59.93 Local acyclicity
In this section we deduce local acyclicity of smooth morphisms from the smooth base change theorem. In SGA 4 or SGA 4.5 the authors first prove a version of local acyclicity for smooth morphisms and then deduce the smooth base change theorem.
We will use the formulation of local acyclicity given by Deligne [Definition 2.12, page 242, SGA4.5]. Let f : X \to S be a morphism of schemes. Let \overline{x} be a geometric point of X with image \overline{s} = f(\overline{x}) in S. Let \overline{t} be a geometric point of \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}). We obtain a commutative diagram
\xymatrix{ F_{\overline{x}, \overline{t}} = \overline{t} \times _{\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})} \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \ar[r] \ar[d] & X \ar[d] \\ \overline{t} \ar[r] & \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \ar[r] & S }
The scheme F_{\overline{x}, \overline{t}} is called a variety of vanishing cycles of f at \overline{x}. Let K be an object of D(X_{\acute{e}tale}). For any morphism of schemes g : Y\to X we write R\Gamma (Y, K) instead of R\Gamma (Y_{\acute{e}tale}, g_{small}^{-1}K). Since \mathcal{O}^{sh}_{X, \overline{x}} is strictly henselian we have K_{\overline{x}} = R\Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}), K). Thus we obtain a canonical map
59.93.0.1
\begin{equation} \label{etale-cohomology-equation-alpha-K} \alpha _{K, \overline{x}, \overline{t}} : K_{\overline{x}} \longrightarrow R\Gamma (F_{\overline{x}, \overline{t}}, K) \end{equation}
by pulling back cohomology along F_{\overline{x}, \overline{t}} \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}).
Definition 59.93.1.reference Let f : X \to S be a morphism of schemes. Let K be an object of D(X_{\acute{e}tale}).
Let \overline{x} be a geometric point of X with image \overline{s} = f(\overline{x}). We say f is locally acyclic at \overline{x} relative to K if for every geometric point \overline{t} of \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) the map (59.93.0.1) is an isomorphism1.
We say f is locally acyclic relative to K if f is locally acyclic at \overline{x} relative to K for every geometric point \overline{x} of X.
We say f is universally locally acyclic relative to K if for any morphism S' \to S of schemes the base change f' : X' \to S' is locally acyclic relative to the pullback of K to X'.
We say f is locally acyclic if for all geometric points \overline{x} of X and any integer n prime to the characteristic of \kappa (\overline{x}), the morphism f is locally acyclic at \overline{x} relative to the constant sheaf with value \mathbf{Z}/n\mathbf{Z}.
We say f is universally locally acyclic if for any morphism S' \to S of schemes the base change f' : X' \to S' is locally acyclic.
Let M be an abelian group. Then local acyclicity of f : X \to S with respect to the constant sheaf \underline{M} boils down to the requirement that
H^ q(F_{\overline{x}, \overline{t}}, \underline{M}) = \left\{ \begin{matrix} M
& \text{if}
& q = 0
\\ 0
& \text{if}
& q \not= 0
\end{matrix} \right.
for any geometric point \overline{x} of X and any geometric point \overline{t} of \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, f(\overline{x})}). In this way we see that being locally acyclic corresponds to the vanishing of the higher cohomology groups of the geometric fibres F_{\overline{x}, \overline{t}} of the maps between the strict henselizations at \overline{x} and \overline{s}.
Proposition 59.93.2. Let f : X \to S be a smooth morphism of schemes. Then f is universally locally acyclic.
Proof.
Since the base change of a smooth morphism is smooth, it suffices to show that smooth morphisms are locally acyclic. Let \overline{x} be a geometric point of X with image \overline{s} = f(\overline{x}). Let \overline{t} be a geometric point of \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, f(\overline{x})}). Since we are trying to prove a property of the ring map \mathcal{O}^{sh}_{S, \overline{s}} \to \mathcal{O}^{sh}_{X, \overline{x}} (see discussion following Definition 59.93.1) we may and do replace f : X \to S by the base change X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}). Thus we may and do assume that S is the spectrum of a strictly henselian local ring and that \overline{s} lies over the closed point of S.
We will apply Lemma 59.86.5 to the diagram
\xymatrix{ X \ar[d]_ f & X_{\overline{t}} \ar[l]^ h \ar[d]^ e \\ S & \overline{t} \ar[l]_ g }
and the sheaf \mathcal{F} = \underline{M} where M = \mathbf{Z}/n\mathbf{Z} for some integer n prime to the characteristic of the residue field of \overline{x}. We know that the map f^{-1}R^ qg_*\mathcal{F} \to R^ qh_*e^{-1}\mathcal{F} is an isomorphism by smooth base change, see Theorem 59.89.2 (the assumption on torsion holds by our choice of n). Thus Lemma 59.86.5 gives us the middle equality in
H^ q(F_{\overline{x}, \overline{t}}, \underline{M}) = H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ S \overline{t}, \underline{M}) = H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \times _ S \overline{t}, \underline{M}) = H^ q(\overline{t}, \underline{M})
For the outer two equalities we use that S = \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}). Since \overline{t} is the spectrum of a separably closed field we conclude that
H^ q(F_{\overline{x}, \overline{t}}, \underline{M}) = \left\{ \begin{matrix} M
& \text{if}
& q = 0
\\ 0
& \text{if}
& q \not= 0
\end{matrix} \right.
which is what we had to show (see discussion following Definition 59.93.1).
