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 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 is assumed to be compactifiable, and give a canonical definition of for 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 -categories. Our approach does not involve -categories at all.

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

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 , and let be a morphism of schemes which is separated and of finite type (for brevity, we say is a *good -scheme *if its structure map is separated and of finite type). The *one-point compactification of over *, denoted , is the contravariant functor from -schemes to sets sending an -scheme to the set of closed subschemes such that the composite map is an open immersion. Equivalently, is the set of pairs where is an open subscheme and is an -scheme map whose graph is a closed immersion.

Usually will be clear from context, and we’ll abbreviate to . Let denote the “structure map”.

(Here and in what follows, we write for the category of -schemes, and we freely “do geometry” in the category of presheaves of sets on in the modern style, since -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 ” in the sense of the Stacks Project.)

Anyway, here are some immediate observations on this thing. First of all, there is a canonical map sending a -point to the pair ; indeed, the separateness of guarantees that is a closed immersion. Moreover, the structure map has a canonical “section at infinity” sending any -scheme to the closed subscheme , and and are “disjoint” in the evident sense.

**Example 1.2.** If is arbitrary and is proper, then . (Hint: For any -point of , the map is a proper open immersion.)

**Example 1.3.** If is arbitrary and , then is the ind-scheme obtained as an “infinite pinching” of along the section at infinity. More precisely, we can assume (e.g. by Theorem 1.5.i below) that is affine. Let be the ring of polynomials such that for all . Set and ; gluing these along their common open in the obvious way, we get an inductive system of schemes , with compatible maps . In the colimit this gives a map . This map turns out to be an isomorphism, but this is not so obvious.

**Example 1.4. **If is arbitrary and , then is NOT an ind-scheme or ind-algebraic space.

This last example is typical: for almost all , the functor 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 , and let and be as above. Then:

**i.** If is any scheme map, there is a canonical isomorphism .

**ii.** The map is (representable in schemes and) an open immersion.

**iii.** The functor is a sheaf for the fpqc topology.

**iv.** The structure map satisfies the valuative criterion of properness.

**v.** If is of finite presentation, then is limit-preserving.

**vi.** The diagonal is representable in formal algebraic spaces: for any scheme with a map , the fiber product is representable by a formal algebraic space whose underlying reduced subspace is a closed subscheme of . If is locally Noetherian then is a countably indexed directed colimit of closed subschemes of along thickenings.

**vii.** If is any proper map of good -schemes, there is a canonical map such that .

**viii.** If is any open immersion of good -schemes, there is a canonical map such that .

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 -point with associated pair , the pullback is just the open subscheme .

For iii., let be an fpqc cover, and suppose given lying in the equalizer of . One first descends the open subscheme to an open subscheme using the fact that is a quotient map, and then one descends the morphism to a map . To see that has the right properties, note that the graph pulls back to the closed immersion along the fpqc cover , so is necessarily a closed immersion.

For iv., one reduces by i. to checking that if is the spectrum of an arbitrary valuation ring with generic point and is any good -scheme, then the evident “restriction” map is a bijection. After showing that the points at infinity match up, this reduces to showing that any section spreads out to a unique point . For this, let be the scheme-theoretic image of in . By the Lemma in my previous post, the composite map is an open immersion, and we’re done.

For v., one takes an arbitrary -point of , where is a limit of affine -schemes, and then uses “standard limit nonsense” to show that everything comes by pullback from a -point for some – 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 be as in the statement. This corresponds to a pair of -points of , i.e. a pair of closed subschemes for such that the induced maps are open immersions. Let , so this is an open subscheme of . Let be the intersection of the ‘s inside , so we get natural closed immersions , and composing either of them with the inclusion realizes as a closed subscheme of the open subscheme . At this point we make the

**Definition. **Let be a scheme, and suppose given an open subscheme together with a closed subscheme . Let be the subfunctor of whose -points are given by scheme maps such that factors over the closed immersion .

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

**Lemma. **Notation as in the previous definition, the functor is a formal algebraic space whose underlying reduced subspace is a closed subscheme of , namely the reduced closed subscheme corresponding to the closed subset . If is Noetherian, is the coherent ideal sheaf corresponding to the scheme-theoretic closure of , and is any coherent ideal sheaf whose vanishing locus cuts out the closed subspace , then .

Intuitively, is the “union” inside of the locally closed subscheme and the formal completion of along the complement of .

For vii., one takes the scheme-theoretic image of along the map and then checks that the resulting closed subscheme has the right properties; in fact .

For viii., one takes the pullback of along the open immersion . 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 is an open embedding of locally compact Hausdorff spaces, then is obtained from by contracting down to the point at infinity, giving a canonical map .

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