The one-point compactification of a scheme, part 1

In this series of posts I want to talk about some joint work in progress with Johan de Jong. The goal here is to give a new and totally canonical definition of the functor $Rf_!$ in the etale cohomology of schemes, and to do some really fun geometry along the way. In fact we go beyond the usual situation where $f$ is assumed to be compactifiable, and give a canonical definition of $Rf_!$ for $f$ any locally finite type morphism of locally Noetherian schemes. Such a definition has also been proposed by Liu-Zheng, by a method which requires a very heavy use of $\infty$-categories. Our approach does not involve $\infty$-categories at all.

For motivation, let $X$ be a locally compact Hausdorff space. The one-point compactification of $X$ is obtained by suitably topologizing the set $\overline{X}=X \cup \{\infty\}$: precisely, one takes the open subsets to be all the open subsets of $X$ together with all subsets of the form $V \cup \{\infty \}$ where $V \subset X$ is such that $X \smallsetminus V$ is a closed compact subset of $X$. Then $\overline{X}$ is a compact Hausdorff space, and $X$ is a dense open subset of $\overline{X}$ if $X$ is non-compact. Quite generally, one can check that the inclusion $X \to \overline{X}$ is final among all open embeddings of locally compact Hausdorff spaces $X \to Y$.

Our first goal is to imitate this compactification in the setting of schemes. The correct definition turns out to be as follows.

Definition 1.1. Fix a base scheme $S$, and let $f: X \to S$ be a morphism of schemes which is separated and of finite type (for brevity, we say $X$ is a good $S$-scheme if its structure map is separated and of finite type). The one-point compactification of $X$ over $S$, denoted $\overline{X}^S$, is the contravariant functor from $S$-schemes to sets sending an $S$-scheme $T\to S$ to the set of closed subschemes $Z \subset X_T = X\times_S T$ such that the composite map $Z \to X_T \to T$ is an open immersion. Equivalently, $\overline{X}^S(T)$ is the set of pairs $(Z,\varphi)$ where $Z \subset T$ is an open subscheme and $\varphi : Z \to X$ is an $S$-scheme map whose graph $\Gamma_{\varphi}: Z \to X\times_S T$ is a closed immersion.

Usually $S$ will be clear from context, and we’ll abbreviate $\overline{X}^S$ to $\overline{X}$. Let $\overline{f} : \overline{X} \to S$ denote the “structure map”.

(Here and in what follows, we write $\mathrm{Sch}/S$ for the category of $S$-schemes, and we freely “do geometry” in the category of presheaves of sets on $\mathrm{Sch}/S$ in the modern style, since $S$-schemes embed fully faithfully into this category in the usual way. There are some small issues here with “big categories” which I will totally ignore; suffice it to say that we “fix a choice of a category $\mathrm{Sch}/S$” in the sense of the Stacks Project.)

Anyway, here are some immediate observations on this thing. First of all, there is a canonical map $j^X : X \to \overline{X}$ sending a $T$-point $\varphi: T \to X$ to the pair $(T,\varphi)$; indeed, the separateness of $f$ guarantees that $\Gamma_{\varphi} : T \to X_T$ is a closed immersion. Moreover, the structure map $\overline{X} \to S$ has a canonical “section at infinity” $\infty: S \to \overline{X}$ sending any $S$-scheme $T$ to the closed subscheme $Z= \emptyset \subset X_T$, and $j^X$ and $\infty$ are “disjoint” in the evident sense.

Example 1.2. If $S$ is arbitrary and $X \to S$ is proper, then $\overline{X} = X \coprod S$. (Hint: For any $T$-point of $\overline{X}$, the map $Z \to T$ is a proper open immersion.)

