Topology generators on general scheme categories

Fix universe size bounds and a specified small category $\mathcal{C}$ of schemes and scheme morphisms. A family below is set-indexed and consists of arrows of this category. The sieve generated by a family consists of the arrows that factor through one of its members. The topology generated by specified families is the least Grothendieck topology in which their generated sieves cover.

An étale covering family $\{U_i\to V\}$ consists of étale morphisms whose images together contain every point of $V$. The étale topology is generated by these families. For the small étale site of $X$, the category is a specified size-bounded category of étale $X$-schemes and $X$-morphisms; a big étale site uses a separately specified category of schemes over its base.

A Nisnevich covering family is an étale family with a residue-field-preserving lift of every point: for each $v\in V$, some $u\in U_i$ lies over $v$ and the induced map $\kappa (v)\to \kappa (u)$ is an isomorphism. Here $\kappa (v)$ is the local ring at $v$ modulo its maximal ideal. The Nisnevich topology is generated by these families.

A scheme is qcqs when it is quasi-compact and quasi-separated. A morphism is finitely presented when it is locally of finite presentation, quasi-compact and quasi-separated. Proper means separated, of finite type and universally closed. For a qcqs target $V$, a cdp generating family $\{Y_i\to V\}$ consists of proper finitely presented morphisms of qcqs schemes and is completely decomposed: each point of $V$ has a residue-field-preserving lift to some $Y_i$. On a specified small category $\mathcal{D}$ of qcqs schemes, let $J_{\mathcal{D}}$ be the topology generated by its available cdp families and Nisnevich covering families. This is the qcqs cdh convention; finite presentation on the cdp generators is retained.

For the extension to schemes that need not be qcqs, require that $\mathcal{C}$ be a full category of schemes, or a full category of schemes over one fixed base, with the following properties: every open subscheme of an object is again an object, and every object admits a set-indexed open covering by qcqs objects of $\mathcal{C}$. All morphisms are taken in the indicated ambient category. Put $\mathcal{D}={\mathcal{C}}_{\mathrm{q}\mathrm{c}\mathrm{q}\mathrm{s}}$, the full category on its qcqs objects, with the topology $J_{\mathcal{D}}$ just defined. These are explicit hypotheses on the chosen category, not consequences of its smallness.

Define the Zariski-local cdh extension $J_{\mathcal{C}}^{\mathrm{l}\mathrm{o}\mathrm{c}}$ as follows. For a sieve $R$ on $V\in \mathcal{C}$ and a morphism $f:T\to V$ with $T\in \mathcal{D}$, define a sieve on $T$ in $\mathcal{D}$ by $R_{f,\mathcal{D}}=\{a:W\to T\mid W\in \mathcal{D},f\circ a\in R\}$. Declare $R$ covering precisely when $R_{f,\mathcal{D}}\in J_{\mathcal{D}}(T)$ for every such $f$. This is a Grothendieck topology. If $V$ is qcqs, the condition is equivalent to $R_{{\mathrm{i}\mathrm{d}}_V,\mathcal{D}}\in J_{\mathcal{D}}(V)$, so it restricts exactly to the stated qcqs topology. For arbitrary $V$, it is equivalent to requiring $R_{j_{\alpha },\mathcal{D}}\in J_{\mathcal{D}}(V_{\alpha })$ for all members of one, equivalently every, qcqs open cover $\{j_{\alpha }:V_{\alpha }\hookrightarrow V\}$ in $\mathcal{C}$.

For a category without these extension hypotheses, the least topology generated by its available Nisnevich families and its cdp families with qcqs targets still exists; call it the available-family generated topology. Its identification with the Zariski-local extension is not asserted. Categories, size bounds and the chosen construction are part of the site data. No identification of small and big sites, ambient-site cohomology comparison, descent theorem or hyperdescent theorem is supplied here.

