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Theorem 37.53.4 (Stein factorization; Noetherian case). Let S be a locally Noetherian scheme. Let f : X \to S be a proper morphism. There exists a factorization

\xymatrix{ X \ar[rr]_{f'} \ar[rd]_ f & & S' \ar[dl]^\pi \\ & S & }

with the following properties:

  1. the morphism f' is proper with geometrically connected fibres,

  2. the morphism \pi : S' \to S is finite,

  3. we have f'_*\mathcal{O}_ X = \mathcal{O}_{S'},

  4. we have S' = \underline{\mathop{\mathrm{Spec}}}_ S(f_*\mathcal{O}_ X), and

  5. S' is the normalization of S in X, see Morphisms, Definition 29.53.3.

Proof. Let f = \pi \circ f' be the factorization of Lemma 37.53.1. Note that besides the conclusions of Lemma 37.53.1 we also have that f' is separated (Schemes, Lemma 26.21.13) and finite type (Morphisms, Lemma 29.15.8). Hence f' is proper. By Cohomology of Schemes, Proposition 30.19.1 we see that f_*\mathcal{O}_ X is a coherent \mathcal{O}_ S-module. Hence we see that \pi is finite, i.e., (2) holds.

This proves all but the most interesting assertion, namely that all the fibres of f' are geometrically connected. It is clear from the discussion above that we may replace S by S', and we may therefore assume that S is Noetherian, affine, f : X \to S is proper, and f_*\mathcal{O}_ X = \mathcal{O}_ S. Let s \in S be a point of S. We have to show that X_ s is geometrically connected. By Lemma 37.53.3 we see that it suffices to show X_ u is connected for every étale neighbourhood (U, u) \to (S, s). We may assume U is affine. Thus U is Noetherian (Morphisms, Lemma 29.15.6), the base change f_ U : X_ U \to U is proper (Morphisms, Lemma 29.41.5), and that also (f_ U)_*\mathcal{O}_{X_ U} = \mathcal{O}_ U (Cohomology of Schemes, Lemma 30.5.2). Hence after replacing (f : X \to S, s) by the base change (f_ U : X_ U \to U, u) it suffices to prove that the fibre X_ s is connected when f_*\mathcal{O}_ X = \mathcal{O}_ S. We can deduce this from Derived Categories of Schemes, Lemma 36.32.7 (by looking at idempotents in the structure sheaf of X_ s) but we will also give a direct argument below.

Namely, we apply the theorem on formal functions, more precisely Cohomology of Schemes, Lemma 30.20.7. It tells us that

\mathcal{O}^\wedge _{S, s} = (f_*\mathcal{O}_ X)_ s^\wedge = \mathop{\mathrm{lim}}\nolimits _ n H^0(X_ n, \mathcal{O}_{X_ n})

where X_ n is the nth infinitesimal neighbourhood of X_ s. Since the underlying topological space of X_ n is equal to that of X_ s we see that if X_ s = T_1 \amalg T_2 is a disjoint union of nonempty open and closed subschemes, then similarly X_ n = T_{1, n} \amalg T_{2, n} for all n. And this in turn means H^0(X_ n, \mathcal{O}_{X_ n}) contains a nontrivial idempotent e_{1, n}, namely the function which is identically 1 on T_{1, n} and identically 0 on T_{2, n}. It is clear that e_{1, n + 1} restricts to e_{1, n} on X_ n. Hence e_1 = \mathop{\mathrm{lim}}\nolimits e_{1, n} is a nontrivial idempotent of the limit. This contradicts the fact that \mathcal{O}^\wedge _{S, s} is a local ring. Thus the assumption was wrong, i.e., X_ s is connected, and we win. \square


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