Example 1.3. If $S$ is arbitrary and $X= \mathbf{A}^1_S$, then $\overline{X}$ is the ind-scheme obtained as an “infinite pinching” of $\mathbf{P}^1_S$ along the section at infinity. More precisely, we can assume (e.g. by Theorem 1.5.i below) that $S=\mathrm{Spec}A$ is affine. Let $B_n \subset A[t^-1]$ be the ring of polynomials $\sum a_i t^{-i}$ such that $a_i=0$ for all $0 < i < n$. Set  $U=\mathrm{Spec}A[t]$ and $V_n = \mathrm{Spec}B_n$; gluing these along their common open $\mathbf{G}_{m,S}$ in the obvious way, we get an inductive system of schemes $X_1=\mathbf{P}^1_S \to X_2 \to X_3 \to \cdots$, with compatible maps $X_i \to \overline{X}$. In the colimit this gives a map $\mathrm{colim} X_n \to \overline{X}$. This map turns out to be an isomorphism, but this is not so obvious.

Example 1.4. If $S$ is arbitrary and $X = \mathbf{A}^2_S$, then $\overline{X}$ is NOT an ind-scheme or ind-algebraic space.

This last example is typical: for almost all $X \to S$, the functor $\overline{X}$ will not be an object of classical algebraic geometry. Nevertheless, it turns out to have very reasonable properties, which we collect in the following theorem:

Theorem 1.5. Fix a base scheme $S$, and let $X \to S$ and $\overline{X}=\overline{X}^S \to S$ be as above. Then:
i. If $S' \to S$ is any scheme map, there is a canonical isomorphism $\overline{X}^S \times_S S' \cong \overline{X \times_S S'}^{S'}$.
ii. The map $j^X : X \to \overline{X}$ is (representable in schemes and) an open immersion.
iii. The functor $\overline{X}$ is a sheaf for the fpqc topology.
iv. The structure map $\overline{X} \to S$ satisfies the valuative criterion of properness.
v. If $X \to S$ is of finite presentation, then $\overline{X} \to S$ is limit-preserving.
vi. The diagonal $\Delta: \overline{X} \to \overline{X} \times_S \overline{X}$ is representable in formal algebraic spaces: for any scheme with a map $T \to \overline{X} \times_S \overline{X}$, the fiber product $W= \overline{X} \times_{\Delta, \overline{X} \times_S \overline{X}} T$ is representable by a formal algebraic space whose underlying reduced subspace is a closed subscheme of $T$. If $S$ is locally Noetherian then $W$ is a countably indexed directed colimit of closed subschemes of $T$ along thickenings.
vii. If $g: X \to Y$ is any proper map of good $S$-schemes, there is a canonical map $\overline{g}: \overline{X} \to \overline{Y}$ such that $j^Y \circ g = \overline{g} \circ j^X$.
viii. If $h: U \to V$ is any open immersion of good $S$-schemes, there is a canonical map $\tilde{h}: \overline{V} \to \overline{U}$ such that $j^U = \tilde{h} \circ j^V \circ h$.

Let me say a few words about the proofs. Part i. is trivial from the definitions. For ii., one checks directly that for a given $T$-point $T \to \overline{X}$ with associated pair $(Z,\varphi)$, the pullback $X \times_{\overline{X}} T$ is just the open subscheme $Z \subset T$.

For iii., let $T' \to T$ be an fpqc cover, and suppose given $(Z',\varphi') \in \overline{X}(T')$ lying in the equalizer of $\overline{X}(T') \rightrightarrows \overline{X}(T' \times_T T')$. One first descends the open subscheme $Z' \subset T'$ to an open subscheme $Z \subset T$ using the fact that $|T'| \to |T|$ is a quotient map, and then one descends the morphism $\varphi'$ to a map $Z \to X$. To see that $(Z,\varphi)$ has the right properties, note that the graph $\Gamma_{\varphi}$ pulls back to the closed immersion $\Gamma_{\varphi'}$ along the fpqc cover $X_{T'} \to X_{T}$, so $\Gamma_{\varphi}$ is necessarily a closed immersion.

