to \mathcal{O}_ X(-dE)/I\mathcal{O}_ X \to \mathcal{O}_ X/I\mathcal{O}_ X \to \mathcal{O}_ X/\mathcal{O}_ X(-dE) \to 0
By Divisors, Lemma 31.15.8 we see that \mathcal{O}_ X(-dE)/I\mathcal{O}_ X is supported in finitely many closed points of X. In particular, this coherent sheaf has vanishing higher cohomology groups (detail omitted). Thus in the following diagram
\xymatrix{ & & A/I \ar[d] \\ 0 \ar[r] & \Gamma (X, \mathcal{O}_ X(-dE)/I\mathcal{O}_ X) \ar[r] & \Gamma (X, \mathcal{O}_ X/I\mathcal{O}_ X) \ar[r] & \Gamma (X, \mathcal{O}_ X/\mathcal{O}_ X(-dE)) \ar[r] & 0 }
the bottom row is exact and the vertical arrow surjective. We have
\text{length}_ A \Gamma (X, \mathcal{O}_ X(-dE)/I\mathcal{O}_ X) < \text{length}_ A(A/I)
since \Gamma (X, \mathcal{O}_ X/\mathcal{O}_ X(-dE)) is nonzero. Namely, the image of 1 \in \Gamma (X, \mathcal{O}_ X) is nonzero as d > 0.
To finish the proof we translate the results above into the statements of the lemma. Since \mathcal{O}_ X(dE) is invertible we have
\mathcal{O}_ X/\mathcal{I}' = \mathcal{O}_ X(-dE)/I\mathcal{O}_ X \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(dE).
Thus \mathcal{O}_ X/\mathcal{I}' and \mathcal{O}_ X(-dE)/I\mathcal{O}_ X are supported in the same set of finitely many closed points, say x_1, \ldots , x_ r \in E \subset X. Moreover we obtain
\Gamma (X, \mathcal{O}_ X(-dE)/I\mathcal{O}_ X) = \bigoplus \mathcal{O}_ X(-dE)_{x_ i}/I\mathcal{O}_{X, x_ i} \cong \bigoplus \mathcal{O}_{X, x_ i}/\mathcal{I}'_{x_ i} = \Gamma (X, \mathcal{O}_ X/\mathcal{I}')
because an invertible module over a local ring is trivial. Thus we obtain the strict inequality. We also get the second because
\text{length}_ A(\mathcal{O}_{X, x_ i}/\mathcal{I}'_{x_ i}) \geq \text{length}_{\mathcal{O}_{X, x_ i}}(\mathcal{O}_{X, x_ i}/\mathcal{I}'_{x_ i})
as is immediate from the definition of length.
\square
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