The Stacks project

Lemma 50.16.2. For $a \geq 0$ there are canonical maps

\[ b^*\Omega ^ a_{X/S} \longrightarrow \Omega ^ a_{L/S} \longrightarrow b^*\Omega ^ a_{X/S} \otimes _{\mathcal{O}_ L} \mathcal{O}_ L((n - 1)E) \]

whose composition is induced by the inclusion $\mathcal{O}_ L \subset \mathcal{O}_ L((n - 1)E)$.

Proof. The first arrow in the displayed formula is discussed in Section 50.2. To get the second arrow we have to show that if we view a local section of $\Omega ^ a_{L/S}$ as a “meromorphic section” of $b^*\Omega ^ a_{X/S}$, then it has a pole of order at most $n - 1$ along $E$. To see this we work on affine local charts on $L$. Namely, recall that $L$ is covered by the spectra of the affine blowup algebras $A[\frac{I}{y_ i}]$ where $I = A_{+}$ is the ideal generated by $y_1, \ldots , y_ n$. See Algebra, Section 10.70 and Divisors, Lemma 31.32.2. By symmetry it is enough to work on the chart corresponding to $i = 1$. Then

\[ A[\frac{I}{y_1}] = R[x_1, \ldots , x_ m, y_1, t_2, \ldots , t_ n] \]

where $t_ i = y_ i/y_1$, see More on Algebra, Lemma 15.31.2. Thus the module $\Omega ^1_{L/S}$ is over the corresponding affine open freely generated by $\text{d}x_1, \ldots , \text{d}x_ m$, $\text{d}y_1$, and $\text{d}t_1, \ldots , \text{d}t_ n$. Of course, the first $m + 1$ of these generators come from $b^*\Omega ^1_{X/S}$ and for the remaining $n - 1$ we have

\[ \text{d}t_ j = \text{d}\frac{y_ j}{y_1} = \frac{1}{y_1}\text{d}y_ j - \frac{y_ j}{y_1^2}\text{d}y_1 = \frac{\text{d}y_ j - t_ j \text{d}y_1}{y_1} \]

which has a pole of order $1$ along $E$ since $E$ is cut out by $y_1$ on this chart. Since the wedges of $a$ of these elements give a basis of $\Omega ^ a_{L/S}$ over this chart, and since there are at most $n - 1$ of the $\text{d}t_ j$ involved this finishes the proof. $\square$

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