## Families of perverse sheaves

In this post I want to talk about some ongoing joint work with Peter Scholze. Since this came up in Scholze’s geometrization lectures, I thought it would be fun to go into a little more detail here. All inaccuracies below are entirely due to me, and the standard caveats about blog-level rigor apply.

The goal, broadly speaking, is to define a relative notion of perversity in etale cohomology, with respect to any finite type morphism $f:X \to S$ of schemes. In order to not make slightly false statements, I will take my coefficient ring to be $\mathbf{F}_\ell$ for some prime $\ell$ invertible on $S$. Everything below also works with more general torsion coefficients killed by an integer invertible on $S$, but then one has to be mindful of the difference between $D^{b}_{c}$ and $D^{b}_{ctf}$. With mild assumptions on $S$, everything below also works with $\mathbf{Q}_\ell$-coefficients.

When $S=\mathrm{Spec}k$ is a point, $X$ is just a finite type $k$-scheme, and we have the familiar perverse t-structure $( \phantom{}^p D^{\leq 0}(X), \phantom{}^p D^{\geq 0}(X))$ on $D(X)=D(X,\mathbf{F}_\ell)$, with all its wonderful properties as usual. The key new definition is the following.

Definition. Given a finite type map of schemes $f:X \to S$, let $\phantom{}^{p/S}D^{\leq 0}(X) \subset D(X)$ be the full subcategory of objects $A$ such that $A|X_{\overline{s}} \in \phantom{}^p D^{\leq 0}(X_{\overline{s}})$ for all geometric points $\overline{s} \to S$.

It is easy to see that $\phantom{}^{p/S}D^{\leq 0}(X)$ is stable under extensions and (after upgrading to derived $\infty$-categories) under filtered colimits, and is set-theoretically reasonable, so it defines the left half of a t-structure on $D(X)$ by Proposition 1.4.4.11 in Lurie’s Higher Algebra. We denote the right half of this t-structure, unsurprisingly, by $\phantom{}^{p/S}D^{\geq 0}(X)$, and call it the relative perverse t-structure (relative to $X\to S$, of course). We write $\phantom{}^{p/S}\tau^{\leq n}$ and $\phantom{}^{p/S}\tau^{\geq n}$ for the associated truncation functors.

This t-structure satisfies a number of good and fairly obvious formal properties which I won’t get into here (it can be glued from any open-closed decomposition of $X$, various operations are obviously left- or right- t-exact, etc.). Less formally, if $S$ is a finite-dimensional excellent Noetherian scheme, then the relative perverse truncation functors preserve $D^{b}_{c}(X) \subset D(X)$, so we get an induced relative perverse t-structure on $D^{b}_{c}(X)$. This follows from some results of Gabber: roughly, one can check that the relative perverse t-structure is the t-structure associated with the weak perversity function $p(x)=-\mathrm{tr.deg}k(x)/k(f(x))$, and that the conditions in Theorem 8.2 are satisfied for excellent $S$. (Nb. Gabber’s methods also reprove the existence of the relative perverse t-structure for any Noetherian $S$, without appealing to $\infty$-categories.)

However, the right half $\phantom{}^{p/S}D^{\geq 0}(X)$ is defined in a very inexplicit way, and it isn’t clear how to get your hands on this at all. The really shocking theorem, then, is the following result.

Key Theorem. An object $A \in D(X)$ lies in $\phantom{}^{p/S}D^{\geq 0}(X)$ if and only if $A|X_{\overline{s}} \in \phantom{}^p D^{\geq 0}(X_{\overline{s}})$ for all geometric points $\overline{s} \to S$.

Note that I really am taking *-restrictions to geometric fibers here, just as in the definition of $\phantom{}^{p/S}D^{\leq 0}(X)$. One might naively guess that !-restrictions should be appearing instead, but no!

This theorem has a number of corollaries.

