######  Corollary B.2.7.

Let
$\mathcal{F}\colon(\mathrm{Sch}^{\mathrm{qq}}_{/S})^{\mathrm{op}}\to\mathcal{A}$
be a presheaf. Then the following are equivalent:

  1. (1)

$\mathcal{F}$ is a sheaf with respect to the Zariski topology.

  2. (2)

$\mathcal{F}$ sends coproducts in $\mathrm{Sch}^{\mathrm{qq}}_{/S}$ to
products in $\mathcal{A}$ and satisfies Čech descent with respect to any
Zariski epi $p\colon X\to Y$.

  3. (3)

$\mathcal{F}$ sends coproducts in $\mathrm{Sch}^{\mathrm{qq}}_{/S}$ to
products in $\mathcal{A}$ and satisfies Čech descent with respect to any
Zariski epi of the form $\coprod_{i=1}^{n}U_{i}\to X$, where $U_{1},...,U_{n}$
are an open cover of $X$.

  4. (4)

$\mathcal{F}(\emptyset)$ is terminal in $\mathcal{A}$ and for every finite
open covering $X=U_{1}\cup\cdots\cup U_{n}$ the induced map

$\mathcal{F}(X)\to\lim_{\emptyset\neq
S\subseteq\\{1,\ldots,n\\}}\mathcal{F}(\cap_{i\in S}U_{i})$

is an equivalence.

  5. (5)

$\mathcal{F}(\emptyset)$ is terminal in $\mathcal{A}$ and for every Zariski
covering of the form $X=U_{1}\cup U_{2}$ of qcqs $S$-schemes, the square

${\mathcal{F}(X)}$${\mathcal{F}(U_{1})}$${\mathcal{F}(U_{2})}$${\mathcal{F}(U_{1}\cap
U_{2})}$

is cartesian.

###### Proof.

The equivalence of (1) and (2) follows from Proposition B.2.6 and the
characterization of [SAG]*Proposition A.3.3.1. The equivalence of (2) and (3)
follows from Corollary B.1.2, since any Zariski epi $Z\to X$ refines to a
Zariski epi of the form $\coprod_{i=1}^{n}U_{i}\to X$. The equivalence of (3)
and (4) is standard and follows from the fact the forgetful functor from the
simplex category of $\mathrm{cosk}_{0}(\\{1,...,n\\})$ to the poset of non-
empty subsets in $\\{1,...,n\\}$ is coinitial. Finally, the equivalence of (4)
and (5) follows in a standard manner by induction. ∎

As for the Nisnevich topology, it is often considered in the literature
exclusively for Noetherian schemes. Since we do not want to make this
assumption at this point, we will work with the following definition in the
general qcqs case (see [SAG]):

######  Definition B.2.8.

Let $X$ be a qcqs scheme. A collection of maps $\\{V_{\alpha}\to
X\\}_{\alpha}$ is said to be a Nisnevich covering if the following holds:

  1. (1)

Each $V_{\alpha}\to X$ is étale.

  2. (2)

There exists a finite sequence of open subsets $\emptyset=U_{0}\subseteq
U_{1}\subseteq\cdots\subseteq U_{n}=X$ of $X$, such that for every
$i=1,\ldots,n$ there exists an index $\alpha_{i}$ such that the dotted lift

${V_{\alpha_{i}}}$${U_{i}\setminus U_{i-1}}$${X}$

exists.

The _Nisnevich topology_ on $\mathrm{Sch}^{\mathrm{qq}}_{/S}$ is then by
definition the Grothendieck topology generated by Nisnevich coverings. In
other words, a sieve on $X\in\mathrm{Sch}^{\mathrm{qq}}_{/S}$ is a covering
sieve if and only if it contains a collection of morphisms $V_{\alpha}\to X$
which form a Nisnevich covering. We note that every Zariski covering of a qcqs
scheme is also a Nisnevich covering, and so the Nisnevich topology on
$\mathrm{Sch}^{\mathrm{qq}}_{/S}$ is finer than the Zariski topology.

######  Remark B.2.9.

In Definition B.2.8, if $\emptyset=U_{0}\subseteq
U_{1}\subseteq\cdots\subseteq U_{n}=X$ is a finite sequence of opens
satisfying (2) of that definition, then any refinement of this sequence (that
is, a sequence obtained by factoring each inclusion $U_{i}\subseteq U_{i+1}$
into a finite sequence $U_{i}=U^{0}_{i}\subseteq
U^{1}_{i}\subseteq\cdots\subseteq U^{m}_{i}=U_{i+1}$) will again satisfy (2).
It hence follows that if $\\{V_{\alpha}\to X\\}$ is a Nisnevich covering such
that $X=\mathrm{spec}(A)$ is affine then we may assume (by passing to a
refinement) that the open subsets $\emptyset=U_{0}\subseteq
U_{1}\subseteq\cdots\subseteq U_{n}=X$ are of the form
$U_{i}=\mathrm{spec}(A[f_{i}^{-1}])$ for $f_{i}\in A$. In particular, a
Nisnevich covering among affine schemes in this sense coincides with the
notion appearing in [SAG]*Definition B.4.1.1.

