The Adic Disk. Drew Moore

Transcription

1 The Adic Disk Drew Moore The Adic Disk D = Spa(C p T, O Cp T ) as a Set The adic disk D is the adic spectrum of the Huber pair (R, R + ) = (C p T, O Cp T ). It is a topologically and locally ringed space equipped with a valuation for each fiber of the structure sheaf. We will study its topological space. It is of course closely related to the rigid analytic disk D = B(0, ). In general, let B(c, r) (resp. B (c, r)) denote the closed (resp. open) disk centered at c with radius r. These are rigid analytic varieties. When r / p Q, we have B (c, r) = B(c, r) as sets. Also, we denote D = B (0, ). The underlying set of D is a set of valuations. For these notes, we will write valuations multiplicatively. As a set, D consists of the continuous valuations v of C p T which satisfy v(f) for all f O Cp T. For x a point of D and f C p T, we use the notation f(x) := x(f). This is merely syntactic sugar. The motivation: if X ad is the adic space associated to a rigid analytic variety X, any point x of X gives a point x of X ad in this way - i.e. x (f) = f(x). Famously, there are 5 types of points x D. We will now give a summary. Type : Let c D be any point of the classical unit disk in C p. For any f C p T, v c : f f(c) R >0 gives a continuous valuation of C p T which is bounded on O Cp T. Type points are called classical points. Types 2 and 3: Fix r R >0 with r, and c D. Set B = B(c, r). Then the map v c,r = v B : a i (T c) i max a i r i R >0 i The valuation v B takes the supremum of the classical valuations evaluated at each point of B. It is type 2 if r is in the value group p Q of C p, otherwise it is type 3.

2 Note that this valuation only depends on the ball B. Type 4: The field C p is not spherically complete. That is, there exists a nested sequence of balls B B 2 whose intersection is nonempty (in fact there exist many such sequences). Necessarily, the limit of the radii of these balls is nonzero. Two such sequences are equivalent if they are cofinal in one another. To an equivalence class of these nested sequences, one can associate a valuation v {Bi } by v {Bi }(f) = inf i v Bi (f) R >0 Type 5: The valuations of types -4 are rank, because their value groups are included in R >0. The value group for a type 5 valuation will be a subgroup of R >0 γ Z. γ is an element infinitesimally smaller than. I.e., if r < is real, then r <, but γ <. Specifically, its value group will be (/p) Q γ Z. This group is abstractly isomorphic to the additive group Q Z, and the induced order on Q Z is the lexicographical order. We define two sorts of type 5 valuations (but they are only distinct because we are studying D instead of P,ad ). First, suppose that c B and 0 < r. Let B = B (c, r) be the corresponding open unit ball. Then we define η B,+ : a i (T c) i max a i r i γ i If r / p Q, then we recover a valuation equivalent to a type 3 valuation. However if we assume r p Q, then this is an honest rank 2 valuation of C p T. Second, suppose c is in the maximal ideal of O Cp, i.e. c D (and r p Q with r < ). Set B = B(c, r) - it is a closed unit disk strictly contained in D. The valuation η B, : a i (T c) i max a i r i γ i is an element of D. If we didn t assume that B was properly contained in D, this valuation would not be bounded by on O Cp T. The proof that the above classification is complete and gives distinct valuations is done as follows. First, show that you can classify valuations on C p T by studying certain valuations on C p (T ). Then break up into cases based on the possibilities for the residue field and value group of the valuation, by comparing with those for C p. 2 The topology on D The topology on D is generated by subsets of the form { } U f,g = x f(x) g(x) 0 2

3 Type Input Notation A point c D c or v c 2 A closed ball B D of radius r O Cp v B 3 A ball B = B D of radius r / O Cp v B 4 A nested sequences {B i } of balls v {Bi } with empty intersection / 5+ An open ball B D of radius r O Cp η B,+ 5 A closed ball B D of radius r m Cp η B, Figure : A summary of the types of points in D for any f, g C p T. We will study one example in depth, the subdisks. Let { x a k } denote the open set U T a,k. As the notation implies, one should think of { x a k } as the adic disk centered at a with radius k. We will determine under what conditions each type of point belongs to { x a k }. Type. It is trivial to verify that c = v c { x a k } c B(a, k ). Types 2 and 3. Let B be a (closed) disk in D, of radius r with a center c. Then v B a = v B (T a) = max{ c a, r} which is less than or equal to if and only if B B(a, k ). Thus, v B { x a k } B B(a, k ) Type 4. By a similar argument to type 2, v {Bi } V a, k if and only if B i B(a, k ) for sufficiently large i. v {Bi } { x a k } B i B(a, k ) i 0 Type 5. Suppose B is a closed disk with radius r < in C p and a center c. Then η B,, a = η B, (T a) = max{c a, rγ } This is less than one if and only if a B and r < k (note the strict inequality). Hence, η B, { x a k } if and only if B B(a, k ). (This is analogous to the fact that η D, is not an element of D, since it senses things infinitesimally outside of D. ) η B, { x a k } B B(a, k ) B B (a, k ) 3