Corollary 1. The heart $\mathrm{Perv}(X/S)$ of the relative perverse t-structure consists of objects $A \in D(X)$ which are perverse after restriction to any geometric fiber of $f$. In particular, the objects with this property naturally have the structure of an abelian category.

This fully justifies the choice of name for this t-structure, and shows that the heart of the relative perverse t-structure gives a completely reasonable notion of a “family of perverse sheaves parameterized by $S$”.

Corollary 2. For any map $T\to S$, the pullback functor $D(X) \to D(X_T)$ is t-exact for the relative perverse t-structures (relative to $S$ and $T$, respectively). In particular, relative perverse truncations commute with any base change on $S$, and pullback induces an exact functor $\mathrm{Perv}(X/S) \to \mathrm{Perv}(X_T / T)$.

Corollary 3. If $X\to S$ is any finitely presented morphism of qcqs schemes, then the relative perverse truncation functors on $D(X)$ preserve $D^{b}_{c}(X)$.

Corollaries 1 and 2 are immediate consequences of the Key Theorem. Corollary 3 then follows from the case where $S$ is Noetherian excellent finite-dimensional by Noetherian approximation arguments, using Corollary 2 crucially.

To prove the key theorem, we make some formal reductions to the situation where $S$ is excellent Noetherian finite-dimensional and $A \in D^{b}_{c}(X)$. In this situation, we argue by induction on $\dim S$, with the base case $\dim S=0$ being obvious. In general, this induction is somewhat subtle, and involves playing off the relative perverse t-structure on $D(X)$ against the perverse t-structures on $D(X_{\overline{s}})$ and the (absolute) perverse t-structure on $D(X)$ (which exists once you pick a dimension function on $S$).

However, when $S$ is the spectrum of an excellent DVR, one can give a direct proof of the key theorem, and this is what I want to do in the rest of this post. Let $i: s \to S$ and $j: \eta \to S$ be the inclusions of the special and generic points, with obvious base changes $\tilde{i}:X_s \to X$ and $\tilde{j}: X_\eta \to X$. By definition, $A \in D(X)$ lies in $\phantom{}^{p/S}D^{\leq 0}(X)$ iff $\tilde{j}^{\ast}A \in \phantom{}^{p}D^{\leq 0}(X_\eta)$ and $\tilde{i}^{\ast}A \in \phantom{}^{p}D^{\leq 0}(X_s)$. By standard results on gluing t-structures (see chapter 1 in BBDG), this implies that $A$ lies in $\phantom{}^{p/S}D^{\geq 0}(X)$ iff $\tilde{j}^{\ast}A \in \phantom{}^{p}D^{\geq 0}(X_\eta)$ and $R\tilde{i}^{!}A \in \phantom{}^{p}D^{\geq 0}(X_s)$. Thus, to prove the key theorem in this case, we need to show that for any $A \in D(X)$ with $\tilde{j}^{\ast}A \in \phantom{}^{p}D^{\geq 0}(X_\eta)$, the conditions $\tilde{i}^{\ast}A \in \phantom{}^{p}D^{\geq 0}(X_s)$ and $R\tilde{i}^{!}A \in \phantom{}^{p}D^{\geq 0}(X_s)$ are equivalent.

To show this, consider the triangle $R\tilde{i}^{!}A \to \tilde{i}^{\ast}A \to \tilde{i}^{\ast}R\tilde{j}_{\ast} \tilde{j}^{\ast} A \to$. The crucial observation is that $\tilde{j}^{\ast}A \in \phantom{}^{p}D^{\geq 0}(X_\eta)$ by assumption, and that $\tilde{i}^{\ast}R\tilde{j}_{\ast}$ carries $\phantom{}^{p}D^{\geq 0}(X_\eta)$ into $\phantom{}^{p}D^{\geq 0}(X_s)$. The italicized result follows from some theorems of Gabber generalizing the classical Artin-Grothendieck vanishing theorem for affine varieties, and is closely related to the well-known fact that nearby cycles are perverse t-exact. This immediately gives what we want: we now know that $\tilde{i}^{\ast}R\tilde{j}_{\ast} \tilde{j}^{\ast} A$ only can only have nonzero perverse cohomologies in degrees $\geq 0$, so $R\tilde{i}^{!}A$ and $\tilde{i}^{\ast}A$ have the same perverse cohomologies in degrees $<0$.

