Lemma 38.38.1. Let Z, X, X', E be an almost blow up square as in Example 38.37.11. Then H^ p(X', \mathcal{O}_{X'}) = 0 for p > 0 and \Gamma (X, \mathcal{O}_ X) \to \Gamma (X', \mathcal{O}_{X'}) is a surjective map of rings whose kernel is an ideal of square zero.
Proof. First assume that A = \mathbf{Z}[f_1, f_2] is the polynomial ring. In this case our almost blow up square is the blowing up of X = \mathop{\mathrm{Spec}}(A) in the closed subscheme Z and in fact X' \subset \mathbf{P}^1_ X is an effective Cartier divisor cut out by the global section f_2T_0 - f_1 T_1 of \mathcal{O}_{\mathbf{P}^1_ X}(1). Thus we have a resolution
0 \to \mathcal{O}_{\mathbf{P}^1_ X}(-1) \to \mathcal{O}_{\mathbf{P}^1_ X} \to \mathcal{O}_{X'} \to 0
Using the description of the cohomology given in Cohomology of Schemes, Section 30.8 it follows that in this case \Gamma (X, \mathcal{O}_ X) \to \Gamma (X', \mathcal{O}_{X'}) is an isomorphism and H^1(X', \mathcal{O}_{X'}) = 0.
Next, we observe that any diagram as in Example 38.37.11 is the base change of the diagram in the previous paragraph by the ring map \mathbf{Z}[f_1, f_2] \to A. Hence by More on Morphisms, Lemmas 37.72.1, 37.72.2, and 37.72.4 we conclude that H^1(X', \mathcal{O}_{X'}) is zero in general and the surjectivity of the map H^0(X, \mathcal{O}_ X) \to H^0(X', \mathcal{O}_{X'}) in general.
Next, in the general case, let us study the kernel. If a \in A maps to zero, then looking on affine charts we see that
a = (f_1x - f_2)(a_0 + a_1x + \ldots + a_ rx^ r)\text{ in }A[x]
for some r \geq 0 and a_0, \ldots , a_ r \in A and similarly
a = (f_1 - f_2y)(b_0 + b_1y + \ldots + b_ s y^ s)\text{ in }A[y]
for some s \geq 0 and b_0, \ldots , b_ s \in A. This means we have
a = f_2 a_0,\ f_1 a_0 = f_2 a_1,\ \ldots ,\ f_1 a_ r = 0, \ a = f_1 b_0,\ f_2 b_0 = f_1 b_1,\ \ldots ,\ f_2 b_ s = 0
If (a', r', a'_ i, s', b'_ j) is a second such system, then we have
aa' = f_1f_2a_0b'_0 = f_1f_2a_1b'_1 = f_1f_2a_2b'_2 = \ldots = 0
as desired. \square
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