4 Type 5+. Suppose B is an open disk with radius r C p and a center c. Then η B,+ a = η B, (T a) = max{c a, rγ} which is less than k if and only if B B(a, k ) We summarize our results in figure 2. η B,+ { x a k } B B(a, k ) Type x x { T a k }? c or v c c B(a, k ) 2 v B B B(a, k ) 3 v B B B(a, k ) 4 v {Bi } B i B(a, k ) i 0 5+ η B,+ B B(a, k ) 5 η B, B B (a, k ) Figure 2: Conditions for when types of points belong to a subdisk. We can similarly define the open sets { x a k } D. It is equal to U f,g with f = k and g = T a. The intersection of these two sets will be denoted (suggestively) { x a = k }. In figure 9, we ve summarized the analysis analagous to what we did in figure 9. Type x x { T a k }? x { T a = k }? c or v c c / B (a, k ) c B(a, k ) \ B (a, k ) 2 v B B B (a, k ) B = B(a, k ) 3 v B B B (a, k ) Never 4 v {Bi } B i B (a, k ) i Never 5+ η B,+ B B(a, k ) Never 5 η B, B B (a, k ) Never Figure 3: Conditions for when types of points belong to certain open sets The prototypical example of an unadmissable cover of the p-adic unit disk is { x = } r<{ x r} In the world of adic spaces, we see immediately why this cover is unadmissable: it doesn t actually cover D! Yes, all classical points are contained in the above union. However, the type 5+ point η corresponding to the open disk B (0, ) is not in this union. Indeed, η = η(t ) = γ, and γ is greater than every r <, but is also not equal to. 4

5 3 Visualizing D Understanding the topological space of D requires knowledge of the lattice of open and closed balls in D. Namely: for any closed ball B D of radius r p Q, the open balls of radius r contained in B are in natural bijection with A (F p ). Namely, fix a center c of B. For any other a B, we have (a c)/r O Cp. Two points a 0, a B are centers of the same open ball of radius r if and only if (a 0 c)/r = (a c)/r mod m, the maximal ideal of O Cp. All types of points, except for type 2, are closed points in D. Suppose B is a closed ball with radius r p Q. The closure {v B } consists of v B along with certain type 5 points: if B D, then η B, {v B } and for any open B D with radius r (the same as B), η B,+ {v B }. Hence, the closure of v B is in bijection with P (F p ) if B D (the extra point corresponding to the type 5 point), while the closure of v D is in bijection with A (F p ). In figure 4 we ve (partially) drawn D. p Q p Q p Q p Type Type 2 Type 3 Type 4 Type 5 Type 2, 3, or 5 p Q p (n )/n 0 0 p Figure 4: The adic disk. 4 The Adic Disk and Formal Models If X is any formal model of the rigid analytic disk, then there exists a continuous specialization map sp: D X 5

6 This is a continuous map of topological spaces. It actually extends to a map of ringed spaces (you have to be careful about which sheaf of rings you take on D). p (n )/n 0 p 0 Generic point Figure 5: The specialization map from D to X = Spf O Cp T In figures 5 and 6, the maps from D to some formal schemes. are depicted. The formal scheme in the first example is X = Spf O Cp T. The second example is the admissable blowup X of X centered at the open ideal (p, T ) of O Cp T. 6

7 0 p Figure 6: The specialization map from D to X, the blowup at the origin in the special fiber of X (from figure 5). References [] B. Conrad. Informal notes for Seminaire Scholze url: http : / / math. stanford.edu/~conrad/perfseminar/. [2] R. Huber. A generalization of formal schemes and rigid analytic varieties. In: Math. Z (994), pp issn: [3] R. Huber. Continuous valuations. In: Math. Z (993), pp issn: [4] F. Martin. Adic Spaces. url: regensburg.de/~maf55605/ intro_adic.pdf. 7

8 [5] Peter Scholze. Perfectoid spaces. In: Publ. Math. Inst. Hautes Études Sci. 6 (202), pp issn: