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 37.53.6. 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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