Institute of Mathematics

Master thesis

The direct summand conjecture

Jon Eugster

June, 2019

supervised by Prof. Dr. Joseph Ayoub

Contents

1 Preliminaries 3

1.1 Homological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Derived category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Distinguished triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.4 Injective and projective objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.5 Milnor’s exact sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Adic spaces 10

2.1 Valuations and non-archimedean fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Tate rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Adic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Almost ring theory 14

3.1 Almost mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Concrete construction of almost modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Almost algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Almost flat, almost projective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5 Etale and almost etale morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Perfectoid spaces 19

4.1 Perfection and tilting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Perfectoid fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Perfectoid algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4 Perfectoid spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.5 Almost purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Pro systems and almost pro systems 25

5.1 Almost-pro-zero modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Quantitative Riemann’s Hebbarkeitssatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 The direct summand conjecture 28

6.1 The DSC in equal characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.2 Almost faithfully flat extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6.3 Additional lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.4 The DSC in mixed characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.5 A derived version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Abstract

The direct summand conjecture in homological algebra says that any finite extension of a regular ringA contains A as a direct summand. It has been proven by Andre in [1] in 2018 and the proof has beensimplified by Bhatt in [4] avoiding a deep result called Abhyankar’s lemma. This Master’s thesis developsthe theoretic basis required for understanding the direct summand conjecture and Bhatt’s proof thereof.

Acknowledgements

I would like to thank my supervisor Joseph Ayoub for suggesting this topic, for patiently answering myquestions and supporting me when needed. Moreover, I would like to thank Prof. Dr. Bhargav Bhattfor a very kind and quick response to questions around corollary 4.5.4 and proposition 6.2.2.

Introduction

The direct summand conjecture gives a statement about finite extensions of regular rings containingthem as direct summands.

Definition 0.0.1 (split). Let A be an abelian category, 0→ Ai↪−→ B

j−→→ Q→ 0 a short exact sequencein A. It is called split if one of the equivalent properties hold:

(I) ∃s : B → A such that s ◦ i = idA.

(II) ∃t : Q→ B such that j ◦ t = idQ.

(III) B ∼= A⊕Q.

Remark. For any injective map A ↪−→ B, there is a canonical choice Q := B/i(A) to complete the shortexact sequence, therefore it makes sense to speak of the injection A ↪−→ B being split. It is also equivalentto say that A is a direct summand of B, due to the 3rd property.

Here is the statement of the direct summand conjecture.

Theorem (direct summand conjecture). Let A be regular, Ai↪−→ B a finite extension of rings. Then i is

split as a map of A-modules.

The conjecture has been put forth by Hochster, who also proved the equal characteristics case in [9] in1973. Recall that A has equal characteristics if it contains Q or Fp. The mixed characteristics case,which means the case where A contains Z but not Q, remained open for a long time and has finally beenproven by Andre in [1] in 2018 using perfectoid spaces, which have been introduced by Scholze in [17] in2012. Later, Bhatt simplified the proof in [4] in 2018. We follow the paper of Bhatt and the main goalof this work is to prove the direct summand conjecture assuming the theory of perfectoid spaces. Wefinish this exposure by giving a dervied variant, which was also proved by Bhatt in [4].

Theorem (derived direct summand conjecture). Let A be a regular ring, let f : X → Spec(A) be aproper surjective map. Then the map A→ RΓ(X,OX) splits in the derived category D(A).

First we collect some well-known preliminary results in chapter 1, then we introduce adic spaces inchapter 2 and almost mathematics in chapter 3. Both concepts are needed to introduce perfectoid spacesin chapter 4, and the pro-systems introduced in chapter 5 are needed to formulate the proof in a waythat still holds when passing to the derived setting. Finally, we prove the direct summand conjecture inequal and mixed characteristics in chapter 6.

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1. Preliminaries

1.1 Homological algebra

Good introductions to homological algebra are for example the book of Gelfand and Manin [7, section III]or Weibel [19]. We recommend in particular the former and will not give a rigorous treatment here.Instead we just recall some of the most fundamental facts we will need later.

1.1.1 Derived category

We will use derived categories and derived functors. The basic problem we observe is that most functorsdo not preserve exact sequences. For example if we take the functor HomA(M,−) for A a ring and Man A-module, then it is only left exact, meaning for a short exact sequences of A-modules

0→ N1 → N2 → N3 → 0

we only get the exact sequence

0→ HomA(M,N1)→ HomA(M,N2)→ HomA(M,N3)

Derived functors were invented to address this problem.

Definition 1.1.1.1 (quasi-isomorphism). A morphism q : A• → B• of cochain complexes is a quasi-isomorphism if the induced morphisms Hn(q) : Hn(A•)→ Hn(B•) in cohomology are isomorphisms forall n ∈ Z.

The idea is to invert all quasi-isomorphisms. An A-module becomes then isomorphic to all its resolutions.

Concretely, we can view an A-module M naturally as cochain complex

C• : · · · → 0→ 0→M → 0→ 0→ . . .

with the only non-trivial group in degree 0. This cochain complex has the following cohomology.

Hi(C•) =

{M, i = 0

0, i 6= 0

If we take any exact sequence0→M → F 0 → F 1 → F 2 → . . .

then the cochain complexC ′• : . . .→ 0→ F 0 → F 1 → F 2 → . . .

has isomorphic cohomology, therefore C• and C ′• become isomorphic if we invert quasi-isomorphisms.

Definition 1.1.1.2 (derived category). Let A be an abelian category, Kom(A) the category of cochaincomplexes on A. Let Q be the set of quasi-isomorphisms in A. The derived category D(A) is the categoryKom(A)[Q−1].

We denote

D+(A) := {C• ∈ D(A) | ∃i0 ∀i < i0 : Hi(C•) = 0}D−(A) := {C• ∈ D(A) | ∃i0 ∀i > i0 : Hi(C•) = 0}Db(A) := D+(A) ∩D−(A)

Remark 1.1.1.3. For all the details and the category-theoretical correct definition of D(A), we recom-mend to look at [7, chapter III]. It turns out that D(A) is an additive category, but it is not abelian.

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1.1.2 Distinguished triangles

As D(A) is not abelian, we can not define short exact sequences in D(A) directly. Instead, we definedistinguished triangles A• → B• → C• → A[1]• which fulfil the analogous function of short exactsequences.

Definition 1.1.2.1 (shifted complex). Given a cochain complex C• and n ∈ Z, we define the shiftedcomplex C[n]• by (C[n])i := Ci+n and differentials diC[n] := di+nC for all i ∈ Z.

Definition 1.1.2.2 (triangle). A triangle in the category A is a sequence of maps of the form

A•a−→ B•

b−→ C•c−→ A[1]• and a morphism of triangles consists of three maps (fA, fB , fC) such that the

following diagram commutes.

A• B• C• A[1]•

A′• B′• C ′• A′[1]•

a

fA

b

fB

c

fC fA[1]

a′ b′ c′

Such a map is an isomorphism if every map fA, fB , fC is an isomorphism.

Definition 1.1.2.3 (cylinder and cone). Let f : A• → B• be a morphism of chain complexes. Thecylinder of f is the complex

Cyl(f)i := Ai ⊕A[1]i ⊕Bi, dCyl(f)(a1, a2, b) := (dA(a1)− a2, −dA(a2), f(a2) + dB(b))

The cone of f is the complex

Cone(f)i := A[1]i ⊕Bi, dCone(f)(a2, b) := (−dA(a2), f(a2) + dB(b))

Remark 1.1.2.4. These notions are inspired from topology. For an inclusion f : X ↪−→ Y we can definethe mapping cylinder Cyl(f) := [0, 1] ×X t Y/(0, x) ∼ f(x) which has the same cohomology as Y andwe have an inclusion x 7→ (x, 1). Moreover, we can take to quotient by identifying (x, 1) ∼ (x′, 1) andwe get the mapping cone.

Cyl(f) Cone(f)

f

X

Y Y

Cyl and Cone defined above behave similarly, meaning that Cyl(f) = A ⊕ A[1] ⊕ B has isomorphiccohomology as the cochain complex B and there is a short exact sequence

0→ A→ Cyl(f)→ Cone(f)→ 0

Concretely the isomorphism is induces by the maps α : B → Cyl(B), b 7→ (0, 0, b) and β : Cyl(B)→ B,(a1, a2, b) 7→ f(a1) + b. Clearly βα = idB , and one simply checks that αβ = idCyl(B)−(dh + hd) forh : Cyl(B)i → Cyl(B)i−1, (a1, a2, b) 7→ (0, a1, 0). See also [7, III.3.3] for more details.

Definition 1.1.2.5 (distinguished triangle). A triangle is distinguished if it is isomorphic to a triangleof the form

A→ Cyl(f)→ Cone(f)→ A[1]

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Remark 1.1.2.6 (triangulated category). Distinguished triangles satisfy the following axioms:

1. Aid−→ A→ 0→ A[1] is distinguished.

2. For all f : A→ B there is a distinguished triangle Af−→ B → C → A[1].

3. A triangle Af−→ B → C → A[1] is distinguished if and only if its rotated triangle

B → C → A[1]−f [1]−−−→ B[1] is distinguished.

4. Given two distinguished triangles and maps α an β as in the diagram below, there exists a map γmaking the diagram commute.

A B C A[1]

A′ B′ C ′ A′[1]

α β ∃γ α[1]

5. (octahedral axiom) Given distinguished triangles

A→ B → C ′ → A[1]

B → C → A′ → B[1]

A→ C → B′ → A[1]

there is a distinguished triangle C ′ → B′ → A′ → C ′[1] such that we get the following diagramswith coinciding brim,

A′ C A′ C

B B′

C ′ A C ′ A

[1]

[1] [1]

[1] [1]

∗

∗

∗

∗

where the triangles with a and the squares coming from B → C → B′ and B → C ′ → B′

respectively B′ → A′ → B and B′ → A→ B commute. Moreover, the triangles marked with ∗ are

required to be distinguished. Arrows like A′[1]−→ B denote arrows A′ → B[1].

1.1.3 Derived functors

Definition 1.1.3.1 (derived functor). Let F : A → B be an additive, left exact functor. Denoteby QA : Kom(A) → D(A) the quotient functor and similarly for QB. Further, denote by K+F :Kom+(A)→ Kom+(B) the induced functor, where we apply F component-wise.

The right derived functor of F is defined to be the right Kan-extension of QB ◦K+F along QA.

D+(A)

Kom+(A) D+(B)

RFQA

QB◦K+F

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This means concretely that it comprises a functor RF : D+(A)→ D+(B) and a natural transformationεF : QB ◦K+F → RF ◦QA, such that they satisfy the following universal property:

For any functor S : D+(A) → D+(B) and any natural transformation α : QB ◦K+F → S ◦ QA thereexists a unique η : RF → S such that α = εF ◦ (η ◦ QA). This can be summarised in the followingcommuting diagram.

D+(A)

Kom+(A) D+(B)

RF

∀S

QA

QB◦K+F

RF◦QAS◦QA

∃!η

εF

η◦QA

∀α

For a right-exact functor G, we define dually the left-derived functor LG : D−(A)→ D−(B) as left Kanextension.

Definition 1.1.3.2 (adapted class of objects). Let F be a left-exact functor. We say a subset R ⊂ ObAis adapted to F if the following three properties are satisfied:

• It is stable under finite direct sums.

• F maps acyclic objects in Kom+(R) to acyclic ones.

• For any X ∈ ObA there is a R ∈ R with X ↪−→ R.

Remark 1.1.3.3. Dually, for a right-exact functor, we require in the last point that for any X ∈ ObAthere is a R ∈ R with R −→→ X. Moreover, if this last condition is satisfied, we say that R is sufficientlylarge.

Theorem 1.1.3.4. Let F be a left-exact functor and assume it admits an adapted class of objects R.Then RF (M) can be computed concretely by applying F to a complex of adapted objects which is quasi-isomorphic to M .

Example 1.1.3.5. For any left-exact functor such as HomA(M,−), a possible adapted class would bethe class of injective objects (see 1.1.4), and we say that the category has enough injectives if this adaptedclass is sufficiently large.

