The Stacks project

Lemma 36.32.8. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume

  1. $f$ is proper, flat, and of finite presentation, and

  2. the fibre $X_ s$ is geometrically reduced.

Then, after replacing $S$ by an open neighbourhood of $s$, there exists a direct sum decomposition $Rf_*\mathcal{O}_ X = f_*\mathcal{O}_ X \oplus P$ in $D(\mathcal{O}_ S)$ where $f_*\mathcal{O}_ X$ is a finite étale $\mathcal{O}_ S$-algebra and $P$ is a perfect of tor amplitude in $[1, \infty )$.

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 aplitude 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

\[ (f_*\mathcal{O}_ X)_ s \longrightarrow H^0(X_ s, \mathcal{O}_{X_ s}) = H^0(E \otimes ^\mathbf {L} \kappa (s)) \]

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. $\square$

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