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 $F_{\overline{x}, \overline{t}} \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}})$.
reference
Definition 59.93.1. 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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