## The six functors for Zariski-constructible sheaves in rigid geometry

In this post I want to talk about my recent paper with Bhargav Bhatt, which you can find here. This paper was a lot of fun to write, and I hope the toolkit we built will be useful for other researchers in this area. In this post I want to make some random remarks on this paper, which probably won’t mean anything if you don’t go read the real introduction to the paper first.

One funny point is that the proof of Theorem 1.6 leans on Theorem 1.7 fairly heavily, but in fact you can prove Theorem 1.6 without appealing to Theorem 1.7, at the price of much more intricate arguments. This was actually the state of the manuscript until mid-December, when we finally figured out how to prove Theorem 1.7.

Another funny point is that the discussion of the “standard” / “constructible” t-structure on $D^{(b)}_{zc}(X,\mathbf{Z}_{\ell})$ turned out to be surprisingly subtle, cf. Theorem 3.39. Note that $D^{(b)}_{zc}(X,\mathbf{Z}_{\ell})$ is by definition a full subcategory of $D(X_v,\mathbf{Z}_{\ell})$, and the latter carries an obvious t-structure. Nevertheless, we weren’t able to settle the question of whether these t-structures are compatible:

Question. Do the cohomological functors $^c \mathcal{H}^n(-)$ on $D^{(b)}_{zc}(X,\mathbf{Z}_{\ell})$ produced by Theorem 3.39 agree with the usual cohomology sheaves on $D(X_v,\mathbf{Z}_{\ell})$?

I would be extremely interested to know the answer to this.

One thing missing from the paper is any discussion of ULA sheaves. (See Fargues-Scholze for the foundations of ULA sheaves in p-adic geometry. In what follows I take $\ell \neq p$, but the case $\ell = p$ should actually also be OK.) The first basic point to make is that for any rigid space $X/K$, any object $A \in D^{(b)}_{zc}(X,\mathbf{F}_{\ell})$ is ULA for the structure map $X \to \mathrm{Spa}K$. Sketch: The claim is local on $X$, so we can assume $X$ is quasicompact. By Proposition 3.6 and stability of ULA sheaves under proper pushforward, we reduce to the special case where $A = \mathbf{F}_{\ell}$ is constant. By an argument with resolution of singularities, we now reduce further to the case where $A$ is constant and $X$ is smooth, which is handled in Fargues-Scholze. Identical remarks apply with $\mathbf{Z}_{\ell}$-coefficients, or with general $\mathbf{Z}/n$ coefficients (but then only for objects of “finite tor-dimension”).

This is already enough to show that the Lefschetz trace formula works as expected for proper rigid spaces in characteristic zero. More precisely, suppose given a correspondence $c=(c_1,c_2): C \to X \times X$ of proper rigid spaces over an algebraically closed field, and a cohomological correspondence $u: c_1^{\ast}A \to Rc_2^{!}A$ on some $A \in D^b_{zc}(X,\mathbf{Z}_{\ell})$. Then the usual recipe to define local terms applies, and the expected equality $\mathrm{tr}(u|R\Gamma(X,A)) = \sum_{\beta \in \pi_0 \mathrm{Fix}(c)} \mathrm{loc}_{\beta}(u,A)$ holds true. (Note that $R\Gamma(X,A)$ is a perfect $\mathbf{Z}_{\ell}$-complex by Theorem 3.35.(3).)  This can be proved by imitating the unpleasant arguments with giant diagrams in SGA5, or by the amazing categorical magic of Lu-Zheng. Of course, the local terms $\mathrm{loc}_{\beta}(u,A)$ are just as mysterious as in the case of schemes.