\square
Lemma 59.93.3. Let f : X \to S be a morphism of schemes. Let \mathcal{F} be a locally constant abelian sheaf on X_{\acute{e}tale} such that for every geometric point \overline{x} of X the abelian group \mathcal{F}_{\overline{x}} is a torsion group all of whose elements have order prime to the characteristic of the residue field of \overline{x}. If f is locally acyclic, then f is locally acyclic relative to \mathcal{F}.
Proof.
Namely, let \overline{x} be a geometric point of X. Since \mathcal{F} is locally constant we see that the restriction of \mathcal{F} to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) is isomorphic to the constant sheaf \underline{M} with M = \mathcal{F}_{\overline{x}}. By assumption we can write M = \mathop{\mathrm{colim}}\nolimits M_ i as a filtered colimit of finite abelian groups M_ i of order prime to the characteristic of the residue field of \overline{x}. Consider a geometric point \overline{t} of \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, f(\overline{x})}). Since F_{\overline{x}, \overline{t}} is affine, we have
H^ q(F_{\overline{x}, \overline{t}}, \underline{M}) = \mathop{\mathrm{colim}}\nolimits H^ q(F_{\overline{x}, \overline{t}}, \underline{M_ i})
by Lemma 59.51.4. For each i we can write M_ i = \bigoplus \mathbf{Z}/n_{i, j}\mathbf{Z} as a finite direct sum for some integers n_{i, j} prime to the characteristic of the residue field of \overline{x}. Since f is locally acyclic we see that
H^ q(F_{\overline{x}, \overline{t}}, \underline{\mathbf{Z}/n_{i, j}\mathbf{Z}}) = \left\{ \begin{matrix} \mathbf{Z}/n_{i, j}\mathbf{Z}
& \text{if}
& q = 0
\\ 0
& \text{if}
& q \not= 0
\end{matrix} \right.
See discussion following Definition 59.93.1. Taking the direct sums and the colimit we conclude that
H^ q(F_{\overline{x}, \overline{t}}, \underline{M}) = \left\{ \begin{matrix} M
& \text{if}
& q = 0
\\ 0
& \text{if}
& q \not= 0
\end{matrix} \right.
and we win.
\square
Lemma 59.93.4. Let
\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S }
be a cartesian diagram of schemes. Let K be an object of D(X_{\acute{e}tale}). Let \overline{x}' be a geometric point of X' with image \overline{x} in X. If
f is locally acyclic at \overline{x} relative to K and
g is locally quasi-finite, or S' = \mathop{\mathrm{lim}}\nolimits S_ i is a directed inverse limit of schemes locally quasi-finite over S with affine transition morphisms, or g : S' \to S is integral,
then f' locally acyclic at \overline{x}' relative to (g')^{-1}K.
Proof.
Denote \overline{s}' and \overline{s} the images of \overline{x}' and \overline{x} in S' and S. Let \overline{t}' be a geometric point of the spectrum of \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S', \overline{s}'}) and denote \overline{t} the image in \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}). By Algebra, Lemma 10.156.6 and our assumptions on g we have
\mathcal{O}^{sh}_{X, \overline{x}} \otimes _{\mathcal{O}^{sh}_{S, \overline{s}}} \mathcal{O}^{sh}_{S', \overline{s}'} \longrightarrow \mathcal{O}^{sh}_{X', \overline{x}'}
is an isomorphism. Since by our conventions \kappa (\overline{t}) = \kappa (\overline{t}') we conclude that
F_{\overline{x}', \overline{t}'} = \mathop{\mathrm{Spec}}\left( \mathcal{O}^{sh}_{X', \overline{x}'} \otimes _{\mathcal{O}^{sh}_{S', \overline{s}'}} \kappa (\overline{t}')\right) = \mathop{\mathrm{Spec}}\left( \mathcal{O}^{sh}_{X, \overline{x}} \otimes _{\mathcal{O}^{sh}_{S, \overline{s}}} \kappa (\overline{t})\right) = F_{\overline{x}, \overline{t}}
In other words, the varieties of vanishing cycles of f' at \overline{x}' are examples of varieties of vanishing cycles of f at \overline{x}. The lemma follows immediately from this and the definitions.
\square
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