The Nisnevich topology is again finitary: indeed, if $\\{V_{\alpha}\to
X\\}_{\alpha}$ is a Nisnevich covering then $\\{V_{\alpha_{i}}\to
X\\}_{i=1,\ldots,n}$ is again a Nisnevich covering, where the $\alpha_{i}$’s
are determined by the finite sequence $\emptyset=U_{0}\subseteq
U_{1}\subseteq\cdots\subseteq U_{n}=X$ as in (2) of Definition B.2.8.

######  Proposition B.2.10.

Let $\mathcal{E}\subseteq\mathrm{Sch}_{/X}$ be a sieve on $X$. Then
$\mathcal{E}$ is covering with respect to the Nisnevich topology if and only
if it contains a finite collection of étale maps $\\{U_{i}\to
X\\}_{i=1,\ldots,n}$ such that the induced map $\coprod_{i}U_{i}\to X$ is a
Nisnevich epi in the sense of Definition B.2.1. In particular, the Nisnevich
topology on $\mathrm{Sch}_{/X}$ coincides with the one induced by the coverage
of Nisnevich epis.

###### Proof.

∎

By [SAG]*Proposition A.3.3.1, we then conclude:

######  Corollary B.2.11.

A presheaf
$\mathcal{F}\colon(\mathrm{Sch}^{\mathrm{qq}}_{/S})^{\mathrm{op}}\to\mathcal{A}$
is a Nisnevich sheaf if and only if it sends coproducts in
$\mathrm{Sch}^{\mathrm{qq}}_{/S}$ to products in $\mathcal{A}$ and satisfies
Čech descent for Nisnevich epis.

###  B.3. Characterization of Nisnevich sheaves

Let $\mathrm{Sch}^{\mathrm{qcs}}_{/S}\subseteq\mathrm{Sch}^{\mathrm{qq}}_{/S}$
be the full subcategory spanned by the morphisms $X\to S$ such that $X$ is
separated, and let
$\mathrm{Aff}_{/S}\subseteq\mathrm{Sch}^{\mathrm{qcs}}_{/S}$ be the full
subcategory therein spanned by those $X\to S$ such that $X$ is affine. Both of
these full subcategories are closed under direct sums and fibre products are
hence themselves extensive. We then define the Nisnevich topologies on
$\mathrm{Sch}^{\mathrm{qcs}}_{/S}$ and $\mathrm{Aff}_{/S}$ to be those
associated to the coverage of Nisnevich epis among separated and affine
schemes, respectively. By the same argument as in the proof of Proposition
B.2.10 we have that the resulting Grothendieck topologies on
$\mathrm{Sch}^{\mathrm{qcs}}_{/S}$ and $\mathrm{Aff}_{/S}$ coincide with the
ones generated by the Nisnevich coverings among separated and affine schemes.

######  Proposition B.3.1.

Let $S$ be a qcqs scheme and let
$\mathcal{F}\colon(\mathrm{Sch}^{\mathrm{qq}}_{/S})^{{}^{\mathrm{op}}}\to\mathcal{A}$
be a functor valued in some presentable $\infty$-category $\mathcal{A}$. Then
the following are equivalent:

  1. (1)

$\mathcal{F}$ is a Nisnevich sheaf.

  2. (2)

$\mathcal{F}$ is a Zariski sheaf and the restriction of $\mathcal{F}$ to
$(\mathrm{Sch}^{\mathrm{qcs}}_{/S})^{\mathrm{op}}\subseteq(\mathrm{Sch}^{\mathrm{qq}}_{/S})^{\mathrm{op}}$
is a Nisnevich sheaf.

  3. (3)

$\mathcal{F}$ is a Zariski sheaf and the restriction of $\mathcal{F}$ to
$(\mathrm{Aff}_{/S})^{\mathrm{op}}\subseteq(\mathrm{Sch}^{\mathrm{qq}}_{/S})^{\mathrm{op}}$
is a Nisnevich sheaf.