It’s also natural to guess that an analogue of Deligne’s generic ULA theorem holds in this setting. Let me formally state this as a conjecture.

Conjecture. Let $f:X \to Y$ be a proper map of characteristic zero rigid spaces, and let $A \in D^{(b)}_{zc}(X,\mathbf{F}_{\ell})$ be any given object. Then there is a dense Zariski-open subset of $Y$ over which $A$ is $f$-ULA.

This should be within reach, but I didn’t think about it very much.

Finally, I want to highlight the open problems mentioned in Remark 4.10, Remark 4.14 and Section 4.5. Conjecture 4.16 is probably (for whatever reason) my favorite open problem right now. Actually, I don’t even know how to prove that $IH^{\ast}(X_C,\mathbf{Q}_p)$ is Hodge-Tate, or the even weaker statement that it has integral Hodge-Tate-Sen weights.

## Euler characteristics and perverse sheaves

Let $X$ be a variety over a separably closed field $k$, and let $A$ be some object in $D^b_c(X,\mathbf{Q}_{\ell})$. Laumon proved the beautiful result that the usual and compactly supported Euler characteristics $\chi(X,A)$ and $\chi_c(X,A)$ are always equal. Recently while trying to do something else, I found a quick proof of Laumon’s result, as well as a relative version, and I want to sketch this here.

Pick an open immersion into a compactification $j:X \to X'$; after a blowup, we can assume that $Z=X' - X$ is an effective Cartier divisor. Write $i:Z \to X'$ for the inclusion of the boundary. By the usual triangle $j_!A \to Rj_*A \to i_*i^* Rj_*A \to$, we reduce to showing that $\chi(X',i_*i^* Rj_*A)=0$. Filtering $A$ by its perverse cohomology sheaves, we reduce further to the case where also $A$ is perverse. Cover $X'$ by open affines $X_n'$ such that $Z_n= Z \cap X_n'$ is the divisor of a function $f_n$. By an easy Mayer-Vietoras argument, it’s now enough to show that for every open $U$ contained in some $X_n'$, $\chi(U,(i_* i^{\ast}Rj_{\ast}A)|U) = 0$.

But now we win: for any choice of such $U \subset X_n'$, there is an exact triangle $R\psi_{f_n}(A|U \cap X) \to R\psi_{f_n}(A|U \cap X) \to (i_* i^{\ast}Rj_{\ast}A)|U \to$ in $D^b_c(U,\mathbf{Q}_{\ell})$ where $R\psi_{f_n}:\mathrm{Perv}(U \cap X) \to \mathrm{Perv}( U \cap Z)$ is the unipotent nearby cycles functor associated with $f_n$, and the first arrow is the logarithm of the unipotent part of the monodromy. Since $\chi(U, -)$ is additive in exact triangles and the first two terms agree, we’re done.

A closer reading of this argument shows that you actually get the following stronger statement: for any $A$, the class $[i_*i^* Rj_*A] \in K_0\mathrm{Perv}(X')$ is identically zero. From here it’s easy to get a relative version of Laumon’s result.

Theorem. Let $f:X \to Y$ be any map of $k$-varieties. Then for any $A\in D^b_c(X,\mathbf{Q}_\ell)$, there is an equality $[Rf_! A]=[Rf_\ast A]$ in $K_0\mathrm{Perv}(Y)$.

## Artin-Grothendieck vanishing, again

A few years ago I started thinking about whether there was a natural rigid analytic version of the Artin-Grothendieck vanishing theorem. Last summer this grew into an obsession, and I managed to prove some general results. In particular, I showed that if $X$ is an affinoid rigid space over a complete algebraically closed field, AND $X$ comes via base change from an affinoid defined over a discretely valued subfield, then $H^i(X,\mathbf{Z}/n)=0$ for all $i > \mathrm{dim}(X)$ and all $n$ prime to the residue characteristic. I also proved a similar result with a non-constant coefficient sheaf, assuming moreover that the base field is of characteristic zero. This all got written up here.

