Lemma 59.48.1. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. Let $f : X \to Y$ be a monomorphism of schemes. Then the canonical map $f_{big}^{-1}f_{big, *}\mathcal{F} \to \mathcal{F}$ is an isomorphism for any sheaf $\mathcal{F}$ on $(\mathit{Sch}/X)_\tau $.
59.48 Big sites and pushforward
In this section we prove some technical results on $f_{big, *}$ for certain types of morphisms of schemes.
Proof. In this case the functor $(\mathit{Sch}/X)_\tau \to (\mathit{Sch}/Y)_\tau $ is continuous, cocontinuous and fully faithful. Hence the result follows from Sites, Lemma 7.21.7. $\square$
Remark 59.48.2. In the situation of Lemma 59.48.1 it is true that the canonical map $\mathcal{F} \to f_{big}^{-1}f_{big!}\mathcal{F}$ is an isomorphism for any sheaf of sets $\mathcal{F}$ on $(\mathit{Sch}/X)_\tau $. The proof is the same. This also holds for sheaves of abelian groups. However, note that the functor $f_{big!}$ for sheaves of abelian groups is defined in Modules on Sites, Section 18.16 and is in general different from $f_{big!}$ on sheaves of sets. The result for sheaves of abelian groups follows from Modules on Sites, Lemma 18.16.4.
Lemma 59.48.3. Let $f : X \to Y$ be a closed immersion of schemes. Let $U \to X$ be a syntomic (resp. smooth, resp. étale) morphism. Then there exist syntomic (resp. smooth, resp. étale) morphisms $V_ i \to Y$ and morphisms $V_ i \times _ Y X \to U$ such that $\{ V_ i \times _ Y X \to U\} $ is a Zariski covering of $U$.
Proof. Let us prove the lemma when $\tau = syntomic$. The question is local on $U$. Thus we may assume that $U$ is an affine scheme mapping into an affine of $Y$. Hence we reduce to proving the following case: $Y = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(A/I)$, and $U = \mathop{\mathrm{Spec}}(\overline{B})$, where $A/I \to \overline{B}$ be a syntomic ring map. By Algebra, Lemma 10.136.18 we can find elements $\overline{g}_ i \in \overline{B}$ such that $\overline{B}_{\overline{g}_ i} = A_ i/IA_ i$ for certain syntomic ring maps $A \to A_ i$. This proves the lemma in the syntomic case. The proof of the smooth case is the same except it uses Algebra, Lemma 10.137.20. In the étale case use Algebra, Lemma 10.143.10. $\square$
Lemma 59.48.4. Let $f : X \to Y$ be a closed immersion of schemes. Let $\{ U_ i \to X\} $ be a syntomic (resp. smooth, resp. étale) covering. There exists a syntomic (resp. smooth, resp. étale) covering $\{ V_ j \to Y\} $ such that for each $j$, either $V_ j \times _ Y X = \emptyset $, or the morphism $V_ j \times _ Y X \to X$ factors through $U_ i$ for some $i$.
Proof. For each $i$ we can choose syntomic (resp. smooth, resp. étale) morphisms $g_{ij} : V_{ij} \to Y$ and morphisms $V_{ij} \times _ Y X \to U_ i$ over $X$, such that $\{ V_{ij} \times _ Y X \to U_ i\} $ are Zariski coverings, see Lemma 59.48.3. This in particular implies that $\bigcup _{ij} g_{ij}(V_{ij})$ contains the closed subset $f(X)$. Hence the family of syntomic (resp. smooth, resp. étale) maps $g_{ij}$ together with the open immersion $Y \setminus f(X) \to Y$ forms the desired syntomic (resp. smooth, resp. étale) covering of $Y$. $\square$
Lemma 59.48.5. Let $f : X \to Y$ be a closed immersion of schemes. Let $\tau \in \{ syntomic, smooth, {\acute{e}tale}\} $. The functor $V \mapsto X \times _ Y V$ defines an almost cocontinuous functor (see Sites, Definition 7.42.3) $(\mathit{Sch}/Y)_\tau \to (\mathit{Sch}/X)_\tau $ between big $\tau $ sites.
Proof. We have to show the following: given a morphism $V \to Y$ and any syntomic (resp. smooth, resp. étale) covering $\{ U_ i \to X \times _ Y V\} $, there exists a smooth (resp. smooth, resp. étale) covering $\{ V_ j \to V\} $ such that for each $j$, either $X \times _ Y V_ j$ is empty, or $X \times _ Y V_ j \to Z \times _ Y V$ factors through one of the $U_ i$. This follows on applying Lemma 59.48.4 above to the closed immersion $X \times _ Y V \to V$. $\square$
Lemma 59.48.6. Let $f : X \to Y$ be a closed immersion of schemes. Let $\tau \in \{ syntomic, smooth, {\acute{e}tale}\} $.
The pushforward $f_{big, *} : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_\tau ) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_\tau )$ commutes with coequalizers and pushouts.
The pushforward $f_{big, *} : \textit{Ab}((\mathit{Sch}/X)_\tau ) \to \textit{Ab}((\mathit{Sch}/Y)_\tau )$ is exact.
Proof. This follows from Sites, Lemma 7.42.6, Modules on Sites, Lemma 18.15.3, and Lemma 59.48.5 above. $\square$
Remark 59.48.7. In Lemma 59.48.6 the case $\tau = fppf$ is missing. The reason is that given a ring $A$, an ideal $I$ and a faithfully flat, finitely presented ring map $A/I \to \overline{B}$, there is no reason to think that one can find any flat finitely presented ring map $A \to B$ with $B/IB \not= 0$ such that $A/I \to B/IB$ factors through $\overline{B}$. Hence the proof of Lemma 59.48.5 does not work for the fppf topology. In fact it is likely false that $f_{big, *} : \textit{Ab}((\mathit{Sch}/X)_{fppf}) \to \textit{Ab}((\mathit{Sch}/Y)_{fppf})$ is exact when $f$ is a closed immersion. If you know an example, please email stacks.project@gmail.com.
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