The Stacks project

Lemma 59.88.6. With $f : X \to S$ and $n$ as in Remark 59.88.1 assume for some $q \geq 1$ we have that $BC(f, n, q - 1)$ is true, but $BC(f, n, q)$ is not. Then there exist a commutative diagram

\[ \xymatrix{ X \ar[d]_ f & X' \ar[d]_{f'} \ar[l] & Y \ar[l]^ h \ar[d]^ e \\ S & S' \ar[l] & \mathop{\mathrm{Spec}}(K) \ar[l]_ g } \]

where $X' = X \times _ S S'$, $Y = X' \times _{S'} \mathop{\mathrm{Spec}}(K)$, $K$ is a field, and $\mathcal{F}$ is an abelian sheaf on $\mathop{\mathrm{Spec}}(K)$ annihilated by $n$ such that $(f')^{-1}R^ qg_*\mathcal{F} \to R^ qh_*e^{-1}\mathcal{F}$ is not an isomorphism.

Proof. Choose a commutative diagram

\[ \xymatrix{ X \ar[d]_ f & X' \ar[l] \ar[d]_{f'} & Y \ar[l]^ h \ar[d]^ e \\ S & S' \ar[l] & T \ar[l]_ g } \]

with $X' = X \times _ S S'$ and $Y = X' \times _{S'} T$ and $g$ quasi-compact and quasi-separated, and an abelian sheaf $\mathcal{F}$ on $T_{\acute{e}tale}$ annihilated by $n$ such that the base change map $(f')^{-1}R^ qg_*\mathcal{F} \to R^ qh_*e^{-1}\mathcal{F}$ is not an isomorphism. Of course we may and do replace $S'$ by an affine open of $S'$; this implies that $T$ is quasi-compact and quasi-separated. By Lemma 59.88.2 we see $(f')^{-1}R^ qg_*\mathcal{F} \to R^ qh_*e^{-1}\mathcal{F}$ is injective. Pick a geometric point $\overline{x}$ of $X'$ and an element $\xi $ of $(R^ qh_*q^{-1}\mathcal{F})_{\overline{x}}$ which is not in the image of the map $((f')^{-1}R^ qg_*\mathcal{F})_{\overline{x}} \to (R^ qh_*e^{-1}\mathcal{F})_{\overline{x}}$.

Consider a morphism $\pi : T' \to T$ with $T'$ quasi-compact and quasi-separated and denote $\mathcal{F}' = \pi ^{-1}\mathcal{F}$. Denote $\pi ' : Y' = Y \times _ T T' \to Y$ the base change of $\pi $ and $e' : Y' \to T'$ the base change of $e$. Picture

\[ \vcenter { \xymatrix{ X' \ar[d]_{f'} & Y \ar[l]^ h \ar[d]^ e & Y' \ar[l]^{\pi '} \ar[d]^{e'} \\ S' & T \ar[l]_ g & T' \ar[l]_\pi } } \quad \text{and}\quad \vcenter { \xymatrix{ X' \ar[d]_{f'} & & Y' \ar[ll]^{h' = h \circ \pi '} \ar[d]^{e'} \\ S' & & T' \ar[ll]_{g' = g \circ \pi } } } \]

Using pullback maps we obtain a canonical commutative diagram

\[ \xymatrix{ (f')^{-1}R^ qg_*\mathcal{F} \ar[r] \ar[d] & (f')^{-1}R^ qg'_*\mathcal{F}' \ar[d] \\ R^ qh_*e^{-1}\mathcal{F} \ar[r] & R^ qh'_*(e')^{-1}\mathcal{F}' } \]

of abelian sheaves on $X'$. Let $P(T')$ be the property

  • The image $\xi '$ of $\xi $ in $(Rh'_*(e')^{-1}\mathcal{F}')_{\overline{x}}$ is not in the image of the map $(f^{-1}R^ qg'_*\mathcal{F}')_{\overline{x}} \to (R^ qh'_*(e')^{-1}\mathcal{F}')_{\overline{x}}$.

We claim that hypotheses (1), (2), and (3) of Lemma 59.88.5 hold for $P$ which proves our lemma.

Condition (1) of Lemma 59.88.5 holds for $P$ because the étale topology of a scheme and a thickening of the scheme is the same. See Proposition 59.45.4.

Suppose that $I$ is a directed set and that $T_ i$ is an inverse system over $I$ of quasi-compact and quasi-separated schemes over $T$ with affine transition morphisms. Set $T' = \mathop{\mathrm{lim}}\nolimits T_ i$. Denote $\mathcal{F}'$ and $\mathcal{F}_ i$ the pullback of $\mathcal{F}$ to $T'$, resp. $T_ i$. Consider the diagrams

\[ \vcenter { \xymatrix{ X \ar[d]_{f'} & Y \ar[l]^ h \ar[d]^ e & Y_ i \ar[l]^{\pi _ i'} \ar[d]^{e_ i} \\ S & T \ar[l]_ g & T_ i \ar[l]_{\pi _ i} } } \quad \text{and}\quad \vcenter { \xymatrix{ X \ar[d]_{f'} & & Y_ i \ar[ll]^{h_ i = h \circ \pi _ i'} \ar[d]^{e_ i} \\ S & & T_ i \ar[ll]_{g_ i = g \circ \pi _ i} } } \]