For iv., one reduces by i. to checking that if $S=\mathrm{Spec}A$ is the spectrum of an arbitrary valuation ring with generic point $\eta \in S$ and $X \to S$ is any good $S$-scheme, then the evident “restriction” map $r: \overline{X}^S(S) \to \overline{X_{\eta}}^{\eta}(\eta)$ is a bijection. After showing that the points at infinity match up, this reduces to showing that any section $s: \eta \to X_{\eta}$ spreads out to a unique point $(Z,\varphi) \in \overline{X}^S(S)$. For this, let $Z \subset X$ be the scheme-theoretic image of $s$ in $X$. By the Lemma in my previous post, the composite map $Z \to S$ is an open immersion, and we’re done.

For v., one takes an arbitrary $T$-point of $\overline{X}$, where $T = \lim T_i$ is a limit of affine $S$-schemes, and then uses “standard limit nonsense” to show that everything comes by pullback from a $T_i$-point for some $i$ – I leave it as an exercise for the reader to fill in the details, making use of Proposition 8.6.3 and Corollaire 8.6.4 in EGA IV_3.

Part vi. is probably the hardest. Let $T \to \overline{X} \times_{S} \overline{X}$ be as in the statement. This corresponds to a pair of $T$-points of $\overline{X}$, i.e. a pair of closed subschemes $Z_i \subset X_T$ for $i=1,2$ such that the induced maps $Z_i \to T$ are open immersions. Let $U=Z_1 \cup_T Z_2$, so this is an open subscheme of $T$. Let $Z=Z_1 \times_{X_T} Z_2$ be the intersection of the $Z_i$‘s inside $X_T$, so we get natural closed immersions $Z \to Z_i$, and composing either of them with the inclusion $Z_i \to U$ realizes $Z$ as a closed subscheme of the open subscheme $U \subset T$. At this point we make the

Definition. Let $T$ be a scheme, and suppose given an open subscheme $U \subset T$ together with a closed subscheme $Z \subset U$. Let $T_{Z \to U}$ be the subfunctor of $T$ whose $V$-points are given by scheme maps $f: V \to T$ such that $f^{-1}(U) \to U$ factors over the closed immersion $Z \to U$.

Now, returning to the situation at hand, you can check directly from the definitions (if you dare) that the fiber product $W$ in vi. is given by the functor $T_{Z \to U}$, for the specific $T,Z,U$ above. This reduces us to a general result:

Lemma. Notation as in the previous definition, the functor $T_{Z\to U}$ is a formal algebraic space whose underlying reduced subspace is a closed subscheme of $T$, namely the reduced closed subscheme corresponding to the closed subset $|Z| \cup (|T| \smallsetminus |U|) \subset |T|$. If $T$ is Noetherian, $\mathcal{I} \subset \mathcal{O}_T$ is the coherent ideal sheaf corresponding to the scheme-theoretic closure $\overline{Z} \subset T$ of $Z$, and $\mathcal{J}$ is any coherent ideal sheaf whose vanishing locus cuts out the closed subspace $|T| \smallsetminus |U|$, then $T_{Z \to U} \cong \mathrm{colim}\, \underline{\mathrm{Spec}}\mathcal{O_X}/(\mathcal{I}\cdot \mathcal{J}^n)$.

Intuitively, $T_{Z \to U}$ is the “union” inside $T$ of the locally closed subscheme $Z$ and the formal completion of $T$ along the complement of $U$.

For vii., one takes the scheme-theoretic image of $Z \subset X_T$ along the map $Z \to X_T \to Y_T$ and then checks that the resulting closed subscheme $Z' \subset Y_T$ has the right properties; in fact $Z' \simeq Z$.

For viii., one takes the pullback of $Z \subset V_T$ along the open immersion $U_T \to V_T$. This clearly has the right properties.

Note that the contravariant functoriality in part viii. might seem strange at first, since it has no analogue for schematic compactifications.  However, this is totally analogous with the situation for one-point compactifications of topological spaces: if $U \to V$ is an open embedding of locally compact Hausdorff spaces, then $\overline{U}$ is obtained from $\overline{V}$ by contracting $\overline{V} \smallsetminus U$ down to the point at infinity, giving a canonical map $\overline{V} \to \overline{U}$.

In part 2, we’ll discuss the applications to etale cohomology.