Completely decomposed proper covers and the cdh topology

PDF p. 5: Lemma 2.4 and cdh covers

Fix a field $k$ and a suitably small category equivalent to the category of separated schemes of finite type over $k$, with the indicated fibre products, closed subschemes and finite disjoint unions represented in it. Its morphisms are $k$-scheme morphisms. As usual, a morphism is proper if it is separated, of finite type and universally closed: every base change has closed image on every closed subset. For $x\in X$, $\kappa (x)$ denotes the residue field of the local ring at $x$.

A morphism $p:Y\to X$ is completely decomposed if, for every point $x\in X$, some $y\in Y$ maps to $x$ and the induced map $\kappa (x)\to \kappa (y)$ is an isomorphism. A completely decomposed proper morphism, or cdp-morphism, is one that is both proper and completely decomposed. Its singleton family $\{Y\to X\}$ is called a proper cdh covering family.

The cdh topology on this category is the least Grothendieck topology for which every Nisnevich covering family and every singleton family associated to a cdp-morphism generates a covering sieve. Thus a family is a cdh covering family when the sieve it generates is covering in this topology. A sieve is a collection of arrows into a fixed object closed under precomposition; a Grothendieck topology specifies covering sieves satisfying maximality, pullback stability and transitivity. A Nisnevich covering family consists of étale morphisms with a residue-field-preserving lift of every point. An arbitrary cdh covering family need not itself consist of proper morphisms.

An abstract blowup square is a cartesian square

$\begin{array}{ccc}E& ⟶& Y\\ \downarrow & & \downarrow p\\ Z& \stackrel{i}{\to }& X\end{array}$

in which $i$ is a closed immersion, $p$ is proper, $E=Y{\times }_{X}Z$, and $p$ restricts to an isomorphism $Y{\times }_X(X\setminus Z)\to X\setminus Z$. Here $X\setminus Z$ is the open complement of the underlying closed subset of $Z$, and cartesian means that the displayed $E$ is the fibre product. The associated family is $\{Z\to X,Y\to X\}$; it is a cdh covering family.

A statement that a presheaf or complex satisfies descent for abstract blowup squares is a statement about those squares. Applying it to all cdh covers requires a separate descent theorem with its hypotheses, including the relevant Nisnevich condition and the intended notion of descent or hyperdescent. This definition does not assert such a theorem.

Nisnevich covers and distinguished squares

Scheme sites and sheafification

The residue-field homomorphism is induced by the local homomorphism of the scheme morphism, so complete decomposition is an intrinsic condition on points. The category is small, and the maximal topology contains the specified sieves. The intersection of all Grothendieck topologies containing them is again a Grothendieck topology, giving the least one. The meanings of Nisnevich covering family and generated topology are those of Nisnevich covers and distinguished squares and Scheme sites and sheafification.

For an abstract blowup square, the map $Z⨿Y\to X$ is proper: a closed immersion is proper, and a finite disjoint union of proper $X$-schemes is proper. Indeed, after any base change, a closed immersion remains a closed immersion, and the image of a closed subset of a finite disjoint union is the finite union of its closed component images; finite type and separatedness are likewise preserved by this finite union. If $x$ lies in the closed subset underlying $Z$, the unique point of $Z$ over $x$ has residue field $\kappa (x)$. If $x$ lies in its open complement, the assumed isomorphism gives a point of $Y$ over $x$ with that residue field. Hence $Z⨿Y\to X$ is completely decomposed. Write $g:W=Z⨿Y\to X$, let $R$ be the sieve generated by $g$, and let $S$ be the sieve generated by the two component maps. These sieves need not be equal, but $S$ is covering if and only if $R$ is covering in any topology containing the Nisnevich covers. Indeed, $S\subseteq R$, so covering of $S$ implies covering of $R$ by upward closure. Conversely, the pullback sieve $g^{\ast }S$ contains both open-and-closed inclusions $Z\to W$ and $Y\to W$. These inclusions form a Nisnevich covering family: they are étale and each point of $W$ lies in one summand with unchanged residue field. Thus $g^{\ast }S$ is covering. Every arrow $f:T\to X$ in $R$ factors as $g\circ h$ for some $h:T\to W$, so $f^{\ast }S=h^{\ast }(g^{\ast }S)$ is covering by pullback stability. If $R$ is covering, local character (the transitivity axiom for covering sieves) therefore implies that $S$ is covering. Here $g$ is completely decomposed proper, so $R$ is a generating cdh covering sieve, and the associated two-member family is consequently cdh covering. This argument uses only the existing Nisnevich and topology axioms, and neither equates these sieves nor assumes a later finite-refinement theorem.