Now, the hypothesis of definability over a discretely valued field is clearly stupid and shouldn’t be there, but I wasn’t able to remove it. So I was extremely happy this morning when Akhil Mathew and Bhargav Bhatt sent me an expanded version of their paper on arc-descent, in which they give a beautiful proof of rigid analytic Artin-Grothendieck vanishing without any superfluous assumptions. Their arguments are phrased in terms of algebraic geometry, rather than rigid analysis; in this post I want to recast (mostly for my own benefit I guess) the essential point of their argument in rigid analytic language.

The key is to prove the following.

Theorem (Bhatt-Mathew). Let $\mathrm{Spa}A$ be an affinoid rigid space over a complete algebraically closed nonarchimedean field $K$. Set $\Lambda = \mathbf{Z}/n$ where $n$ is any integer prime to the residue characteristic. Then $R\Gamma(\mathrm{Spa}A,\Lambda) \in D^{\leq \mathrm{dim}A}(\Lambda)$.

This implies the characteristic zero case of Conjecture 1.2 in my paper.

The proof proceeds in three steps.

Step One: Treat the case where $\mathrm{Spa}A$ is smooth. This was already done by Berkovich in the 90’s and I’ll take it for granted, although BM give their own nice argument for it. (Both arguments eventually appeal to the classical Artin-Grothendieck vanishing theorem.)

Step Two: Prove the weaker statement that $R\Gamma(\mathrm{Spa}A,\Lambda) \in D^{\leq 1+\mathrm{dim}A}(\Lambda)$ in general.

For this we use induction on $\mathrm{dim}A$. I’ll assume for simplicity that $K$ has characteristic zero. Without loss of generality we can assume that $A$ is reduced. Then by excellence of affinoid algebras, we can pick some non-zero-divisor $f \in A$ such that $A[1/f]$ is regular. Fix a nonzero nonunit $\pi \in \mathcal{O}_K$, and for any $n \geq 1$ consider the rational subsets $U_n = \{ x\,with\,|f(x)| \geq |\pi|^n \}$ and $V_n = \{x\,with\,|f(x)| \leq |\pi|^n \}$ inside $\mathrm{Spa}A$. Set $W_n = U_n \cap V_n$, so we get a Mayer-Vietoras distinguished triangle

$R\Gamma(\mathrm{Spa}A,\Lambda) \to R\Gamma(U_n,\Lambda)\oplus R\Gamma(V_n,\Lambda) \to R\Gamma(W_n,\Lambda)\to$

for any $n$. Note that $U_n$ and $W_n$ are smooth affinoids, so their etale cohomology is concentrated in degrees $\leq \mathrm{dim}A$ by Step One. Therefore, truncating the above Mayer-Vietoras sequence we get a quasi-isomorphism

$\tau^{\geq \mathrm{dim}A+2}R\Gamma(\mathrm{Spa}A,\Lambda) \simeq \tau^{\geq \mathrm{dim}A+2}R\Gamma(V_n,\Lambda)$

for any $n$. Moreover, $\mathrm{Spa}(A/f) \sim \lim_{n} V_n$ in the sense of adic spaces, which implies that the etale cohomology of the left-hand side is the colimit of the etale cohomologies of the right-hand sides. Therefore, passing to the colimit over $n$, the previous quasi-isomorphism gives a quasi-isomorphism

$\tau^{\geq \mathrm{dim}A+2} R\Gamma(\mathrm{Spa}A,\Lambda) \simeq \tau^{\geq \mathrm{dim}A+2}R\Gamma(\mathrm{Spa}(A/f),\Lambda)$.

But now we win, because $A/f$ is an affinoid of dimension $\dim(A)-1$, so by the induction hypothesis its etale cohomology is concentrated in degrees $\leq \mathrm{dim}A$.

