The Stacks project

Lemma 54.8.1. Let $(A, \mathfrak m, \kappa )$ be a Noetherian normal local domain of dimension $2$. Consider a commutative diagram

\[ \xymatrix{ X' \ar[rd]_{f'} \ar[rr]_ g & & X \ar[ld]^ f \\ & \mathop{\mathrm{Spec}}(A) } \]

where $f$ and $f'$ are modifications as in Situation 54.7.1 and $X$ normal. Then we have a short exact sequence

\[ 0 \to H^1(X, \mathcal{O}_ X) \to H^1(X', \mathcal{O}_{X'}) \to H^0(X, R^1g_*\mathcal{O}_{X'}) \to 0 \]

Also $\dim (\text{Supp}(R^1g_*\mathcal{O}_{X'})) = 0$ and $R^1g_*\mathcal{O}_{X'}$ is generated by global sections.

Proof. We will use the observations made following Situation 54.7.1 without further mention. As $X$ is normal and $g$ is dominant and birational, we have $g_*\mathcal{O}_{X'} = \mathcal{O}_ X$, see for example More on Morphisms, Lemma 37.52.6. Since the fibres of $g$ have dimension $\leq 1$, we have $R^ pg_*\mathcal{O}_{X'} = 0$ for $p > 1$, see for example Cohomology of Schemes, Lemma 30.20.9. The support of $R^1g_*\mathcal{O}_{X'}$ is contained in the set of points of $X$ where the fibres of $g'$ have dimension $\geq 1$. Thus it is contained in the set of images of those irreducible components $C' \subset X'_ s$ which map to points of $X_ s$ which is a finite set of closed points (recall that $X'_ s \to X_ s$ is a morphism of proper $1$-dimensional schemes over $\kappa $). Then $R^1g_*\mathcal{O}_{X'}$ is globally generated by Cohomology of Schemes, Lemma 30.9.10. Using the morphism $f : X \to S$ and the references above we find that $H^ p(X, \mathcal{F}) = 0$ for $p > 1$ for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$. Hence the short exact sequence of the lemma is a consequence of the Leray spectral sequence for $g$ and $\mathcal{O}_{X'}$, see Cohomology, Lemma 20.13.4. $\square$


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