Scheme sites and sheafification

Nisnevich covers and distinguished squares

Étale morphism of schemes

Proper morphisms

Locally finite presentation morphisms

Quasi-compact and quasi-separated morphisms

Quasi-compact and quasi-separated schemes

Open immersions of schemes

Morphisms of schemes

Grothendieck universe closure convention

Reference 0 supplies the étale covering and small-category conventions. Reference 1 supplies only the qcqs cdp/cdh generator convention and its finite-presentation qualification; its assertion that cdh is defined on all schemes is not used as a proof of an extension construction. Reference 2 supplies finite-presentation terminology. The generated-topology argument and the explicitly specified Zariski-local extension, including its restriction and open-cover criteria, are justified locally below.

Given: The category and generators of the Definition, with F1–F4; the additional category hypotheses are imposed whenever $J_{\mathcal{C}}^{\mathrm{l}\mathrm{o}\mathrm{c}}$ is used.

1.1 For each specified family the arrows factoring through its members form a sieve, since precomposition preserves such a factorization. By F1 the size bounds make the sieves and topology assignments sets in the chosen larger universe. The assignment declaring every sieve covering is a Grothendieck topology containing the generators. Intersect all topologies containing them, object by object: maximality and pullback stability hold in every member, and if a sieve is locally covering along an intersection-covering sieve, transitivity in every member makes the tested sieve belong to every member. Thus the intersection is the unique least containing topology. This constructs the étale and Nisnevich topologies, $J_{\mathcal{D}}$, and the available-family generated topology on any specified category. F2 and F3 specify exactly their families; no extension or comparison follows merely from this intersection argument. [F1, F2, F3, given]

2.1 For the extension, every $R_{f,\mathcal{D}}$ is a sieve by precomposition. The maximal sieve passes every test. Pullback stability follows because testing $g^{\ast }R$ at $f$ is testing $R$ at $g\circ f$. For transitivity, let $S$ pass every test and suppose $h^{\ast }R$ passes every test for each $h$ in $S$. Fix $f:T\to V$ with $T\in \mathcal{D}$. The sieve $S_{f,\mathcal{D}}$ covers $T$ in $J_{\mathcal{D}}$. For each $a:W\to T$ in that sieve, $f\circ a$ lies in $S$, so testing $(f\circ a{)}^{\ast }R$ at ${\mathrm{i}\mathrm{d}}_W$ shows that $a^{\ast }{R}_{f,\mathcal{D}}=R_{f\circ a,\mathcal{D}}$ covers $W$. Transitivity in $J_{\mathcal{D}}$ gives $R_{f,\mathcal{D}}$ covering. This proves all three topology axioms. When $V\in \mathcal{D}$, the identity test is necessary and, by pullback stability in $J_{\mathcal{D}}$, sufficient; moreover a sieve on $V$ in $\mathcal{D}$ generates a sieve in $\mathcal{C}$ whose restriction is exactly the original sieve, since $\mathcal{D}$ is full. Hence the restriction is exactly $J_{\mathcal{D}}$. Finally, fix a qcqs open cover $\{j_{\alpha }:V_{\alpha }\hookrightarrow V\}$. Passing every test implies passing its $j_{\alpha }$ tests. Conversely assume these tests pass and fix $f:T\to V$ with $T$ qcqs. By F4 the inverse images $f^{-1}(V_{\alpha })$ are open objects and admit qcqs open covers. Take all qcqs open subschemes $W\subseteq T$ whose images lie in at least one $V_{\alpha }$; they form a set-indexed open covering of $T$, so F2 makes their generated sieve covering in $J_{\mathcal{D}}$. For every such $W$, select one witnessing index only for this individual argument. The arrow $W\to V$ factors through $V_{\alpha }$, and pullback stability in $J_{\mathcal{D}}$ shows that $R_{f,\mathcal{D}}$ restricts to a covering sieve on $W$. Further pullback gives the same conclusion for every arrow of the generated covering sieve. Transitivity now makes $R_{f,\mathcal{D}}$ covering on $T$. Thus $R$ passes every test. This proves the criterion for one, hence every, qcqs open cover, with no simultaneous choices and no cohomological comparison. [F1, F2, F4, step 1.1, given] ∎