The proof of Proposition B.3.1 will require the following lemma:

######  Lemma B.3.2.

Let

${W}$${Y}$${Z}$${X}$

be a pullback square of qcqs schemes all of whose legs are Nisnevich epis, and
let $p\colon W^{\prime}\to W$ be a map of qcqs schemes. Then $p$ is a
Nisnevich epi if and only if the composites $W^{\prime}\to W\to Z$ and
$W^{\prime}\to W\to Y$ are Nisnevich epis.

###### Proof.

The only if direction follows from the fact that Nisnevich epis are closed
under composition. To prove the if the direction, let
$\emptyset=U_{0}\subseteq U_{1}\subseteq\cdots\subseteq U_{n}=Z$ and
$\emptyset=V_{0}\subseteq V_{1}\subseteq\cdots\subseteq V_{m}=Y$ be ascending
sequences of open subschemes such that each of the inclusions $U_{i}\setminus
U_{i-1}\hookrightarrow Z$ and $V_{j}\setminus V_{j-1}\hookrightarrow Y$ lifts
to $W^{\prime}$. We construct a sequence of open subschemes

$W_{1,1}\subseteq W_{1,2}\subseteq\cdots\subseteq W_{1,m}\subseteq
W_{2,1}\subseteq\cdots\subseteq W_{1,m}\subseteq\cdots\subseteq W_{n,m}=W$

inductively as follows. First we set $W_{1,1}=U_{1}\times_{X}V_{1}$. Let
$1\leq i\leq n$ and $1\leq j\leq m$ be such that $W_{i^{\prime},j^{\prime}}$
has been defined for every $(i^{\prime},j^{\prime})$ such that either
$i^{\prime}<i$ or $i^{\prime}=i$ and $j^{\prime}<j$, that is, all
$(i^{\prime},j^{\prime})$ lexicographically smaller than $(i,j)$. Let
$(i^{\prime},j^{\prime})$ be the largest element smaller than $(i,j)$ in the
lexicographical order. Then we define $W_{i,j}$ by

$W_{i,j}=W_{i^{\prime},j^{\prime}}\cup(U_{i}\times_{X}V_{j}).$

We now observe that by construction, both $U_{i-1}\times_{X}V_{j}$ and
$U_{i}\times V_{j-1}$ are contained in $W_{i^{\prime},j^{\prime}}$, and hence
$W_{i,j}\setminus W_{i^{\prime},j^{\prime}}$ is contained in $(U_{i}\setminus
U_{i-1})\times_{X}(V_{j}\setminus V_{j-1})$ for every lexicographically
consecutive pair $(i^{\prime},j^{\prime})<(i,j)$ in
$\\{1,\ldots,n\\}\times\\{1,\ldots,m\\}$ (this is also true if $i$ or $j$ are
$1$, since $U_{0}=V_{0}=\emptyset$) It then follows that the inclusion
$W_{1,1}\hookrightarrow W$, as well as each of the inclusions
$W_{i,j}\setminus W_{i^{\prime},j^{\prime}}$ for every lexicographically
consecutive pair $(i^{\prime},j^{\prime})<(i,j)$ in
$\\{1,\ldots,n\\}\times\\{1,\ldots,m\\}$, lifts to $W^{\prime}$, and so
$W^{\prime}\to W$ is a Nisnevich epi, as desired.

∎

###### Proof of Proposition B.3.1.

Assume first that
$\mathcal{F}\colon(\mathrm{Sch}^{\mathrm{qq}}_{/S})^{\mathrm{op}}\to\mathcal{A}$
is a Nisnevich sheaf. By Corollary B.2.11 $\mathcal{F}$ satisfies čech descent
for Nisnevich epis. Since any Zariski epi is a Nisnevich epi $\mathcal{F}$ is
also a Zariski sheaf. In addition, the restriction of $\mathcal{F}$ to
$\mathrm{Sch}^{\mathrm{qcs}}_{/S}$ satisfies Čech descent for Nisnevich epis
among separated and affine schemes and so (1) $\Rightarrow$ (2), and similarly
(2) $\Rightarrow$ (3) for the same reason.

We now prove that (3) $\Rightarrow$ (2) $\Rightarrow$ (1). Suppose that
$\mathcal{F}$ is a Zariski sheaf whose restriction to
$(\mathrm{Aff}_{/S})^{\mathrm{op}}$ is a Nisnevich sheaf. We prove that the
restriction of $\mathcal{F}$ to $\mathrm{Sch}^{\mathrm{qcs}}_{/S}$ is also a
Nisnevich sheaf. Now since $\mathcal{F}$ is a Zariski sheaf it takes
coproducts of schemes to products in $\mathcal{A}$ by Corollary B.2.7. It will
hence suffice to show that it satisfies Čech descent with respect to Nisnevich
epis among separated schemes. Let $q\colon V\to X$ be a Nisnevich epi of
quasi-compact $S$-schemes such that $X$ and $V$ are separated. Choose a finite
open covering $X=\cup_{i=1,\ldots,n}U_{i}$ by affines and write
$U=\coprod_{i}U_{i}$, so that $p\colon U\to X$ is a Zariski epi and $U$ is
affine (and in particular quasi-compact and separated). Let $Y=U\times_{X}V$.
Being a fibre product of quasi-compact and separated schemes, $Y$ is also
quasi-compact separated. Let us now choose again a finite open covering
$Y=\cup_{j=1,..,m}W_{i}$, and set $W=\coprod_{j}W_{j}$, so that $W\to Y$ is a
Zariski epi and $W$ is affine. Since Nisnevich epis are closed under base
change and composition, we have that the composed map $W\to Y\to U$ is a
Nisnevich epi, and similarly, the composed map $W\to Y\to V$ is a Zariski epi.
Consider the commutative square

(58)
${W}$${V}$${U}$${X}$$\scriptstyle{p^{\prime}}$$\scriptstyle{q^{\prime}}$$\scriptstyle{q}$$\scriptstyle{p}$

whose vertical arrows are Nisnevich epis and whose horizontal arrows are
Nisnevich epis. We claim that this square satisfies the assumptions of Lemma
B.1.1. For this, first note that since Nisnevich epis are closed under base
change and composition we have that all the maps in Čech resolutions of $p$
and $p^{\prime}$ are Zariski epis. Combined with Lemma B.3.2, this implies
that each of the maps $U_{n}(p^{\prime})\to U_{n}(p)$ is a Nisnevich epi. At
the same time, $U_{n}(p)$ decomposes as a finite disjoint union of the form

$U_{n}(p)=\coprod_{i_{1},\ldots,i_{n+1}}[U_{i_{1}}\cap\cdots\cap U_{i_{n+1}}]$

and each $U_{i_{1}}\cap\cdots\cap U_{i_{n+1}}$ is affine, being the
intersection of affine open subschemes inside a separated scheme. By
assumption we thus have that $\mathcal{F}$ satisfies Čech descent with respect
to each $U_{n}(p^{\prime})\to U_{n}(p)$. We now need to show that
$\mathcal{F}$ also satisfies Čech descent with respect to each
$U_{m}(q^{\prime})\to U_{m}(q)$. Now we claim that each of these maps is a
Zariski epi. To see this, let us factor the above square as the pasting of two
squares as follows

${W}$${Y}$${V}$${U}$${U}$${X}$$\scriptstyle{p^{\prime\prime}}$$\scriptstyle{q^{\prime}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{q^{\prime\prime}}$$\scriptstyle{q}$$\scriptstyle{p}$

We claim that the maps $U_{m}(q^{\prime})\to U_{m}(q^{\prime\prime})$ and
$U_{m}(q^{\prime\prime})\to U_{m}(q)$ are both Zariski epis (and hence the
composite is such as well). For the latter, by construction this is the map

$W\times_{U}\cdots\times_{U}W=\coprod_{i_{1},\ldots,i_{m+1}}[W_{i_{1}}\times_{U}\cdots\times_{U}W_{i_{m+1}}]\to
Y\times_{U}\cdots\times_{U}Y$

which is a Zariski epi since each of the maps

$W_{i_{1}}\times_{U}\cdots\times_{U}W_{i_{m+1}}\to
Y\times_{U}\cdots\times_{U}Y$

is an open embedding.

We have thus shown that the square (58) satisfies the assumption of Lemma 58,
and hence we may conclude that $\mathcal{F}$ satisfies Čech descent with
respect to $p$ if and only if it satisfies Čech descent with respect to $q$.
Since $p$ is a Zariski epi $\mathcal{F}$ satisfies descent with respect to $p$
and hence we conclude that $\mathcal{F}$ satisfies Čech descent with respect
to $q$.

This shows that (3) $\Rightarrow$ (2). The proof that (2) $\Rightarrow$ (1)
proceeds in exactly the same manner, where we note that the only place we used
the assumption that $X$ is separated was in order to conclude that each
$U_{i_{1}}\cap\cdots\cap U_{i_{n+1}}$ is affine, being the intersection of
affine open subschemes inside a separated scheme. In order to prove (2)
$\Rightarrow$ (1) we no longer need each $U_{i_{1}}\cap\cdots\cap U_{i_{n+1}}$
to be affine, just separated. But this automatic from the fact that these are
open subschemes of the affine (and in particular separated) scheme
$U_{i_{1}}$, and being separated is a property which is inherited by open
subschemes. ∎

## References

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