So we can calculate RHomA(M,N) by taking a injective resolution F • of N and compute

RHomA(M,N)i = HomA(M,F i)

Remark 1.1.3.6. Derived functors are triangulated, i.e. they map a distinguished triangle to a distin-guished triangle.

1.1.4 Injective and projective objects

Injective and projective objects are dual concepts, so we will only introduce one of the notions.

Projective objects are important because the can be used to calculate left-derived functors. Moreover,projective objects help us finding direct summands, as illustrated in 1.1.4.2.

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Definition 1.1.4.1 (projective). Let A be a ring and P an A-module. The A-module P is projectiveif for any pair of maps of A-modules f : P → M and g : N → M with g surjective, there exists anh : P → N such that g ◦ h = f .

N

P M

g

f

∃h

Proposition 1.1.4.2. Let A be a ring, P an A-module. The following are equivalent:

(I) P is projective.

(II) Every short exact sequence 0→M → N → P → 0 of A-modules splits.

(III) P is a direct summand of a free A-module Q.

(IV) HomA(P,−) is an exact functor.

(V) There exist sets {pi ∈ P} and {ϕi ∈ P∨ := HomA(P,A)} such that every x ∈ P can be written asx =

∑i∈I ϕi(x)pi with only finitely many ϕi(x) 6= 0.

Proof. ”(I)⇒ (II)” For a short exact sequence 0→M → Ng−→→ P → 0 we have g surjective, and together

with idP : P → P the definition of projectivity gives us an h : P → N such that g ◦ h = idP , thus thesequence is split.

P

0 M N P 0

idP∃hg

”(II)⇒ (III)”: We can take a free module Q with s : Q −→→ P , thus we have a short exact sequence0→ ker(s)→ Q→ P → 0. This sequence splits by (II), thus P is a direct summand of Q.

”(III)⇒ (V)”: We can write every element as such a sum in the free module Q and then we get the resultin P by just applying the projection. π : Q→ P .

”(V)⇒ (I)”: P being projective means that for all A-modules M , N and f : P → M , g : N −→→ M wefind a h : P → N making the following diagram commute.

N

P M

g

f

h

Let x ∈ P . We can write x =∑i=1...n ϕi(x)pi and f(x) =

∑i=1...n ϕi(x)f(pi). Now as g is surjective,

we have for every i an element ni ∈ N with g(ni) = f(pi), so we can define h(x) :=∑i ϕi(x) · ni.

”(I)⇔ (IV)”: HomA(P,−) is always left-exact, so asking it to be exact is equivalent to asking that forevery g : N −→→ M surjective the map HomA(P,N) −→→ HomA(P,M) is surjective, which means that forevery f : P →M there is an h : P → N with g ◦ h = f , which is the definition of projectivity.

Definition 1.1.4.3 (projective dimension). The projective dimension of a module M is the minimallength n of a finite projective resolution . . .→ 0→ Pn → . . .→ P1 → P0 →M → 0, or ∞ if there is nosuch resolution.

Injective and projective objects are especially important because of the following fact, which is presentedin [7, III.6.12].

Theorem 1.1.4.4. Projective objects form an adapted class of objects for any right-exact functor.

Dually, injective objects form an adapted class of objects for any left-exact functor.

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1.1.5 Milnor’s exact sequence

Later, we will need the so-called Milnor exact sequence. This is explained in [19, 3.5.8]. We first recallthe following definition.

Definition 1.1.5.1. (Mittag-Leffler condition) A tower . . .p32−→ G2

p21−→ G1p10−→ G0 satisfies the Mittag-

Leffler condition if∀n ∈ N ∃m0 > n ∀m > m0 : pmn (Gm) = pm0

n (Gm0) ⊂ Gn

Theorem 1.1.5.2 (Milnor’s exact sequence). Let . . .→ C2 → C1 → C0 be a tower of chain complexes ofabelian groups, which satisfies degree-wise the Mittag-Leffler condition. Then we have the exact sequence

0→ lim1

nHq+1(Cn)→ Hq(lim

nCn)→ lim

nHq(Cn)→ 0

Remark 1.1.5.3. In particular, this can be used to construct an isomorphism of the latter two byshowing that the lim1 is zero. This is also what we use it for later, just in the setup of almost mathematics.

1.2 Witt vectors

Witt vectors can be seen as a generalisation of the formation of the p-adic integers Zp from the finitefield Fp = Z/(p). One of the big advantages of Witt vectors is that using the ghost components, one getsclosed forms for addition and multiplication, making calculations easier.

For example, an intrinsic notation for p-adic integers over the finite field Fp would be the power seriesnotation. Concretely we may take as an example p = 3 and x = 46. We would have the notation

(1, 2, 0, 1)3 := 1 · 33 + 2 · 32 + 0 · 31 + 1 · 30 = 46

But using this notation, the operations + and · are not particularly easy to calculate, especially whenwe work with numbers of infinite length like 2−1, which is represented as (. . . , 1, 1, 1, 2)3.

But there is a much nicer presentation of p-adic integers which allows us to calculate multiplicationcomponent-wise. This is the representation as lim found in every algebra course. For example, we havein this representation

46 = (1, 1, 19, 46, 46, . . . )

2 = (2, 2, 2, 2, 2, 2, 2, . . . )

2−1 = (2, 5, 14, 41, 122, . . . ) ∈ Z3 = limn

Z/(3n)

While calculating 2 · 2−1 in the power series presentation is long and tedious, it can be done component-wise in this second representation.

The latter representation is simultaneously an example of Witt vectors, which have been originallyintroduced in [20] and we follow the elaborations of [16] for this short introduction.

Definition 1.2.1 (divisor stable set). A set P ⊂ N is divisor stable if it is not empty and for all d ∈ Nand all n ∈ P , d | n implies that d ∈ P .

Definition 1.2.2 (Witt polynomial, Witt vectors, ghost component). For any n ∈ N, we define the Wittpolynomial

wn :=∑d|n

d · (Xd)nd

and for P ⊂ N a divisor-stable set and A any ring, we define the set of Witt vectors as WP (A) :=∏n∈P A.

Therefore the evaluation of wn can be viewed as a map wn : WP (A) → A. For any Witt vectorx ∈WP (A) we call wn(x) the nth ghost component of x. Moreover, all those maps together define a mapw∗ : WP (A)→ AP .

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Proposition 1.2.3. There is a unique covariant functor WP of algebras with

(I) WP (A) = AP as sets.

(II) for any ring homomorphism f : A→ B we have WP (f)((an)n∈P ) = (f(an))n∈P .

(III) All wn : WP (A)→ A are ring homomorphisms.

(IV) (1, 0, 0, . . . ) is the unit element in WP (A) and (0, 0, 0, . . . ) is the zero element.

For the proof we refer to [16].

Remark 1.2.4. One can check that if no element of P is a zero-divisor, then the map w∗ is an injectionand if every element in P has an inverse in A, then it is a bijection.

If w∗ is an injection, then we can represent every element x ∈WP (A) by its ghost componentsw∗(x) = (wn(x))n∈P , and multiplication and addition can be computed component wise.

Example 1.2.5. Let P := {1, p, p2, p3, . . . } and A := Z/(p). Then WP (A) = Zp are the p-adic integers.

Example 1.2.6. Previously, we looked at the number

x := (1, 2, 0, 1)3 = 1 · 33 + 2 · 32 + 0 · 31 + 1 · 30 = 46

This number’s representation with ghost components looks like

(w0(x), w1(x), . . . ) = (1, 1, 19, 46, 46, 46, . . . ) ∈ Z3

which is the usual notation for p-adic numbers.

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2. Adic spaces

2.1 Valuations and non-archimedean fields

Definition 2.1.1 (valuation ring). Let A be a ring with one. A valuation on A is a map | · | : A→ Γ∪{0},with (Γ, ·) being a totally ordered abelian group, such that

• |xy| = |x| · |y|

• |x+ y| ≤ max(|x|, |y|)

• |0| = 0, |1| = 1Γ

A valuation is discrete if Γ ' Z.

Remark 2.1.2. Other sources write valuations additively. In this case we have a valuation| · | : A→ Γ ∪ {∞} with

• |xy| = |x|+ |y|

• |x+ y| ≥ min(|x|, |y|)

• |0| =∞, |1| = 0Γ

The standard example would be the valuation on Z(p), which could be either defined as ν(u · pn) := p−n

(multiplicative) or ν′(u · pn) := n (additive).

For Γ := R>0, the multiplicative definition given above would usually be called semi-norm and a valuationwould be a constant multiple of the map x 7→ − log|x|, but it seems that most literature on perfectoidspaces uses this notion of multiplicative valuations, therefore it is also used in this work. Probably oneof the most important sources being Huber’s work on adic spaces [12].

Definition 2.1.3 (rank and height). Let | · | : A→ Γ ∪ {∞} be a valuation and assume Γ is generatedby the image of | · | (one can always make Γ smaller for this). The rank of a valuation is the Q-rank ofΓ, that is the dimension of Γ⊗Q as vector space.

The height of the valuation is the Krull dimension of its valuation ring {a ∈ Frac(A) : |a| ≤ 1}.

Definition 2.1.4 (equivalence of valuations). Let A be a ring and |·| : A→ Γ ∪ {0}, |·|′ : A→ Γ′ ∪ {0}be two valuations on A. |·| and |·|′ are equivalent if one of the following equivalent conditions is satisfied:

(I) There is an isomorphism of ordered groups α : Γ∼=−→ Γ′ with |·|′ = α ◦ |·|

(II) The support supp(|·|) = supp(|·|′) and valuation rings{a ∈ Frac(A) : |a| ≤ 1} = {a ∈ Frac(A) : |a|′ ≤ 1} agree.

(III) ∀a, b ∈ A : |a| ≤ |b| ⇐⇒ |a|′ ≤ |b|′.

Definition 2.1.5 (NA field). A non-archimedean field (NA field) is a field K with a non-trivial valuation|·| : K → R≥0 of height 1.

The subset K◦ := {x ∈ K : |x| ≤ 1} is the called valuation ring of K. This is a local ring with maximalideal K◦◦ := {x ∈ K : |x| < 1}.The field k := K◦/K◦◦ is called the residue field of K. Any $ ∈ K◦◦ \ {0} is called pseudo-uniformiser.

Remark 2.1.6. Every NA field is naturally a topological field with topology induced by the valuation.Moreover, each element x ∈ K◦ can be represented as x = u ·$, u ∈ (K◦)×, $ ∈ K◦◦.

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Example 2.1.7. Let K = Qp. Every element is of the form x = u · pn with |x| = p−n for some n ∈ Z.Moreover, we have K◦ = Zp, K◦◦ = mp := (p), k = Z/(p).

Next we recall the definition of Banach algebras, which are algebras over a NA field that are completewith respect to the valuation.

Definition 2.1.8 (Banach algebra). Let K a NA field. A Banach K-algebra is a pair (A, |·|) with Abeing a K-algebra and |·| : A→ R>0 with the following properties:

(I) (extends) |·| extends the norm on K.

(II) (norm) |f | = 0 ⇐⇒ f = 0

(III) (sub-multiplicativity) |fg| ≤ |f | · |g|

(IV) (NA property) |f + g| ≤ max(|f |, |g|)

(V) (complete) A is complete for the metric d(f, g) := |f − g|

2.2 Tate rings

Definition 2.2.1 (Tate ring).

(I) A Tate ring is a topological ring A where there exists a subring A0 ⊂ A with an element$ ∈ A0∩A×such that {$nA0 | n ∈ N} forms a basis of open neighbourhoods of 0.We call A0 a ring of definition, $ a pseudo-uniformiser and (A0, $) a couple of definition.

(II) A subset M ⊂ A is bounded if there is a n ∈ Z with M ⊂ $nA0.

(III) a ∈ A is power-bounded if {an}n∈N ⊂ A is bounded. A◦ ⊂ A is the subring of power-boundedelements.

(IV) An element t ∈ A◦ is topologically nilpotent if tnn→∞−−−−→ 0. We denote with A◦◦ the ideal of

topologically nilpotent elements.

Remark 2.2.2. We list some properties of Tate rings that are not really needed later but help under-standing them. All of them are discussed in [3, chapter 7]

(I) For a couple of definition, we have A = A0[ 1$ ].

(II) A◦ is open and integrally closed in A. A◦ contains all possible rings of definition and it is the totalintegral closure in A of any ring of definition A0.

(III) For an arbitrary commutative ring A with t ∈ A \ {0}, we get a Tate ring A[ 1t ] with couple of

definition (A/{a ∈ A | at = 0}, t).

(IV) We can either take algebraic completion A0 = lim←−nA0/$n and A := A0[ 1

$ ] with {$nA0} being

a basis of open neighbourhoods of 0 ∈ A, or we can take topological completion with respect toCauchy sequences. The two coincide.

Speaking of completion, we use the following notation for completions of polynomial algebras:

Notation 2.2.3. Let A be any ring, A its completion. Consider a polynomial algebra A[T1, . . . , Tn] overA. We denote with

A 〈T1, . . . , Tn〉 :=

∑i1...in≥0

xi1...inTi11 · · ·T inn

∣∣∣∣∣∣xi1...in ∈ A, xi1...in i1...in→∞−−−−−−−→ 0

its completion.

11

Definition 2.2.4 (affinoid Tate ring). A affinoid Tate ring is a pair (A,A+) comprising a Tate ring Aand an open and integrally closed subring A+ ⊆ A◦.Maps of affinoid Tate rings are continuous maps A→ B that carry A+ into B+.

Definition 2.2.5 (Tate algebra). Let K be a NA field. A Tate K-algebra is a K-algebra which is aTate ring and which has a topology as Tate ring which is compatible with the topology induced by theone on K.

Concretely this means there exists a ring of definition A0 ⊂ A such that {rA0 | r ∈ K×} forms a basisof open neighbourhoods of 0.

Definition 2.2.6 (affinoid algebra). Let K be a NA field. An affinoid K-algebra (or affinoid TateK-algebra) is a K-algebra that is an affinoid Tate ring which has a topology as affinoid Tate ring thatis compatible with the topology induced by the one on K..

An affinoid K-algebra is topologically of finite type (tft.) if A+ = A◦ and if there is a n ∈ N>0 such thatA is a quotient of K 〈T1, . . . , Tn〉.

2.3 Adic spaces

Definition 2.3.1 (adic spectrum, rational subsets). Let (A,A+) be an affinoid Tate ring. Define

Spa(A,A+) :={|·| : A→ Γ ∪ {0} continuous valuation

∣∣ ∀f ∈ A+ : |f | ≤ 1}/'

with the equivalence of valuations as defined in 2.1.4. For x ∈ Spa(A,A+), write f 7→ |f(x)| for thecorresponding valuation on A.

Equip X := Spa(A,A+) with the topology which has as a basis of open subsets

X

⟨f1, . . . , fn

g

⟩:={x ∈ Spa(A,A+)

∣∣ ∀i : |fi(x)| ≤ |g(x)|}

with f1, . . . , fn, g ∈ A, and A = (f1, . . . , fn) as ideal. These are called rational subsets.

Spa(A,A+) is supposed to be an analogue of the spectrum Spec(A) for commutative rings, and sharessome similar properties.

Next, we want to define a structure sheaf on Spa(A,A+).

Definition 2.3.2 (structure presheaf). Let (A,A+) be an affinoid Tate ring, and let

U := X

⟨f1, . . . , fn

g

⟩⊂ Spa(A,A+) =: X

be a rational subset. Choose a couple of definition (A0, $) of A.

Endow the algebra A[ f1g , . . . ,fng ] ⊂ A[g−1] with the topology where {$nA0[ f1g , . . . ,

fng ]}n∈N is a basis

of open neighbourhoods of 0. Let B be the integral closure of A+[ f1g , . . . ,fng ] in A[ f1g , . . . ,

fng ] and

define (OX(U), O+

X(U))

:=

(A

⟨f1

g, . . . ,

fng

⟩, B

)For a general open subset W ⊂ X, we define

OX(W ) := lim←−U⊂W rational

OX(U)

and similar for O+.

Details can be found in [17, 2.13–2.15] and [11, chapter 1]. This is in general only known to be a presheaf.However, if we impose some constrains on A, we know that it will be a sheaf.

12

Definition 2.3.3 (strongly noetherian). A Tate ring A is strongly noetherian if A 〈T1, . . . , Tn〉 is noethe-rian for all n ∈ N.

Remark 2.3.4. If (A,A+) is an affinoid Tate ring withA strongly noetherian, thenOX , X = Spa(A,A+)is actually a sheaf.

Definition 2.3.5 (affinoid adic space). Let (A,A+) be an affinoid Tate ring for which OX is a sheaf,X := Spa(A,A+). Assume that (X,OX) is a locally ringed topological space and for each x ∈ X takethe induced continuous valuation | · (x)| on OX,x. Then the triple (X,OX , {| · (x)|}x∈X) is an affinoidadic space over K.

Definition 2.3.6 (adic space). An adic space is a triple (X,OX , {| · (x)|}x∈X) that is locally on Xisomorphic to an affinoid adic space.

Example 2.3.7. Let K be a complete, algebraically closed NA field, A := K 〈T 〉, A+ = A◦ = K◦ 〈T 〉.Let us fix a norm | · | : K → R≥0. Then there are 5 types of points in X := Spa(A,A+):

Gausspoint

(1)

(5)

(5)

(5)

(3)

(4)

(2)

(2)

(2)

(1) The classical points. Let x ∈ K◦. For any f =∑anT

n ∈ K 〈T 〉, we can evaluate f at x, so we geta norm | · (x)| : f 7→ |

∑anx

n|.

(2), (3) The rays. Let 0 < r ≤ 1, x ∈ K◦. Then

f =∑

an(T − x)n 7→ supy∈K◦:|x−y|≤r

(|f(y)|)

defines another valuation. if r ∈ |K×|, then the point given by this valuation is of type (2),otherwise of type (3).Note that for r=0, the points of these types would coincide with the ones from (1). For r=1,D(x, 1) := {y ∈ K◦ : |x − y| ≤ 1} is independent of x, we call the corresponding valuationGausspoint. Moreover, a branching occurs exactly at all points of type (2).

(4) Dead ends. There might be sequences of disks D1 ⊃ D2 ⊃ . . . with⋂Di = ∅. In this case we have

another valuationf 7→ inf

isupy∈Di|f(y)|

(5) rank 2 valuations. Fix 0 < r < 1. Take Γ := R>0 × γZ, where we require r′ < γ < r for all r′ < rand look at the valuation ∑

an(T − x)n 7→ max|an|γn

If r ∈ |K×|, then this gives a new point infinitesimally close to the point of type (2) defined by(x, r). One checks that those are the only rank-2 valuations that are not equivalent to a rank-1valuation of one of the other four types.

All points but the one of type (2) are closed, the latter have a closure comprising the point of type (2)together with all points of type (5) around it. More details can be found in [17, 2.20].

13

3. Almost ring theory

3.1 Almost mathematics

We aim to give a very brief introduction into almost ring theory. A complete treatment of the topic canbe found in [6] by Gabber and Ramero.

Definition 3.1.1 (setup of almost mathematics). Let A be a ring, and assume there is a flat ideal I ⊂ Awith I2 = I.

Remark 3.1.2. The most basic facts of almost mathematics can be developed without I being flat, butquickly it becomes a necessary requirement, see [6, 2.5.14]. Therefore we require it from the beginning,following the notation of [3, 4.1.2].

Remark 3.1.3. Basically we can construct the category of almost modules, denoted by ModaA as a quo-tient of ModA by ModA/I . So we think of almost mathematics as a weakening of ordinary (commutative)algebra where we systematically ignore I-torsion.

Of course, we get ordinary commutative algebra by taking I := A.

Notation 3.1.4. For the rest of the section, let A be a ring and consider almost mathematics withrespect to I ⊂ A.

Remark 3.1.5. In particular we have I ⊗A I ' I2 ' I.

Definition 3.1.6 (almost zero). Let M be an A-module. An element f ∈M is almost zero (with respectto I) if I · f = 0. The module M is almost zero (wrt. I) if every element is almost zero.

Usually we suppress the ”with respect to I” if it is clear which ideal I is meant and we tend to write

”(wrt. t1p∞ )” when we mean with respect to the ideal generated by all p-power roots of t, as defined

in 4.0.1. This will also be our main usage of almost mathematics.

3.2 Concrete construction of almost modules

Generally speaking, we want to ignore I-torsion, so we want to define the category of almost A-modulesas ModaA := ModA /ModA/I . We follow the discussion in [3, 4.1.5]

We start with describing how the three categories ModaA, ModA and ModA/I relate to each other, whilewe define the first one on the way.

Consider the restriction of scalars along A −→→ A/I. This gives us a fully faithful functori∗ : ModA/I → ModA, sending an A/I-module to itself as A-module. This functor has a right-adjoint

i! : M → HomA(A/I,M) and a left-adjoint i∗ : M →M ⊗A A/I

Lemma 3.2.1. The image of i∗ : ModA/I → ModA is closed under extensions, therefore ModA/I formsa Serre-subcategory of ModA and we can define the quotient.

Proof. Concretely we need to show that for a short exact sequence 0 → M → N → P → 0 in ModAwith M and N in (the image of) ModA/I , P also lies in ModA/I . M and N are both killed by I, so Pis killed by I2, but I2 = I, so P is indeed in ModA/I .

This means that we can really define ModaA := ModA /ModA/I . Now we can also define functorsbetween ModA and ModaA. The first one is the quotient functor, which can be realised explicitly byj∗ : ModA → ModaA,M 7→ I ⊗M . It too has a right-adjoint j∗ : M 7→ HomA(I,M) and a left adjointj! : M 7→M , which turns out to be just the inclusion. We arrive at the following picture:

14

ModA/I ModA ModaA

M ⊗A A/I M M

M M I ⊗M

HomA(A/I,M) M ; Hom(I,M) Mi!

i∗

i∗

j∗ / (−)∗

j∗ / (−)a

j! / (−)!

``

``

with i∗ a i∗ a i! and j! a j∗ a j∗ adjunctions and the composition in each horizontal row being zero.

Notation 3.2.2. For a module M ∈ ModA one usually writes Ma := j∗(M) = I ⊗ M ∈ ModaA,M∗ := j∗M

a = HomA(I,M) and M! := j!Ma = I⊗M . Elements in M∗ ⊂M are called almost elements

of M and we note that after applying (−)a, the maps M! → M → M∗ become almost isomorphisms.Sometimes we also write M [I] := i!(M) = HomA(A/I,M).

Lemma 3.2.3. The map j∗ : ModA → ModaA,M 7→ I ⊗M is an explicit realisation of the quotientfunctor q : ModA → ModA /ModA/I .

Proof. We must show that j∗(ModA/I) = 0, that j∗ is exact, and that it is universal with those twoproperties.

For j∗(ModA/I) = 0, let M ∈ ModA/I ⊂ ModA. We have I⊗AM = I⊗AA/I⊗A/IM = I/I2⊗AM = 0as I = I2. j∗ is exact because I is flat.

In order to show universality, let q′ : ModA → ModaA be any other functor of abelian categories withthose two properties. Fix M ∈ ModA. We have the canonical map I ⊗AM 7→M , which has kernel andcokernel identified with TorAi (A/I, I) for i = 1, 0. Particularly, both are killed by I, thus by q′, too. q′ isexact, which tells us that q′(I ⊗AM) ∼= q′(M). We also have that I ⊗AM = j!j

∗(M), so we have shownthat q′ ∼= q′j!j

∗, so q′ factors through j∗ as wanted.

3.3 Almost algebras

One can define the category of Aa-algebras (denoted CAlg(ModaA)). As (−)a clearly commutes withtensor products, one gets a functor

(−)a : CAlg(ModA)→ CAlg(ModaA)

This functor admits a right adjoint

(−)∗ : CAlg(ModaA)→ CAlg(ModA)

However, (−)! does not preserve commutative algebras as it does not preserve units. For this note that(Aa)! = I 6= A. To fix this, one defines B!! for an Aa-algebra B as the pushout of

I ∼= Aa! B!

A B!!

y

which results in a functor(−)!! : CAlg(ModaA)→ CAlg(ModA)

that is left-adjoint to (−)a.

As we only aim to introduce the basic notations, we refrain from giving more details and refer theinterested reader to [3] and [6] instead.

15

3.4 Almost flat, almost projective

In order to generalise the notions of flatness and projectivity, we simply take the usual definitions andreplace every requirement of being zero with the requirement of being almost zero. For example for a flatmodule M , we ask that the functor − ⊗AM is exact, in particular that TorAi (M,N) = 0 for all i > 0,so in almost mathematics, we ask TorAi (M,N) to be almost zero.

Definition 3.4.1. Let M be a A-module.

(I) M is almost projective if ExtiA(M,N) is almost zero for all A-modules N and all i > 0.

(II) M is almost flat if TorAi (M,N) is almost zero for all A-modules N and all i > 0.

(III) M is almost faithfully flat if it is almost flat and the natural map

HomA(N1, N2)→ HomA(N1 ⊗M,N2 ⊗M)

has an almost zero kernel for all A-modules N1, N2.

Later we will use faithfully flatness in the following equivalent formulation:

Lemma 3.4.2 (almost faithfully flat). Let A, B be rings with a map A → B. This map is almostfaithfully flat (i.e. makes B an almost faithfully flat A-module) if and only if it satisfies the followingtwo conditions:

(I) TorAi (M,B) =a 0, ∀i > 0

(II) M ⊗A B =a 0 =⇒ M = 0

Proof. ”⇒”. Let M be an A-module with M ⊗B =a 0. For every other A-module N we haveHomA(M ⊗B,N ⊗B) =a 0, therefore by assumption HomA(M,N) =a 0. As this holds for arbitrary Nwe get that M = 0.

”⇐”. For the other direction, let N1, N2 be two A-modules and take a map f : N1 → N2 such thatf ⊗B : n⊗m 7→ f(n)⊗m is the zero map.

This means f(N1)⊗B = 0 so by the assumption we get that f(N1) = 0.

Next we need finitely presented almost-modules in order to define almost etale maps. However, the naivedefinition of taking an almost-module which is finitely presented, does not work because it is way toorestrictive.

Definition 3.4.3 (almost finitely generated/presented). Let M be an A-module. M is almost finitelygenerated (almost finitely presented) if for all ε ∈ I there is a finitely generated (finitely presented)A-module Mε with a morphism ϕε : Mε →M such that both kernel and cokernel are annihilated by ε.

M is uniformly almost finitely generated if there is a n0 ∈ N such that all Mε are generated by at mostn0 elements.

Usually we will be working over valuation rings. Being almost finitely generated there takes then thefollowing explicit form:

Example 3.4.4. Let V be a valuation ring of height 1 whose value group Λ is dense in R, let A be aV -algebra, and M an A-module. Denote with m ⊂ V the maximal ideal, let π ∈ V be a nonzero elementof valuation |π| < 1 and let mα := {v ∈ V | |v| > α} for each α ∈ Λ.

M is almost finitely generated if for all α ∈ Λ>0 there is a finitely generated A-module Nα and morphismsϕα : Nα →M , ψα : M → Nα such that

ϕα ◦ ψα = πα · idM , ψα ◦ ϕα = πα · idNα

Finally we also state a lemma from [6, 4.1.5], which we need later.

Lemma 3.4.5. Let V be a valuation ring, A be an almost V -algebra. If M ⊗AN is an almost projectiveand almost faithfully flat A-module, then so are M and N .

16

3.5 Etale and almost etale morphisms

In this chapter we introduce almost (finite) etale morphism by adapting the notions of etale morphisms toalmost mathematics. We refer to [14] as a general introduction to etale morphisms and etale cohomologyand use [3] for the generalisation to almost mathematics.

Etale cohomology was introduced, because of the realisation that the Zariski topology on a variety orscheme was too coarse. For example, if X is an irreducible variety over an algebraically closed field,then the cohomology groups Hr(X,Z), computed using Zariski topology, are zero for all r > 0. Etalemorphisms capture the idea of being locally an isomorphism and etale cohomology is much more useful.For example, for differential manifolds, we have the inverse mapping theorem.

Definition 3.5.1 (etale morphisms of manifolds). A C∞-map ϕ : M → N of differentiable manifolds iscalled etale at m ∈M if the map on tangent spaces dϕ : Tgtm(M)→ Tgtϕ(m)(N) is an isomorphism.

Theorem 3.5.2 (inverse mapping theorem). A C∞-map is a local isomorphism at every etale point.

For varieties we want to have the same, so we would define etale maps with the same geometric motivation

Definition 3.5.3 (etale morphisms of algebraic varieties). Let X, Y be non-singular algebraic varietiesover an algebraically closed field. A regular map ϕ : Y → X is called etale at y ∈ Y ifdϕ : Tgty(Y )→ Tgtϕ(y)(X) is an isomorphism.

This can be generalised to general schemes, such that it agrees with the above definition in this specialcase

Definition 3.5.4 (etale morphisms of schemes).

(I) A ring homomorphism A→ B is flat if the functor M 7→ B ⊗AM from A-modules to B-modulesis exact.

(II) A local homomorphism ϕ : A→ B of local rings is unramified if

(a) ϕ(mA)B = mB , and

(b) The field B/mB is finite and separable over A/mA.

(III) A morphism ϕ : Y → X of schemes is flat (rsp. unramified) if the map OX,ϕ(y) → OY,y is flat (rsp.unramified) for every y ∈ Y .

(IV) A morphism ϕ : Y → X of schemes is etale if it is flat, unramified, and of finite presentation.

Recall (see Hartshorne [8, section II.3]) that a morphism of schemes Y → X is finite, if Y has an opencovering of affine schemes X =

⋃Vi =

⋃Spec(Ai) where each f−1(Vi) is affine, equal to Spec(Bi),

where Bi is a finite Ai-algebra (i.e. finitely generated as Ai-module). Moreover, Y → X is dominant ifϕ(Y ) ⊆ X is dense.

Notation 3.5.5. For a scheme X we write Xfet for the category of all finite etale maps Y → X. If theyare surjective, we call such maps also finite etale coverings of X.

Analogously, one can define almost etale maps (rsp. coverings) in the setup of almost mathematics.

Definition 3.5.6 (almost finite etale morphisms). Let V be a valuation ring whose value group Λ isdense in R. A morphism A→ B of V -algebras is almost finite etale, or an almost finite etale covering ofB, if it satisfies the following two properties:

(I) B is almost finitely presented, almost faithfully flat, and almost projective as an A-module.

(II) B is almost finitely presented and almost projective as B ⊗A B-module.

With the definition of almost finite etale coverings, we have a way to almost split extensions of V -algebrasA→ B if they can be dominated by an almost finite etale cover of A, as demonstrated in the next lemma,3.5.8, which has been presented by Bhatt in [2, 2.7].

17

Definition 3.5.7 (dominated). Let f : A→ B be a morphism of rings. We say that f can be dominatedby an almost finite etale cover of A if there is a map B → C such that the composition A→ B → C isan almost finite etale cover.

Lemma 3.5.8. Let f : A → B be an inclusion of V -algebras that makes B an almost projective andalmost faithfully flat A-module. Then the cokernel coker(f) is an almost projective A-module.

Proof. For any two A-modules M and N we have the following facts:

• RHom(M ⊗LA N,−) ∼= RHom(M,RHom(N,−))

• If M and N are both almost projective, then so is M ⊗L N

• If one of M and N is almost flat, then M ⊗LA N ∼= M ⊗A N .

Hence we have that B⊗AB is almost projective and almost faithfully flat. If we tensor the exact sequence

0→ A→ B → coker(f)→ 0

with −⊗A B, we get0→ B → B ⊗A B → coker(f)⊗A B → 0

which is split by the multiplication map µ : B⊗AB → B. Therefore, coker(f)⊗AB is almost projectiveas it is a direct summand of the almost projective A-module B ⊗A B, and by lemma 3.4.5 coker(f) isalmost projective.

Corollary 3.5.9. Let f : A→ B be a morphism of V -algebras that can be dominated by an almost finiteetale cover of A. Then f is almost split.

Proof. f being dominated by an almost finite etale cover of A means that there is a map g : B → Csuch that g ◦ f : A→ C is almost finite etale.

This means C is per definition an almost projective, almost faithfully flat A-module. Hence, we havethe short exact sequence 0→ A→ C → coker(g ◦ f)→ 0 with coker(g ◦ f) almost projective by lemma3.5.8, and proposition 1.1.4.2 implies that g ◦ f is almost split.

A B Cf g

∃ϕ

, ϕ ◦ (g ◦ f) = idA

So ϕ ◦ g gives us an almost splitting for f : A→ B.

18

4. Perfectoid spaces

For this entire chapter, we fix a prime p and we denote with φ the map φ : x 7→ xp. If the ring hascharacteristics p, this is the Frobenius endomorphism, if it does not, then this is simply a multiplicativemap which respects multiplication but not addition.

Perfect rings are rings that contain all p-power roots, and in some sense perfectoid rings will be ringsthat contain a lot of p-power roots.

In perfectoid rings we have a natural notion of almost mathematics with respect to a given element.

Definition 4.0.1 (p-power roots). Let A be a ring, t ∈ A. We say t admits a compatible system of

p-power roots if there are elements t1

pk ∈ A, k ∈ N such that for all k ∈ N we have(t

1

pk

)p= t

1

pk−1 . Such

an element is also called a perfect element. We denote with (t1p∞ ) the ideal generated by {t

1

pk }k∈N.

For any perfect element t we have (t1p∞ )2 = (t

1p∞ ), so we can look at almost mathematics with respect

to that ideal.

We are following the lecture notes of Bhatt [3] and Scholze’s paper on perfectoid spaces [17].

4.1 Perfection and tilting

We start by introducing tilting, which is one of the key features for perfectoid spaces, over general rings.First, we recall the definition of spaces where every element has arbitrary p-power roots.

Definition 4.1.1 (perfect ring). A ring A of characteristic p is perfect if the Frobenius morphismφ : x 7→ xp is an isomorphism. It is semi-perfect if φ is surjective.

There are universal constructions from and into the subcategory of perfect rings.

Definition 4.1.2 (tilting). Let A be a ring with char(A) = p, φ : x 7→ xp the Frobenius morphism. SetAperf := lim−→φ

(A) and Aperf := lim←−φ(A).

Now let A be a ring of any characteristic and chose any $ ∈ A with $ | p. We set A[ := (A/ ($))perf

and call it tilt of A.

Remark 4.1.3. For a ring A of characteristic p, we get canonical maps A→ Aperf and Aperf → A.

A→ Aperf is universal for maps from A to perfect rings and Aperf → A is universal for maps from perfectrings to A.

The next lemma shows, that the tilt A[ is independent of the choice of $.

Lemma 4.1.4. Let $ ∈ A \ {0} with $ | p. Assume A is $-adically complete, and recall the notationφ : x 7→ xp. The projection A −→→ A/($) induces a bijection

lim←−φ

A'−→ lim←−

φ

A/($) = A[

Proof. First we show injectivity. Let (an)n, (bn)n ∈ lim←φA with an ≡ bn (mod $) for all n ∈ N. For

all k, n ∈ N, an+k ≡ bn+k (mod $) implies apk

n+k ≡ bpk

n+k (mod $k) because $ | p. Further we have

apk

n+k = an and bpk

n+k = bn, thus we get an ≡ bn (mod $k) for all n, k ∈ N. Because A is $-adicallycomplete, we conclude that an = bn for all n ∈ N.

Now for surjectivity, let (an)n ∈ lim←φA/($) and choose arbitrary lifts an ∈ A. These an do not

necessarily satisfy that apk

n+k?= an, but we can find some bn with an ≡ bn (mod $) that do have

19

this relation. For this, note that apn+k+1 ≡ an+k (mod $) for all k, n ∈ N, and consider the sequence(ap

k

n+k

)k∈N

. For all k1 ≤ k2 we have the relation apk1

n+k1− ap

k2

n+k2≡ 0 (mod pk1) and hence (mod $k1).

Therefore the sequence is Cauchy and has a limit bn ∈ R. These bn’s satisfy bpn+1 ≡ bn (mod $k) forall k ∈ N>0, thus bpn+1 = bn. We end up with (bn)n ∈ lim←φR which maps to (an)n.

Remark 4.1.5. So far one can think of choosing $ := p, but we make the distinction here because weneed it for defining perfectoid fields.

Next we construct a map (−)] that goes in the opposite direction of (−)[.

Definition 4.1.6 (sharp map). The bijection of lemma 4.1.4 gives us a multiplicative map

] : R[ → R, f 7→ f ]

by projecting to the last term of lim←φR.

The image consists of exactly those f ∈ R that admit a compatible system of p-power roots {f1

pk }.

4.2 Perfectoid fields

Next, we look at perfection over a NA field.

Definition 4.2.1 (perfectoid field). A NA field K with residue characteristic p is perfectoid if thevaluation has height 1, the value group |K×| is non-discrete and K◦/(p) is semi-perfect.

Lemma 4.2.2. Let K NA field of characteristics p. Then K is perfectoid exactly if it is perfect.

Proof. ”⇐”. Let K be perfect. Necessarily this means that the value group |K×| is non-discrete, for

example the subgroup {|z1pn |}n∈Z ⊂ |K×| for a nonzero z ∈ K is non-discrete.

Take x ∈ K◦. As the Frobenius is an isomorphism, there exists y ∈ K with yp = x. Moreover|y|p = |yp| = |x| ≤ 1 hence |y| ≤ 1 and y ∈ K◦.”⇒”. For the other direction, being perfectoid means that the Frobenius K◦/(p)→ K◦/(p) is surjectivebut as we have characteristics p, (p) is the zero ideal, so φ : K◦ → K◦ is surjective. For injectivity, ifxp = 0 then 0 = |0| = |xp| = |x|p so |x| = 0 and we have x = 0.

Now we construct the tilt K[ of a perfectoid field K. Fix a pseudo-uniformiser $ ∈ K with |p| ≤ |$| < 1.First we define its ring of integers as (K[)◦ := K◦[. Lemma 4.1.4 gives us a map

] : lim←φ

K◦/($)→ K◦

Lemma 4.2.3. There exists $[ ∈ K◦[ such that |($[)]| = |$|.

Proof. Pick an element $1 with |$p1 | = |$|. In particular $1 ∈ (K◦/($)) \ {0}. Choose any element

$[ := (0, $1, . . . ) ∈ lim←−φK◦/($). Such an element exists by the surjectivity of the Frobenius morphism.

Now the way we defined $[ gives us that |($[)] −$p1 | ≤ |$|2, so we get |$| = |($[)]|.

Remark 4.2.4. The choice of $[ is not unique. Usually one addresses this problem by redefining

$ := ($[)]. Then $ admits a compatible system of p-power roots $1

pk = (($[)1

pk )].

Definition 4.2.5 (tilt of a perfectoid field). Let K be a perfectoid field, $ ∈ K with |p| ≤ |$| < 1.Define K[ := K◦[

[($[)−1

]using the element $[ ∈ K◦[ constructed in lemma 4.2.3.

20

Lemma 4.2.6.

(I) There is a homeomorphism K[ ∼= lim←−φK. Therefore we have a multiplicative map ] : K[ → K.

(II) K[ is a perfectoid field of characteristic p.

(III) |K[×| = |K×|

(IV) K[◦/($[) ∼= K◦/($)

(V) If char(K) = p then K = K[

(VI) The ideal (($[)1p∞ ) ⊂ K◦[ generated by

{($[)

1

pk | k ∈ Z}

is maximal.

Proof. ”(I)”. We already have a map ] : K◦[ = lim←φK◦/($)→ K◦. As it is multiplicative, it extends

to a map ]′ : K[ → lim←−φK, so by projecting to the last term we get ] : K[ → K.

The inverse map to ]′ can be given explicitly by x 7→ (x], (x1p )], . . . ) which extends similarly to an inverse

of ]. It is clear that both maps are continuous, hence ] defines a homeomorphism.

”(II)”. K[ is perfect, complete and of characteristic p. All of this follows directly from K◦[ =lim←−φK

◦/($). In particular, it is perfectoid.

”(III)”. Follows from the fact that the valuation on K[ is induced from the one on K.

”(IV)”. $[ ∈ K[◦ maps under ] to $ = 0 ∈ K◦/$, hence we get a map K[◦/$[ → K◦/$. It is certainlysurjective but we need to check injectivity. Take an element g ∈ K[◦ that maps to zero in K◦/$. Thismeans concretely that g] ∈ ($) ⊂ K◦. But we know that |$| = |($[)]|, so there is an a ∈ K◦ such thatg] = a · ($[)]. We can find a lift a = (a0, a1, . . . ) of a in K[◦ by setting

an :=

(g]) 1pn(

$[]) 1pn

and thus we have g = a ·$[ and g = 0 ∈ K[◦/$[ give us the injectivity.

”(V)”. We have K[ ∼= lim←−φK. In characteristic p, the perfectoid field K is perfect (see lemma 4.2.2), so

the Frobenius φ is an isomorphism.

”(VI)”. As K◦ is a valuation ring of height 1, the maximal ideal of K◦/$ is its nilradical, and the same

holds for K◦[/$[. But the nilradical in K◦[/$[ is the image of (($[)1p∞ ), because this ideal lies in

the nilradical and the quotient K◦[/$[ is reduced because it is perfect. Therefore (($[)1p∞ ) must be

maximal.

The most important theorem here is an almost purity theorem which will be presented in section 4.5,saying that K and K[ behave similar in some sense, respectively that there is an equivalence of (almost)etale covers.

4.3 Perfectoid algebras

For this section fix a perfectoid field K with tilt K[. Choose $ ∈ K with |p| ≤ |$| < 1 and such that$ admits a compatible system of p-power roots, as described in section 4.2.

Definition 4.3.1 (perfectoid algebra). A perfectoid K-algebra is a Banach K-algebra A such that A◦ ⊂ Ais open and bounded and the Frobenius

φ : A◦/$ → A◦/$

is surjective. Morphisms between perfectoid K-algebras are continuous K-algebra morphisms.

Definition 4.3.2 (perfectoid affinoid algebra). An affinoid (Tate) K-algebra (A,A+) is perfectoid if Ais a perfectoid K-algebra.

21

Definition 4.3.3 (integral perfectoid algebra). Let K be a perfectoid field, K◦ ⊂ K its ring of integers.

A K◦-algebra A is integral perfectoid if it is flat, $-adically complete and satisfies the following properties

(I) If f ∈ A[ 1$ ] satisfies f ·$

1

pk ∈ A for all k ∈ N, then f ∈ A. †

(II) The Frobenius morphism induces an isomorphism A/$1p ∼= A/$

Definition 4.3.4 (perfectoid K◦a-algebra). A K◦a-algebra A is perfectoid if A is $-adically complete,flat over K◦, and the map K◦/$ → A/$ is relatively perfect, i.e. the Frobenius induces an isomorphism

A/$1p ∼= A/$.

4.4 Perfectoid spaces

Fix a perfectoid field K with tilt K[.

Definition 4.4.1 (perfectoid space). Let (A,A+) be a perfectoid affinoid K-algebra. The adic spaceX := Spa(A,A+) is called an affinoid perfectoid space.

A perfectoid space is an adic space over a perfectoid field K that is locally isomorphic to an affinoidperfectoid space. Morphisms of perfectoid spaces are morphisms of adic spaces.

Remark 4.4.2. If A is perfectoid then the presheaf OX with X = Spa(A,A′) mentioned in 2.3.2 isactually a sheaf.

We define the tilt of a perfectoid space by gluing:

Definition 4.4.3 (tilt). Let X be a perfectoid space over K, X[ one over K[. We say that X[ is the tiltof X if for all perfectoid affinoid K-algebras (A,A+) we have a functorial isomorphism

Hom(Spa(A[, A[+), X[) ∼= Hom(Spa(A,A+), X)

Perfectoid spaces are covered in great detail in Scholze’s work [17] and we do not intent to cover ev-erything. Instead we present just a few facts from their paper which are needed to prove the directsummand conjecture. First, we have an approximation lemma for perfectoid spaces [17, 6.7].

Lemma 4.4.4 (approximation lemma). Let (A,A+) be a perfectoid affinoid K-algebra with tilt (A[, A[+)and let X = Spa(A,A+), X[ = Spa(A[, A[+) be the associated affinoid perfectoid spaces.

(I) ∀g ∈ A, ∀c ≥ 0, ∀ε > 0 there exists gc,ε ∈ A[such that:

|g(x)− g]c,ε(x)| ≤ |$|1−ε max(|g(x)|, |$|c), ∀x ∈ X

(II) The map X → X[ induces a homeomorphism identifying rational subsets.

Remark 4.4.5. (I) says that every element of perfectoid affinoid K-algebra A can be approximated byperfect elements coming from the tilt.

† This could be written in short as A = (Aa)∗ using the notations from section 3.2

22

Next, we also need a concrete description of OX for perfectoid spaces, coming from [17, 6.4].

Lemma 4.4.6 (description of OX). Let (A,A+) be a perfectoid affinoid K-algebra with tilt (A[, A[+)and let X = Spa(A,A+), X[ = Spa(A[, A[+) be the associated affinoid perfectoid spaces.

Further, let U = X[⟨f1...fng

⟩⊂ X[ be a rational subset. Assume all fi, g ∈ A[◦ and that fn = $[N for

some N ∈ N. ‡

(I) Recall that A◦⟨(

f]1g]

) 1p∞

, . . . ,(f]ng]

) 1p∞⟩

denotes the $-adic completion of A◦[(

f]1g]

) 1p∞

, . . . ,(f]ng]

) 1p∞]

.

Then A◦⟨(

f]1g]

) 1p∞

, . . . ,(f]ng]

) 1p∞⟩a

is a perfectoid K◦a-algebra

(II) OX(U ]) is a perfectoid K-algebra with associated K◦a-algebra

OX(U ])◦a = A◦

⟨(f ]1g]

) 1p∞

, . . . ,

(f ]ng]

) 1p∞⟩a

(III) For the tilt we have OX(U ])[ = OX[(U ]).

Another result from the same paper is the vanishing theorem [17, 6.14], which we use in the derivedversion of the direct summand conjecture in section 6.5.

Theorem 4.4.7 (vanishing theorem). Let K be of any characteristic and let (A,A+) be a perfectoidaffinoid K-algebra, X = Spa(A,A+). For any finite covering X =

⋃i=1...n Ui with rational subsets, the

sequence

0→ OX(X)◦a →∏i

OX(Ui)◦a →

∏i,j

OX(Ui ∩ Uj)◦a

is exact. Moreover, OX is a sheaf and Hi(X,O◦aX ) = 0 for i > 0.

Remark 4.4.8. For the last statement about the cohomology one uses Banach’s Open Mapping Theoremas explained in [18, 7.4.4].

4.5 Almost purity

At the end of section 3.5 in corollary 3.5.9 we have seen a way to almost split an extension of algebrasA → B in the case where it can be dominated by an almost finite etale cover of A. In this section wewould like to explain how Falting’s almost purity theorem can be used to create such an almost finiteetale cover. For this we need A to be a perfectoid algebra, so we first need to extend the definition ofetale morphisms.

Definition 4.5.1 (finite etale maps). A map f : (A,A+)→ (B,B+) of affinoid Tate rings is finite etaleif A→ B is finite etale and B+ is the integral closure of A+ in B. We write (A,A+)fet for the categoryof all such maps.

A map f : X → Y of adic spaces is finite etale if there exists a cover Y =⋃i∈I Vi such that for each

i ∈ I, Vi and Ui = f−1(Vi) are affinoid and (OY (Vi),O+Y (Vi))→ (OX(Ui),O+

X(Ui)) is finite etale. Againwe write Yfet for the category of all those maps.

Definition 4.5.2 (strongly finite etale). A map f : (A,A+)→ (B,B+) of affinoid perfectoid K-algebrasis strongly finite etale if it is finite etale and B+a is almost finite etale over A+a. We denote the categoryof all such maps by (A,A+)sfet.

A map f : A→ B of perfectoid spaces is strongly finite etale if there exists a cover Y =⋃i∈I Vi such that

for each i ∈ I, Vi and Ui = f−1(Vi) are perfectoid affinoid and (OY (Vi),O+Y (Vi)) → (OX(Ui),O+

X(Ui))is strongly finite etale. Again we write Ysfet for the category of those maps.

‡ The assumption fn = $[N can always be fulfilled without changing the rational subset U .

23

Theorem 4.5.3 (strong almost purity). Let (A,A+) be a perfectoid affinoid K-algebra, X := Spa(A,A+)with tilt X[.

(I) Let U ⊂ X be an affinoid perfectoid subspace. There is a fully faithful functor from the category ofstrongly finite etale covers of U to the category of finite etale covers of OX(U).

(II) For every U , this functor is an equivalence of categories.

(III) For any finite etale cover B/A, B is perfectoid and B◦a is finite etale over A◦a. Moreover, B◦a isuniformly almost finitely generated A◦a-module.

Theorem 4.5.3 has been presented by Scholze in [17, 7.9] and a direct consequence is the following:

Corollary 4.5.4. Let A0 be a perfectoid affinoid K◦-algebra and assume we have a map A0 → B0 thatis almost finite etale after inverting p. Then A0 → B0 can be dominated by an almost finite etale coverA0 → C. Moreover, the induced B0 → C is an isomorphism after inverting p.

Proof. Being finite etale after inverting p means concretely that for A := A0[ 1p ] and B := B0[ 1

p ] that

A→ B is a finite etale cover of A. Part (III) of the almost purity theorem 4.5.3 says then that A◦ → B◦

is almost finite etale.We have an almost isomorphism A0∼=a A◦, so A0

∼=a−−→ A◦ → B◦ is an almost finiteetale cover of A◦ which dominates A0 → B0.

Together with the result from the previous section this gives us an almost splitting if we haveA0[ 1

p ]→ B0[ 1p ] almost finite etale.

24

5. Pro systems and almost pro systems

For this chapter, let A be a ring, t ∈ A a non-zerodivisor and assume t admits a compatible system

of p-power roots {t1

pk }k. All almost mathematics in this chapter are with respect to (t1p∞ ). We follow

again Bhatt [4].

5.1 Almost-pro-zero modules

A projective system M1 ← M2 ← M3 ← . . . of A-modules should be almost zero if its limit is almostzero, so we have the following definition:

Definition 5.1.1 (almost-pro-zero). A pro-A-module {Mn}n≥1 is almost-pro-zero (wrt. t1p∞ ) if

∀k ∈ N, ∀n ∈ N>0, ∃m ∈ N≥n : Im(Mm →Mn) is killed by t1

pk .

A map of pro-objects in Db(A) is an almost-pro-isomorphism if the cohomology groups of its cone forman almost-pro-zero system.

Notation 5.1.2. Let M be an A-module, k ∈ N. For this chapter we use the following notation

M [t1

pk ] := HomA(A/(t1

pk ),M). † Concretely this is the subset M [t1

pk ] ⊂ M comprising all elements

that are killed by t1

pk ; the inclusion is given by f 7→ f(1A).

With this we present some basic stability properties about almost-pro-zero modules and especially abouttheir Rlim.

Lemma 5.1.3. Let {Mn}n≥1 be an almost-pro-zero pro-A-module. Then Rlim({Mn}n≥1) is almost zero.

Proof. Let k ∈ N. As {Mn}n≥1 is almost-pro-zero, the inclusion {Mn[t1

pk ]}n≥1 → {Mn}n≥1 is an pro-isomorphism, so both sides have the same Rlim. Therefore the cohomology groups of

Rlim({Mn}n≥1) = Rlim({Mn[t1

pk ]}n≥1) are killed by t1

pk , so Rlim({Mn}n≥1) is almost-zero.

Lemma 5.1.4. Let {Nn}n≥1 → {Mn}n≥1 be an almost-pro-isomorphism in Db(A). ThenRlim({Nn}n≥1)→ Rlim({Mn}n≥1) is an almost isomorphism.

Proof. Apply lemma 5.1.3 to the cone.

Lemma 5.1.5. Let {Mn}n≥1 be an almost-pro-zero pro-A-module, F : ModA → ModA an A-linearfunctor. Then {F (Mn)}n≥1 is almost-pro-zero.

Proof. Fix k ≥ 0 and n ≥ 1. As {Mn}n≥1 is almost-pro-zero, there is a m > n, such that the map

Mm →Mn factors through Mn[t1

pk ]→Mn. Then F (Mm)→ F (Mn) factors through

F (Mn[t1

pk ])→ F (Mn) and thus through F (Mn)[t1

pk ]→ F (Mn) because F is A-linear.

† Compare this to the notation 3.2.2, where we defined M [I] := i!(M)

25

5.2 Quantitative Riemann’s Hebbarkeitssatz

Another crucial part of the proof of the direct summand conjecture will be presented in this section.The quantitative form of the Riemann’s Hebbarkeitssatz for almost-pro-systems gives us an almost-isomorphism modulo tm that allows us to reduce the problem to rational subsets.

Let A be an integral perfectoid K◦-algebra, t ∈ A \ {0} which admits a compatible system of p-powerroots, X := Spa(A[t−1], A). Fix a g ∈ A which admits a compatible system of p-power roots.

We recall from definition 2.3.1 that

X

⟨tn

g

⟩:= {x ∈ X : |tn(x)| ≤ |g(x)|}

A

⟨tn

g

⟩= O+

X

(X

⟨tn

g

⟩)Theorem 5.2.1 (quantitative Riemann’s Hebbarkeitssatz for pro-systems). Let m ∈ N. Consider theprojective system of maps {

fn : A/tm → A

⟨tn

g

⟩/tm

}

(I) each ker(fn) is almost-zero (wrt. t1p∞ ).

(II) {coker(fn)}n≥1 is uniformly almost-pro-zero (wrt. g1p∞ ).

i.e. ∀k ∈ N, ∃c ∈ N, ∀n ∈ N>0 coker(fn+c)→ coker(fn) is killed by g1

pk .

In particular this means that {fn} is an almost-pro-isomorphism (wrt. (tg)1p∞ ).

Proof. Assume first that g is a non-zerodivisor modulo tm. For any n we define formally an indeterminantun with a compatible system of p-power roots and define

Mn := A[un1p∞ ]/(tm, ∀k : (un · g)

1

pk − tn

pk )

Note that this is like adjoining formally(tn

g

) 1p∞

to A/tm. We have Mn∼=a A

⟨tn

g

⟩/tm (wrt. t

1p∞ ) by

4.4.6 and its proof in [17, 6.4], so we show the statement for {fn : A/tm →Mn}n.

g is not a zerodivisor modulo tm and therefore neither is g1

pk . Hence fn is injective and therefore thekernel is zero.

Now for the cokernels we fix k ∈ N and show that c := pk ·m gives us the desired uniform almost-pro-

zeroness. In particular we want to show that under the transition map Mn+cτn,c−−→ Mn, each element

g1

pk · uen+c, e ∈ N[ 1p ] maps into A/tm = Im(fn) ⊂ Mn. If that is the case, then for every n, the map on

cokernels induced by τn,c gets killed by g1

pk , resulting in almost-pro-zeroness (wrt. g1p∞ ).

By construction τn,c

(g

1

pk · uen+c

)= g

1

pk · uen · tce. If e ≥ 1pk

then tm | tce and the expression becomes

zero modulo tm. If e < 1pk

, then

g1

pk · tce · uen = g1

pk−e · tce · (g · un)e = g

1

pk−e · tce · (tn)e = g

1

pk−e · t(c+n)e ∈ A/tm ⊂Mn

This means that the c-fold transition map on cokernels is killed by g1

pk .

Now for the general case where we do not impose that g is a non-zerodivisor modulo tm, we can consider

R := K◦⟨T

1p∞⟩

with g := T . Then g is not a zerodivisor modulo tm, so we have from the first part an

almost-pro-isomorphism (wrt. (tT )1p∞ )

{R/tm}n →{R

⟨tn

T

⟩/tm

}n

26

There is a unique map R → A with T1

pk 7→ g1

pk for all k, so base change gives us an almost-pro-isomorphism

{A/tm}n →{R

⟨tn

T

⟩⊗LR A/tm

}n

and applying H0 to this almost-pro-isomorphism we have

{A/tm}n →{R

⟨tn

T

⟩⊗R A/tm

}n

Finally, the description of OX in lemma 4.4.6 tells us that we have an almost isomorphism

R

⟨tn

T

⟩⊗R A/tm ∼=a A

⟨tn

g

⟩/tm.

27

6. The direct summand conjecture

6.1 The DSC in equal characteristic

The case of equal characteristic zero is simple using the trace map and the case of equal characteristicp > 0 has been proved by Hochster in [9]. For completeness we repeat Hochster’s argument here.

First we need some lemmas that allow us to make some reductions.

Lemma 6.1.1. Let A ↪−→ B be rings with B finitely presented as A-module, T a faithfully flat A-algebra.The following are equivalent:

(I) A ↪−→ B is a direct summand

(II) Am ↪−→ Bm is a direct summand for all maximal ideal m ∈ A.

(III) T ↪−→ B ⊗A T is a direct summand.

Proof. Consider 0→ A→ B → B/A→ 0.

Saying A is a direct summand of B is equivalent to saying that

HomA (B/A,B)→ HomA (B/A,B/A)

is surjective.

We show the equivalences ”(I) ⇐⇒ (II)” and ”(I) ⇐⇒ (III)” at the same time. For the first equivalencewe assume that S := Am and for the second one, we take S := T .

In both cases we get thatHomA (B/A,B)→ HomA (B/A,B/A)

is surjective if and only if

HomA (B/A,B)⊗A S → HomA (B/A,B/A)⊗A S

is surjective (for all S = Am in the first case). For this we use either that surjectivity is a local property(case 1) or that S is faithfully flat (case 2).

Since B, and therefore B/A, are finitely presented, we get the commutative diagram

HomA (B/A,B)⊗A S HomS ((B ⊗A S)/(A⊗A S), B ⊗A S)

HomA (B/A,B/A)⊗A S HomS ((B ⊗A S)/(A⊗A S), (B ⊗A S)/(A⊗A S))

∼=

∼=

For both cases, the surjectivity of the right-hand column is exactly the condition in either (II) or (III),so the claim follows.

Lemma 6.1.2 (DSC in char. (0, 0)). Let A be an integrally closed domain containing Q, B a domainthat is a finite integral extension of A. Then A ↪−→ B is a direct summand.

Proof. We can always enlarge B. Therefore – and because B is integral over A – we can assume that Bis the integral closure of A in some finite normal field extension L of K := Frac(A). Let d := [L : K] bethe degree of that extension. Then we can verify that the trace map 1

d TrL/K : B → A splits A ↪−→ B.Indeed, as b ∈ B is integral over A, TrL/K(b) ∈ K is integral over A, hence it lies in A. Moreover, for

a ∈ A, we have 1d TrL/K(a) = 1

d · da = a.

28

Lemma 6.1.3. Let A be a domain and B an integral extension of A. Then there exists a prime idealp ⊂ B such that p ∩A = (0). If A ↪−→ B/p is a direct summand, then A ↪−→ B is one, too.

Proof. This is trivial using the ”lying over”-property of integral extensions and looking at

A B

B/p

π

f

If f splits A ↪−→ B/p, then fπ splits A ↪−→ B.