as in the previous paragraph. It is clear that $\mathcal{F}'$ on $T'$ is the colimit of the pullbacks of $\mathcal{F}_ i$ to $T'$ and that $(e')^{-1}\mathcal{F}'$ is the colimit of the pullbacks of $e_ i^{-1}\mathcal{F}_ i$ to $Y'$. By Lemma 59.51.8 we have

\[ R^ qh'_*(e')^{-1}\mathcal{F}' = \mathop{\mathrm{colim}}\nolimits R^ qh_{i, *}e_ i^{-1}\mathcal{F}_ i \quad \text{and}\quad (f')^{-1}R^ qg'_*\mathcal{F}' = \mathop{\mathrm{colim}}\nolimits (f')^{-1}R^ qg_{i, *}\mathcal{F}_ i \]

It follows that if $P(T_ i)$ is true for all $i$, then $P(T')$ holds. Thus condition (2) of Lemma 59.88.5 holds for $P$.

The most interesting is condition (3) of Lemma 59.88.5. Assume $T'$ is a quasi-compact and quasi-separated scheme over $T$ such that $P(T')$ is true. Let $Z \subset T'$ be a closed subscheme with complement $V \subset T'$ quasi-compact. Consider the diagram

\[ \xymatrix{ Y' \times _{T'} Z \ar[d]_{e_ Z} \ar[r]_{i'} & Y' \ar[d]_{e'} & Y' \times _{T'} V \ar[l]^{j'} \ar[d]^{e_ V} \\ Z \ar[r]^ i & T' & V \ar[l]_ j } \]

Choose an injective map $j^{-1}\mathcal{F}' \to \mathcal{J}$ where $\mathcal{J}$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules on $V$. Looking at stalks we see that the map

\[ \mathcal{F}' \to \mathcal{G} = j_*\mathcal{J} \oplus i_*i^{-1}\mathcal{F}' \]

is injective. Thus $\xi '$ maps to a nonzero element of

\begin{align*} & \mathop{\mathrm{Coker}}\left( ((f')^{-1}R^ qg'_*\mathcal{G})_{\overline{x}} \to (R^ qh'_*(e')^{-1}\mathcal{G})_{\overline{x}} \right) = \\ & \mathop{\mathrm{Coker}}\left( ((f')^{-1}R^ qg'_*j_*\mathcal{J})_{\overline{x}} \to (R^ qh'_*(e')^{-1}j_*\mathcal{J})_{\overline{x}} \right) \oplus \\ & \mathop{\mathrm{Coker}}\left( ((f')^{-1}R^ qg'_*i_*i^{-1}\mathcal{F}')_{\overline{x}} \to (R^ qh'_*(e')^{-1}i_*i^{-1}\mathcal{F}')_{\overline{x}} \right) \end{align*}

by part (2) of Lemma 59.88.2. If $\xi '$ does not map to zero in the second summand, then we use

\[ (f')^{-1}R^ qg'_*i_*i^{-1}\mathcal{F}' = (f')^{-1}R^ q(g' \circ i)_*i^{-1}\mathcal{F}' \]

(because $Ri_* = i_*$ by Proposition 59.55.2) and

\[ R^ qh'_*(e')^{-1}i_*i^{-1}\mathcal{F} = R^ qh'_*i'_*e_ Z^{-1}i^{-1}\mathcal{F} = R^ q(h' \circ i')_*e_ Z^{-1}i^{-1}\mathcal{F}' \]

(first equality by Lemma 59.55.3 and the second because $Ri'_* = i'_*$ by Proposition 59.55.2) to we see that we have $P(Z)$. Finally, suppose $\xi '$ does not map to zero in the first summand. We have

\[ (e')^{-1}j_*\mathcal{J} = j'_*e_ V^{-1}\mathcal{J} \quad \text{and}\quad R^ aj'_*e_ V^{-1}\mathcal{J} = 0, \quad a = 1, \ldots , q - 1 \]

by $BC(f, n, q - 1)$ applied to the diagram

\[ \xymatrix{ X \ar[d]_ f & Y' \ar[l] \ar[d]_{e'} & Y \ar[l]^{j'} \ar[d]^{e_ V} \\ S & T' \ar[l] & V \ar[l]_ j } \]

and the fact that $\mathcal{J}$ is injective. By the relative Leray spectral sequence for $h' \circ j'$ (Cohomology on Sites, Lemma 21.14.7) we deduce that

\[ R^ qh'_*(e')^{-1}j_*\mathcal{J} = R^ qh'_*j'_*e_ V^{-1}\mathcal{J} \longrightarrow R^ q(h' \circ j')_* e_ V^{-1}\mathcal{J} \]

is injective. Thus $\xi $ maps to a nonzero element of $(R^ q(h' \circ j')_* e_ V^{-1}\mathcal{J})_{\overline{x}}$. Applying part (3) of Lemma 59.88.2 to the injection $j^{-1}\mathcal{F}' \to \mathcal{J}$ we conclude that $P(V)$ holds. $\square$

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