Perfectoid spaces 1
Daniel Miller
February 14, 2014
For a field k, I will write Gk for the absolute Galois group Gal(ksep /k), where ksep ⊂ k¯ is the separable
closure of k. The group Gk carries a natural topology under which it is compact and totally disconnected.
One way to realize this topology is the isomorphism
Gk '
Gal(K/k ).
lim
←−
K/k Galois
[K:k]<∞
Each group in the projective limit is given the discrete topology, and we give Gk the inverse limit topology.
1
The tilting correspondence for fields
Consider the following fields:
k1 = Q p ( p p
−∞
) = Qp(
√
pn
p : n > 1)
k2 = F p ((t)) = Frac(F p JtK).
A motivating result is the following theorem of Fontaine and Wintenberger:
Theorem 1. There is a (canonical) isomorphism of topological groups Gk1 ' Gk2 .
Proof. Start by defining the following “big” fields:
k = kb1 = completion of Q p (ζ p∞ ) under the p-adic norm
[
n
perf = completion of
k[ = kd
F p ((t1/p )) under the t-adic norm.
n>1
Note that Gk1 ' Gk and Gk2 ' Gk[ . Both of these facts are easy – basically consequences of Krasner’s lemma
and the fact that purely inseparable extensions don’t change the absolute Galois group. The proof consists
of constructing a functor
(−)[ : {finite extensions of k} → {finite extensions of k[ }
and then proving this functor is an equivalence of categories. We’ll content ourselves with constructing
(−)[ .
Let K be a finite extension of k. Since k is henselian, the valuation (or, equivalently, the norm) on k
extends uniquely to one on K. So we can write
K ◦ = O K = { x ∈ K : | x | 6 1} = { x ∈ K : v ( x ) > 0}.
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This is a (non-noetherian) local ring, and p is an element of its maximal ideal. The quotient K ◦ /p is a ring
of characteristic p, so the Frobenius Φ : K ◦ → K ◦ , x 7→ x p , is a (surjective) ring homomorphism. Define
(
)
K [◦ = OK[ = lim K ◦ /p =
←−
( xn ) ∈
Φ
∏ OK /p : xn+1 = xn
p
.
n>0
It’s obvious that K [◦ is a topological ring (give it the inverse limit topology, where each K ◦ /p is discrete).
To establish some more properties of K [◦ , consider the following map
n
( xn )n 7→ lim f
xn p .
(−)] : K [ → K
n→∞
Here f
xn ∈ K ◦ is an arbitrary lift of xn ∈ K ◦ /p. It’s not immediately obvious that the limit converges, or
that it is independent of the choice of lifts. But consider the following fact: if x ≡ y mod pn , then x p ≡ y p
mod pn+1 . Indeed, we can write x = y + pn · t, whence
p p −1 n
p
p
p
x =y +
y
( p t) + · · · +
y( pn t) p−1 + ( pt) p
1
1
p −1
p
n +1 1 p
y
t+··· .
=y +p
p 1
xn , f
So give ( xn ) ∈ K [◦ and different lifts f
xn0 , we have
n
p
f
xn p ≡ f
xn0
n
mod pn+1 .
n
It easily follows that f
xn p is a Cauchy sequence with limit independent of the lifts. It turns out that (−)] :
K [◦ → K ◦ induces an isomorphism of topological monoids K [ ' limΦ K. Thus K [◦ is a domain, and we can
←−
put K [ = Frac(K [◦ ).
−∞
Something I didn’t show is that for k = Q p\
( p p ), one has k[ = F p ((t p
p
−∞
)). Simply notice that
p
O k = Z p [ X n : X n + 1 = X n , X1 = p ] ∧
h −∞ i
p
p
Ok /p = F p [ Xn : Xn+1 = Xn , X1 = 0] = F p X p
/ ( X p ).
The rest is simple “book-keeping.”
Now we can state a much more general version of the Fontaine-Wintenberger theorem. A perfectoid
field is a complete valued field k with residue characteristic p > 0, whose valuation is rank-one and nondiscrete, such that Frobenius is surjective on k◦ /p. For such a field, one can define k[ exactly as above, and
the general theorem is that the “tilting operation” K 7→ K [ induces an isomorphism of topological groups
Gk ' Gk[ .
2
The general tilting correspondence
Let k be a perfectoid field. Scholze was able to generalize the isomorphism Gk ' Gk[ to a relative context.
Let A be a Banach k-algebra. Just as with k, write A◦ = { a ∈ A : | a| 6 1} for the subring of elements
a ∈ A such that { an : n > 1} is bounded. We say that A is perfectoid if A◦ is open, and if Frobenius
Φ : A◦ /p → A◦ /p is surjective.
A typical example is as follows. Start with the Tate algebra kh T i consisting of power series ∑ ai T i such
that ai → 0. The norm | ∑ ai T i | = max{| ai |} makes this a Banach k-algebra. If we instead start with
2
−∞
−∞
k[ T p ], we can complete with respect to the same norm, and we get a ring A = kh T p i. Since A◦ /p =
−∞
(k◦ /p)h T p i, we see that A is perfectoid.
If k is a perfectoid field, write k-Perf for the category of perfectoid algebras over k. If A is a perfectoid
k-algebra, then write A[ for the ring
A[ = lim A◦ /p.
←−
Φ
(−)[
k[ -Perf
An important theorem is that
: k-Perf →
is an equivalence of categories. One proves this
via some “almost mathematics.”
Start with the category k◦ -Mod of k◦ -modules. Let m ⊂ k◦ be the unique maximal ideal. Since k is
perfectoid, its valuation is non-discrete, so m2 = m. Let Σ ⊂ k◦ -Mod be the subcategory of k◦ -modules
killed by m. Since m2 = m, the subcategory Σ is full, in the sense that whenever
0 → M0 → M → M00 → 0
is exact and M0 , M00 ∈ Σ, we also have M ∈ Σ. For, ordinarily mM0 = mM00 = 0 would only imply
m2 M = 0, but m2 = m. We define the category of almost k◦ -modules, or k◦ a -modules, to be the Serre quotient
(k◦ -Mod)/Σ. The basic idea is that one formally inverts morphisms f : M → N with kernel and cokernel
killed by m. Fortunately, I don’t have to go into details because we can describe hom-sets in k◦ a -Mod
explicitly. Write (−) a : k◦ -Mod → k◦ a -Mod for the quotient functor. One has
homk◦a ( M a , N a ) = homk◦ (m ⊗ M, N ).
There is actually a triple of functors
(−)!
|
k◦ -Mod
a
(−) a
/ k◦ a -Mod
(−)∗
where
( M a )! = M ⊗ m
( M a )∗ = homk◦ (m, M).
We have adjunctions (−)! a (−) a a (−)∗ . This is similar to the situation when i : U ⊂ X is an open
subscheme. In that situation, there is a triple of functors
i!
{
Qcoh( X )
c
i∗
/ Qcoh(U )
i∗
with i! a i∗ a i∗ . So we should think of k◦ a -Mod as being the category of quasicoherent sheaves on some
subscheme of Spec(k◦ ) that doesn’t actually exist.
The category k◦ a -Mod inherits the structure of a tensor category from k◦ -Mod, i.e. M a ⊗ N a = ( M ⊗ N ) a .
So it makes sense to talk about “algebra objects” in k◦ a -Mod, which we call k◦ a -algebras. It turns out that
we can use the triple of functors (−) a , (−)∗ , (−)! to show that any k◦ a -algebra R is of the form A a , for some
“honest” k◦ -algebra A. Choose some π ∈ k◦ with |π | = 1/p. Even if π 1/p may exist as an element in k◦ ,
the ideal (π 1/p ) makes sense, so we can talk about the quotient M/π 1/p , for M a k◦ -module.
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We say that a k◦ a -algebra A is perfectoid if A is π-adically complete, and if Frobenius induces an isomorphism A/π 1/p → A/π. We can even talk about k◦ a /π-modules in k◦ a -Mod, and we say that a k◦ a /π∼
algebra A is perfectoid if it is flat, and if Frobenius induces an isomorphism A/π 1/p −
→ A. One of the main
theorems is that the obvious functors induce equivalences of categories:
k-Perf
∼
/ k◦ a -Perf
∼
/ (k◦ a /π )-Perf.
Note that k◦ /π = k[◦ /π. Since k[ is also perfectoid, we have equivalences of categories
k[ -Perf
∼
/ k[◦ a -Perf
∼
/ (k[◦ a /π )-Perf.
It follows that there is a natural equivalence of categories k-Perf ' k[ -Perf. This equivalence respects e´ tale
extensions, so we obtain the original result Gk ' Gk[ .
∼
Even better, if A is a perfectoid k-algebra, then the tilting equivalence k-Perf −
→ k[ -Perf respects finite
[ .
e´ tale A-algebras, so we get an equivalence Af´et ' Af´
et
I don’t want to go into more details right now, but one can “glue” these equivalences to obtain a tilting
equivalence between “perfectoid spaces over k,” and “perfectoid spaces over k[ .” This equivalence respects
e´ tale sites, i.e. Xe´ t ' Xe[´ t .
3
The Weight-Monodromy conjecture
Let k be a local field, and X a proper smooth variety over k. :et Fq be the residue field of k. Let ` be a prime
not dividing q, and let V = Hi ( Xks , Q` ). This is a finite-dimensional Q` -vector space on which Gk acts
continuously. Recall that Gk contains subgroups I ⊃ P. The group I (inertia) is the kernel of the reduction
b The group P is the p-Sylow subgroup of I.
map Gk → GFq ' Z.
Define a character t` : I → Z×
` by
n
n
σ (π 1/` ) = t(σ )π 1/` .
It turns out (Grothendieck `-adic monodromy theorem) that there is an open subgroup J ⊂ I and a nilpotent
operator N on V such that for all σ ∈ J, we have ρ(σ ) = exp( N · t` (σ )). Some linear algebra tells us that
there is a canonical filtration Fil• V such that N Filr V ⊂ Filr−2 V and such that N r : grr V → gr−r V is an
isomorphism for all r > 0. One has
Filr V =
∑
ker( N s+1 ) ∩ im( N t ).
s − t =r
Conjecture (Deligne). For any j ∈ Z, the eigenvalues of Frobenius on gr j V are q-Weil of weight i + j.
Recall that x ∈ Q` is q-Weil of weight w if for all isomorphisms Q` ' C, the image of w in C is algebraic
and has absolute value qw/2 .
Theorem 2 (Deligne). Suppose k has characteristic p. Then the Weight-Monodromy conjecture holds for X.
Theorem 3 (Scholze). Suppose X is a projective smooth toric variety over a local field k. Then the WeightMonodromy conjecture is true for X.
Hopefully, I will describe the details of Scholze’s proof next time. For now, the basic idea is that he first
∞
base-changed to k(π 1/p )∧ , a perfectoid field. Frobenius and the monodromy operator do not change, so
it suffices to prove the theorem after the base-change. Scholze then uses the tilting functor to “move the
problem to characteristic p,” where it is known.
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