The direct summand conjecture is equivalent to the monomial conjecture, which asserts that for a systemof parameters there is no relation (x1 · . . . · xn)k /∈ (xk+1

1 , . . . , xk+1n )B. Hochster gave a good overview

of more recent homological conjectures in [10], and proved the direct summand conjecture in equalcharacteristic via this monomial conjecture in [9].

Proposition 6.1.4. Let A be a regular local ring, x1, . . . , xn a regular system of parameters for A, andB a A-algebra which is finitely generated as an A-module. The following are equivalent:

(I) A ↪−→ B is a direct summand

(II) (x1 · · ·xn)k /∈ (xk+11 , . . . , xk+1

n )B, ∀k ∈ N>0

Proof. Let m = (x1, . . . , xn) ⊂ A be the maximal ideal, Ik := (xk1 , . . . , xkn) and uk := (x1 · · ·xn)k−1.

Further we define Ak := A/Ik, mk := m/Ik and Bk := B/IkB.

As x1, . . . , xn is a system of parameters, so is xk+11 , . . . , xk+1

n because√(xk+1

1 , . . . , xk+1n ) =

√(x1, . . . , xn) = m. Therefore, Ak is Gorenstein, see [13, 18.1]. Moreover, it

is also zero-dimensional. The annihilator ideal is generated by uk as for every y ∈ m = (x1, . . . , xn),y ·uk = y · (x1 · · ·xn)k−1 ∈ Ik. Therefore AnnAk(mk) ∼= R/m and every ideal I ⊂ A with Ik ( I containsuk.

The direction ”(I) ⇒ (II)” is immediate. If A is a direct summand of B, then every ideal of A will becontracted and hence uk /∈ IkA implies that uk /∈ IkB.

The converse direction ”(I)⇐ (II)” requires more work. Assume uk /∈ IkB for all k ∈ N>0. By (III) oflemma 6.1.1 it is enough to show that A ⊂ B ⊗A A is a direct summand, A being the completion of A.

As A ↪−→ A is faithfully flat, so is B ↪−→ B⊗A A, and we deduce that uk /∈ (Ik(B⊗A A))∩ A, which allowsus to assume without loss of generality that A is complete.

As the ideals {Ik}k and {mk}k are cofinal and A complete, we have

HomA(B,A) = limk

HomA(Bk, Ak) ∼= limk

HomAk(Bk, Ak) (?)

In order to prove that Ai↪−→ B splits we need to find ϕ ∈ HomA(B,A) such that ϕ ◦ i = idA, so if we

define h : HomA(B,A)→ A, ϕ 7→ ϕ(1B) then H = h−1(1A) is the set of all such splittings and we needto show that this is non-empty.

Similarly we can define hk : HomA(Bk, Ak) → Ak, ϕ 7→ ϕ(1Bk) and Hk = h−1k (1Ak) and the inverse

limit in (?) gives us H = limkHk.

Now, each Hk is non-empty as we can show that each Ak ↪−→ Bk splits. For this, we have that IkB∩A = Ikas Ik ( IkB ∩ A would imply that uk+1 ∈ IkB ∩ A, which would contradict the hypothesis. Thereforethe inclusion A ↪−→ B induces an inclusion Ak ↪−→ Bk for all k and because Ak is a zero-dimensionalGorenstein ring, it is injective as Ak-module and thus Ak ↪−→ Bk splits.

With this we are done, because the Hk are coset of a submodule of HomA(B,Ak) and an inverse limitof nonempty cosets in an Artinian module is non-empty.

29

For that we can take the decreasing sequence {Im(Hi+k → Hk)}i of non-empty cosets. This sequencestabilises in a coset Ek of Hk. All the Ek form together a subsystem of nonempty subcosets and surjectivemaps, therefore limEk 6= ∅, and thus H 6= ∅.

With those preliminary results, we can now prove the direct summand conjecture in equal characteristic.

Theorem 6.1.5 (Hochster, DSC in equal char.). Let A be a regular ring which contains a field, B anA-algebra, finitely presented as A-module. Then A ↪−→ B is a direct summand.

Proof. By 6.1.1 we can assume that A is a complete regular local ring, with maximal ideal m. Moreover,it can be assumed that B is a domain by 6.1.3.

The case where A contains a field of characteristic zero is covered in lemma 6.1.2.

Let us assume that A contains a field of characteristic p > 0. By proposition 6.1.4 we just have to checkthat there is no relation of the form

(x1 · . . . · xn)k =

n∑i=1

bixk+1i (6.1)

with k ∈ N>0, bi ∈ B. As B is a domain, it is torsion-free over A, hence it can be embedded into a freeA-module. Consequently, there is an A-homomorphism ψ : B → A such that ψ(1) 6= 0. Therefore we

can choose t ∈ N large enough so that ψ(1) /∈ mpt

.

But if there were a k satisfying equation (6.1), we could raise both sides of the equation (6.1) to the pt

power and apply ψ:

(x1 · . . . · xn)kpt

ψ(1) =

n∑i=1

ψ(bi)x(k+1)pt

i (6.2)

Hence we would have ψ(1) ∈ (xk′+pt

1 , . . . , xk′+pt

n ) : (x1 · . . . ·xn)k′

with k′ := kpt This ideal quotient turns

out to be (xpt

1 , . . . , xpt

n ), so we get finally ψ(1) ∈ (xpt

1 , . . . , xpt

n ) ⊂ mpt

, which is a contradiction.

For the calculation of the ideal quotient, we note that x1, . . . , xn is a regular sequence – which means(x1, . . . , xn) 6= A and xi+1 ∈ A/(x1, . . . , xi) not a zerodivisor – whereby we can calculate it as if the xi’swere indeterminants over Z (see for example [5]).

6.2 Almost faithfully flat extension

The goal of this section is to prove 6.2.4, which gives us an extension A → A∞ to a perfectoid algebraA∞ which is almost faithfully flat modulo p, and adjoins all p-power roots of a discriminant g ∈ A, whichwe will introduce later in the proof. The following two results can be found in [4, 5.1–5.2].

Lemma 6.2.1. Let ϕ : A→ B be a map of commutative rings, A noetherian. Assume there is a π ∈ Asuch that A and B are π-adically complete and π-torsionfree and that ϕ : A/π → B/ϕ(π) is (faithfully)flat. Then ϕ is (faithfully) flat.

Proof. First we prove flatness.

We need to check that for all finitely generated A-modules M , we have that M ⊗LA B lies in D≥0.

Let M be a finitely generated A-module. As A is noetherian, we can choose a resolution by finitelygenerated free A-modules P • →M . By definition M ⊗LA B = P • ⊗A B.

Since B is ϕ(π)-adically complete we have B = limnB/ϕ(π)n and hence P •⊗AB ' limn P•⊗AB/ϕ(π)n.

Therefore, we have:

M⊗LAB ' Rlim(M⊗LAB/ϕ(π)n) ' Rlim((M⊗LAA/πn)⊗LA/πnB/ϕ(π)n) ' Rlim(M/πn⊗LA/πnB/ϕ(π)n)

30

The first isomorphism comes from replacing M with its resolution P •, the second isomorphism comesfrom the fact that πn⊗1 = 1⊗ϕ(π)n = 0. For the third one we note that the pro-A-complex {M⊗LAA/πn}is pro-isomorphic to {M/πn}. Indeed, the obstruction is the pro-system {M [πn]}, which is pro-zero asthe π∞-torsion is bounded by M being finitely generated.

π and ϕ(π) are non-zerodivisors and ϕ : A/π → B/ϕ(π) is flat, hence we see that A/πn → B/ϕ(π)n isflat, because π and ϕ(π) are both non-zerodivisors.

Therefore, M/πn ⊗LA/πn B/ϕ(π)n lies in D≥0 for all n ∈ N>0 and so does the limit.

Next, we treat the case where ϕ : A/π → B/ϕ(π) is faithfully flat. By the previous part we know alreadythat ϕ : A→ B is flat.

We know that ϕ is faithfully flat exactly when Spec(ϕ) : Spec(B)→ Spec(A) is surjective, and becauseA → B is flat, the image of Spec(B) is stable by generisation. Therefore it is enough to check thatSpec(ϕ) is surjective on closed points.

So let m ∈ Spec(A) be a closed point.

A is π-adically complete, therefore π ∈ m, and we have A/m⊗A B ' A/m⊗A/π B/ϕ(π) 6= 0, where theinequality is by faithfully flatness of ϕ.

We conclude that A/m⊗A B 6= 0 and hence m lies in the image of Spec(ϕ).

First, we construct an extension to an integral perfectoid algebra which is almost faithfully flat over A,then from there on we make sure to have a compatible system of p-power roots of g in a future step.

Proposition 6.2.2. Let A be a p-torsionfree regular local ring with residue characteristic p. Then thereis an extension A→ A0,∞ such that

(I) A0,∞ is an integral perfectoid K◦a-algebra, K := Qp(p

1p∞)∧

.

(II) A→ A0,∞ is almost faithfully flat (wrt. p1p∞ ). (In the meaning of lemma 3.4.2)

Proof. We can replace A by a regular local ring that is faithfully flat over it. Therefore we may assumethat A has an algebraically closed residue field k and that A is complete for the topology given by powersof the maximal ideal m ⊂ A. Let d = dim(A) be its dimension. Let W := W (k) be the Witt vectorsover k.

First we look at the unramified case, by which we mean p /∈ m2. We can then pick a basis (p, x2, . . . , xd)of m/m2, so we get an isomorphism A ∼= W Jx2, . . . , xdK. Take A0,∞ to be the p-adic completion of

A[p1p∞ , x2

1p∞ , . . . , xd

1p∞ ]. By proposition 6.2.1, A0,∞ is faithfully flat over A and thus almost faithfully

flat.

Now assume p ∈ m2 (i.e. ramified case). We can choose a regular sequence (x1, . . . , xd) generating m andget a surjection ψ : P0 := W Jx1, . . . , xdK −→→ A. Since A is regular and p ∈ m2, we get that ker(ψ) = (p−f)as ideal, where f = f(x1, . . . , xd) ∈ (p, x1, . . . , xd)

2 is a power series. As A is p-torsionfree, we get thatp - f .

Moreover, we can assume that f has no constant term. For this we note that p2 divides the constantterm of f , so we can write p− f = p− p2f0 − f1 with f1 having no constant term. Since A is p-adicallycomplete, there is a unit u ∈ A× such that p− p2f0 = u · p, so the ideal (p− f) is generated by p−u−1f1

and u−1f1 has no constant term.

Now we take A′ to be the p-adic completion of (colimm Pm)⊗P0A ∼= (colimm Pm)/(p− f) with

Pm := P0[x1pm

1 , . . . , x1pm

d ]. The map P0 → Pm is faithfully flat, so P0 → colimm Pm is faithfully flat, andso is A→ A′. Moreover, g := Φ−1(f)(x1, . . . , xd) ∈ P1, with Φ being the unique lift of the Frobenius on kto W (k) and Φ−1(f) being the power series where Φ−1 is applied to the coefficients, satisfies gp = f +phfor some h ∈ P1. g and f have no constant term, so h has none, either. Therefore h+ 1 is a unit in A′.In A′ we have thus gp = p+ ph = p(1 + h) = p · u, u ∈ (A′)×.

A′′ := A′[ 1p ], topologised by making {pnA′}n an open neighbourhood of zero, is a perfectoid Tate ring (in

the sense of definition [18, 6.1.1]) with A′′◦ = A′. Lemma 4.2.3 works likewise for perfectoid Tate rings(see also [18, 6.2.2]), so we get an element $ ∈ A′ that admits a compatible system of p-power roots and

31

which satisfies $p = p · u′ for some u′ ∈ (A′)×. ($1p∞ ) coincides with

√(p), hence it is independent of

the choice of $ or its roots.

Look at the extension A′[

1p

]→ A′

[1p , u′ 1p∞], obtained by formally extracting p-power roots of u′, and

let A0,∞ be the $-adic completion of the integral closure of A′ in this extension.