Step Three. Bootstrap from the result of Step Two by a trick. More precisely, let $X=\mathrm{Spa}A$ and $\Lambda=\mathbf{Z}/n$ be as in the statement of the main theorem. By Step Two, we just have to show that $H^{\mathrm{dim}+1}(X,\Lambda)=0$. By another application of Step Two, the complex $R\Gamma(X,\Lambda) \otimes_{\Lambda}^{\mathbf{L}} R\Gamma(X,\Lambda)$ has cohomology in degree $2\mathrm{dim}A+2$ given by $H^{\mathrm{dim}A+1}(X,\Lambda)^{\otimes 2}$, and its enough to show that the latter module is zero. But

$R\Gamma(X,\Lambda) \otimes_{\Lambda}^{\mathbf{L}} R\Gamma(X,\Lambda) \simeq R\Gamma(X \times X,\Lambda)$

by the Kunneth formula*, and $X \times X$ is a $2\mathrm{dim}A$-dimensional affinoid, so its cohomology is concentrated in degrees $\leq 2\mathrm{dim}A+1$ by yet another application of Step Two. This gives the result.

*The necessary result is that if $X$ and $Y$ are $K$-affinoid spaces, then $R\Gamma(X \times Y, \Lambda) \simeq R\Gamma(X,\Lambda) \otimes_{\Lambda}^{\mathbf{L}} R\Gamma(Y,\Lambda)$. I’m not sure if this is in the literature; Bhargav and Akhil prove (an algebraic form of) it in their paper. However, it is easy to deduce this from the results in Huber’s book. The point is that $X, Y$ have canonical adic compactifications $\overline{X},\overline{Y}$, and etale cohomology (with constant coefficients) doesn’t change if you replace $X$ or $Y$ by its compactification. But then $\overline{X}$ and $\overline{Y}$ are proper over $\mathrm{Spa}K$ (in the sense of Huber’s book), so it’s easy to show that

$R\Gamma(\overline{X} \times \overline{Y},\Lambda) \simeq R\Gamma(\overline{X},\Lambda) \otimes_{\Lambda}^{\mathbf{L}} R\Gamma(\overline{Y},\Lambda)$

by the usual combination of proper base change and the projection formula.

## A stupid remark on cohomological dimensions

Let $Y$ be a finite-dimensional Noetherian scheme, and let $\ell$ be a prime invertible on $Y$. Gabber proved that if $f:X \to Y$ is any finite type morphism, then there is some integer $N$ such that $R^n f_{\ast} F$ vanishes for all $\ell$-torsion etale sheaves $F$ and all $n > N$, cf. Corollary XVIII.1.4 in the Travaux de Gabber volume. In particular, if the $\ell$-cohomological dimension $\mathrm{cd}_{\ell}(Y)$ is finite, then so is $\mathrm{cd}_{\ell}(X)$. It’s natural to ask for a quantitative form of this implication.

Claim. One has the bound $\mathrm{cd}_{\ell}(X) \leq \mathrm{cd}_{\ell}(Y)+2\mathrm{dim}f+2\mathrm{dim}(X)+s(X)$.

Here $\mathrm{dim}f$ is the supremum of the fibral dimensions of $f$, and $s(X) \in \mathbf{Z}_{\geq 0}$ is defined to be one less than the minimal number of separated open subschemes required to cover $X$. In particular, $s(X)=0$ iff $X$ is separated.

Can this be improved?

Anyway, here’s a cute consequence:

Corollary. Let $Y$ be a finite-dimensional Notherian scheme, and let $\ell$ be a prime invertible on $Y$ such that $\mathrm{cd}_{\ell}(Y) < \infty$. Then all affine schemes $X \in Y_{\mathrm{et}}$ have uniformly bounded $\ell$-cohomological dimension. In particular, the hypotheses of section 6.4 of Bhatt-Scholze hold, so the arguments therein show that $D(Y,\mathbf{F}_{\ell})$ is compactly generated and that its compact objects are exactly the objects of $D^{b}_{c}(Y,\mathbf{F}_{\ell})$.

## 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.