## 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
$$\label{etale-cohomology-equation-alpha-K} \alpha _{K, \overline{x}, \overline{t}} : K_{\overline{x}} \longrightarrow R\Gamma (F_{\overline{x}, \overline{t}}, K)$$

by pulling back cohomology along $F_{\overline{x}, \overline{t}} \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}})$.

Definition 59.93.1. Let $f : X \to S$ be a morphism of schemes. Let $K$ be an object of $D(X_{\acute{e}tale})$.

1. 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.

2. 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 $X$.

3. 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 $X'$.

4. 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 $\mathbf{Z}/n\mathbf{Z}$.

5. 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. $\square$

$\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S }$

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

1. $f$ is locally acyclic at $\overline{x}$ relative to $K$ and

2. $g$ 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. $\square$

[1] We do not assume $\overline{t}$ is an algebraic geometric point of $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$. Often using Lemma 59.90.2 one may reduce to this case.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GJM. Beware of the difference between the letter 'O' and the digit '0'.