The Stacks project

Proof. The proof of this lemma is similar to the proof of Lemma 36.32.5 which we suggest the reader read first. By cohomology and base change (Lemma 36.30.4) the complex $E = Rf_*\mathcal{O}_ X$ is perfect and its formation commutes with arbitrary base change. This first implies that $E$ has tor amplitude in $[0, \infty )$.

We claim that after replacing $S$ by an open neighbourhood of $s$ we can find a direct sum decomposition $E = H^0(E) \oplus \tau _{\geq 1}E$ in $D(\mathcal{O}_ S)$ with $\tau _{\geq 1}E$ of tor amplitude in $[1, \infty )$. Assume the claim is true for now and assume we've made the replacement so we have the direct sum decomposition. Since $E$ has tor amplitude in $[0, \infty )$ we find that $H^0(E)$ is a flat $\mathcal{O}_ S$-module. Hence $H^0(E)$ is a flat, perfect $\mathcal{O}_ S$-module, hence finite locally free, see More on Algebra, Lemma 15.74.2 (and the fact that finite projective modules are finite locally free by Algebra, Lemma 10.78.2). Of course $H^0(E) = f_*\mathcal{O}_ X$ is an $\mathcal{O}_ S$-algebra. By cohomology and base change we obtain $H^0(E) \otimes \kappa (s) = H^0(X_ s, \mathcal{O}_{X_ s})$. By Varieties, Lemma 33.9.3 and the assumption that $X_ s$ is geometrically reduced, we see that $\kappa (s) \to H^0(E) \otimes \kappa (s)$ is finite étale. By Morphisms, Lemma 29.36.17 applied to the finite locally free morphism $\underline{\mathop{\mathrm{Spec}}}_ S(H^0(E)) \to S$, we conclude that after shrinking $S$ the $\mathcal{O}_ S$-algebra $H^0(E)$ is finite étale.

It remains to prove the claim. For this it suffices to prove that the map

is surjective, see More on Algebra, Lemma 15.76.2. Choose a flat local ring homomorphism $\mathcal{O}_{S, s} \to A$ such that the residue field $k$ of $A$ is algebraically closed, see Algebra, Lemma 10.159.1. By flat base change (Cohomology of Schemes, Lemma 30.5.2) we get $H^0(X_ A, \mathcal{O}_{X_ A}) = (f_*\mathcal{O}_ X)_ s \otimes _{\mathcal{O}_{S, s}} A$ and $H^0(X_ k, \mathcal{O}_{X_ k}) = H^0(X_ s, \mathcal{O}_{X_ s}) \otimes _{\kappa (s)} k$. Hence it suffices to prove that $H^0(X_ A, \mathcal{O}_{X_ A}) \to H^0(X_ k, \mathcal{O}_{X_ k})$ is surjective. Since $X_ k$ is a reduced proper scheme over $k$ and since $k$ is algebraically closed, we see that $H^0(X_ k, \mathcal{O}_{X_ k})$ is a finite product of copies of $k$ by the already used Varieties, Lemma 33.9.3. Since by Lemma 36.32.7 the idempotents of this $k$-algebra are in the image of $H^0(X_ A, \mathcal{O}_{X_ A}) \to H^0(X_ k, \mathcal{O}_{X_ k})$ we conclude. $\square$