Next, we show (II). This means showing TorAi (M,A0,∞) =a 0 for i > 0 and(M ⊗A A0,∞ =a 0 =⇒ M = 0) for all A-modules M . First the map A→ A0,∞ splits as

Aα−→ A′

β−→ (A0,∞)!!γ−→ A0,∞

where by construction α is faithfully flat and γ is an injective almost isomorphism. This means that(A0,∞)!! is p-torsionfree and p-adically complete inheriting the properties from (A0,∞). Moreover, as forany m ≥ 0, A′ → (A0,∞) is almost faithfully flat modulo pm, β is faithfully flat modulo pm.

We can now apply proposition 6.2.1 to conclude that β ◦ α is faithfully flat. Together with γ being analmost isomorphism, we get the first part, namely TorAi (M,A0,∞) =a 0 for i > 0.

For the second part, take an A-module M with M ⊗A A0,∞ =a 0. Equivalently, M ⊗A (A0,∞)!! =a 0as γ is an almost isomorphism. β ◦ α is faithfully flat, therefore we can filter M and assume that

M = A/I, I ∈ A an ideal. So the hypothesis says then that for all n ∈ N, $1pn ∈ I · (A0,∞)!!, therefore

p = $pv−1 ∈ Ipn · (A0,∞)!!. By faithfully flatness of β ◦ α, we conclude that p ∈ Ipn . Hence p ∈⋂n I

pn

and we can use Krull’s intersection theorem. As p 6= 0 we conclude that⋂n I

pn 6= 0 and therefore I = A,yielding M = 0.

Proposition 6.2.3. (adjoining roots to the discriminant) Let K be a perfectoid field, K◦ ⊂ K its ring

of integers, t ∈ K◦ an element with |t| = |p| which admits a compatible system of p-power roots (t1p∞ ).

Let A be a integral perfectoid K◦-algebra, fix g ∈ A.

There is an extension A→ A∞ such that

• A∞ is a perfectoid K◦-algebra.

• A→ A∞ is almost faithfully flat modulo t (wrt. t1p∞ ).

• g admits a compatible system of p-power roots (g1p∞ ) ⊂ A∞.

Proof. Essentially we formally adjoin a perfect element T and set T = g inside A⟨T

1p∞⟩

. So let

Y := Spa(A⟨T

1p∞⟩ [p−1], A⟨T

1p∞⟩)

Bj := O+Y

(Y

⟨T − gtj

⟩)A∞ :=

(colimj∈N

Bj

)∧where the completion is t-adically.

For any y ∈ Y⟨T−gtj

⟩we have |T (y)− g(y)| ≤ |tj |, therefore tj | T − g ∈ Bj and T − g = 0 ∈ A∞. Thus

g has a distinct system of p-power roots in A∞ given by g1

pk := T1

pk .

The approximation lemma 4.4.4 for perfectoid spaces gives us a f ∈(A⟨T

1p∞⟩)[

such that

• f ] ≡ T − g (mod t1p )

• Y⟨T−gtj

⟩= Y

⟨f]

tj

⟩

32

Moreover, lemma 4.4.6 gives us a concrete description of Bj as

Bj = O+Y

(Y

⟨f ]

tj

⟩)=a A

⟨T

1p∞ ,

(f ]

g]

) 1p∞⟩

=

(colimk

A⟨T

1p∞⟩ [u

1

pk

]/(

(u · tj)1

pk − (f ])1

pk

))∧=(A⟨T

1p∞⟩ [u

1p∞]/(∀k : (u · tj)

1

pk − (f ])1

pk

))∧=

(colimk

A⟨T

1p∞⟩ [u

1p∞]/(

(u · tj)1

pk − (f ])1

pk

))∧=:

(colimk

Cj,k

)∧where Cj,k := A

⟨T

1p∞⟩ [u

1p∞]/(

(u · tj)1

pk − (f ])1

pk

). We show that A/tε → Cj,k/t

ε is faithfully flat

for ε = 1pk+1 . In Cj,k we have (u · tj)

1

pk − (f ])1

pk = 0 and tj

pk ≡ 0 (mod tε), hence (f ])1

pk ≡ 0 (mod tε).The k-fold Frobenius induces an isomorphism

Cj,k/tε = A0,∞

[T

1p∞ , u

1p∞]/(tε, (f ])

1

pk

)−→ A0,∞

[T

1p∞ , u

1p∞]/(t1p , T − g

).

A[T1p∞ , u

1p∞ ]/(t

1p , T − g) is a free A/t

1p -algebra, in particular it is also faithfully flat.

Therefore each Bj/tε is almost faithfully flat over A/tε and so is A∞/t

ε. Finally, as tε | t, we concludethat A/t→ A∞/t is almost faithfully flat.

Combining the two previous propositions we can extend a regular ring A in an almost faithfully flat way(modulo p) to a perfectoid algebra. This will then allow us to almost split the inclusion A ↪−→ B after abase change to A∞.

Corollary 6.2.4 (almost faithfully flat extension). Let A be a p-adically complete, regular local ring ofmixed characteristics (0, p) and let g ∈ A. Then there is an extension A→ A∞ such that

• A∞ is a perfectoid Qp(p1p∞ )-algebra.

• g admits a compatible system of p-power roots (g1p∞ ) ⊂ A∞.

• A→ A∞ is almost faithfully flat modulo p (wrt. (pg)1p∞ ).

Proof. From proposition 6.2.2 we get an almost faithfully flat extension A → A0,∞, with A0,∞ beingintegral perfectoid. We can then apply proposition 6.2.3 to A0,∞ with t := p to get the result.

33

6.3 Additional lemmas

In this section we cover some lemmas that contain part of the proof of the direct summand conjecture.They are stated here isolated from the rest of the proof, in order to make the proof more readable.

First we explain how we can reduce the problem to the setup of almost mathematics.

Lemma 6.3.1. Let A be a ring, Q an element in the derived category of A-modules with a mapα : Q → A[1], m ∈ N≥3, g ∈ A coprime to p. Assume we have an extension A → A∞ which is almost

faithfully flat modulo p (wrt. p1p∞ ), A∞ a perfectoid algebra and assume g admits a compatible system

of p-power roots in A∞. Denote by Q∞, α∞ the respective base changes to A∞.

If α∞/pm ∈ HomA∞(Q∞, A∞/p

m[1]) is almost zero (wrt. (pg)1p∞ ), then α/pm = 0 ∈ HomA(Q,A/pm[1]).

Proof. We proof the contraposition: Assume α/pm 6= 0.

For all k ∈ N≥3 we have that gp2 6= 0 in A/pk since g and p are coprime. A direct corollary of Krull’s

intersection theorem states that for any proper ideal I ( A of the local regular ring A we have⋂k I

pk = 0.Therefore, as gp2 6= 0, there is a k′ such that

gp2 /∈ AnnA/pm(α/pm)pk′

using the extension A→ A∞ which is almost faithfully flat modulo p , we get

gp /∈ AnnA∞/pm(α∞/pm)p

k′

where we loose one power of p by passing to almost mathematics. (Concretely gp2·x 6= 0 then (gp·x)·p 6= 0so (gp · x) is not almost zero.) As p and g are both perfect elements in A∞, this is equivalent to say

(gp)1

pk′ /∈ AnnA∞/pm(α∞/p

m)

and hence α∞/pm is not almost zero.

Lemma 6.3.2. Let A be complete regular, A∞ be a perfectoid algebra Q an A-module, Q∞ := Q⊗LAA∞,g ∈ A∞ coprime to p. Then there exists

can : HomA∞(Q∞, A∞/pm[1])→ lim

nHomA∞(Q∞, A∞〈

pn

g〉/pm[1])

which is an almost isomorphism (wrt. p1p∞ ).

Proof. Recall that for X := Spa(A∞[t−1], A∞), we set A∞

⟨tn

g

⟩:= O+

X(X⟨tn

g

⟩).

Consider the pro-system of maps{fn : A∞/p

m → A∞

⟨pn

g

⟩/pm

}.

The quantitative Riemann’s Hebbarkeitssatz 5.2.1 tells us that {fn} is an almost-pro-isomorphism (wrt.

(pg)1p∞ ) and therefore the tower of chain complexes {Cn} :=

{RHom

(Q∞, A∞

⟨pn

g

⟩/pm[1]

)}n

has

limit RHom (Q∞, A∞/pm[1]), so we can use Milnor’s exact sequence and get

0→ lim1

nH1(Cn)→ H0(RHom (Q∞, A∞/p

m[1]))→ limnH0(Cn)→ 0

which is

0→ lim1

n

{Hom(Q∞, A∞

⟨pn

g

⟩/pm)

}n

→ Hom (Q∞, A∞/pm[1])→ lim

nHom(Q∞, A∞

⟨pn

g

⟩/pm[1])→ 0

Therefore, we need to show that lim1{HomA∞(Q∞, A∞

⟨pn

g

⟩/pm)}n is almost-zero. Again by the quan-

titative Riemann’s Hebbarkeitssatz 5.2.1, the pro-system {HomA∞(Q∞, A∞

⟨pn

g

⟩/pm)}n is almost-pro-

isomorphic to the constant pro-system {HomA∞(Q∞, A∞/pm)}n which has a pro-zero lim1, thus by

applying lemma 5.1.4 we get that lim1{HomA∞(Q∞, A∞

⟨pn

g

⟩/pm)}n is almost-pro-zero.

34

6.4 The DSC in mixed characteristic

Finally we can explain the proof of the direct summand conjecture. The proof of this result is originallydue to Andre [1] and we follow the exposition of Bhatt [4].

Theorem 6.4.1 (direct summand conjecture). Let A be regular, Ai−→ B a finite extension of rings.

Then i is split as a morphism of A-modules.

Proof. We can directly reduce to the case where A is local of mixed characteristic (0, p) as the othercase has been presented in section 6.1. By lemma 6.1.1 we can further reduce to the case where A is ap-adically complete, local regular ring.

Choose g ∈ A coprime to p such that A[(pg)−1] → B[(pg)−1] is finite etale. Let A → A∞ be theextension that has been constructed in 6.2.4. Saying that A→ B splits is equivalent to showing that forthe canonical exact triangle

A→ B → Qα−→ A[1]

the boundary map α is zero. (See lemma [15, 1.2.7])

As A is p-adically complete, we can show this modulo pm for all m ∈ N>0 large enough, which reducesby lemma 6.3.1 to showing that the base change

α∞/pm := (α/pm)⊗LA A∞ ∈ HomA∞(Q∞, A∞/p

m[1])

is almost zero (wrt. (pg)1p∞ ) for m large enough.

By lemma 6.3.2 we have an almost isomorphism (wrt. (pg)1p∞ )

HomA∞(Q∞, A∞/pm[1]) ∼= lim

nHomA∞(Q∞, A∞

⟨pn

g

⟩/pm[1])

Now, we have that A→ B is finite etale after inverting pg and g divides pn in A∞

⟨pn

g

⟩, thus the base

change A∞

⟨pn

g

⟩→ B ⊗A A∞

⟨pn

g

⟩of A → B is finite etale after inverting p. By the almost purity

theorem 4.5.3 this can be dominated by an almost finite etale cover of A∞〈pn

g 〉, as explained in corollary

4.5.4, and thus it is almost split by 3.5.9. Hence, the same holds modulo pm, and the image of α∞/pm

in limn HomA∞(Q∞, A∞〈pn

g 〉/pm[1]) is almost zero (wrt. (pg)

1p∞ ), so we are done.

35

6.5 A derived version

Theorem 6.5.1 (derived version). Let A be a regular ring, let f : X → Spec(A) be a proper surjectivemap. Then the map A→ RΓ(X,OX) splits in the derived category D(A).

Proof. By 6.1.1 we can assume that A is a regular local ring. We may also assume that X is integraland f is generically finite by taking the closure of a suitable generically defined multisection. We canchoose g ∈ A coprime to p such that f is finite etale after inverting pg. We can now proceed exactly asin the proof of 6.4.1. First, we use 6.2.4 to construct an extension A → A∞ and using the exact samearguments, we arrive at

A∞

⟨pn

g

⟩/pm → RΓ(X,OX)⊗LA A∞

⟨pn

g

⟩/pm

for which we must show that it is almost split (wrt. p1p∞ ).

The base change X ×Spec(A) Spec(A∞

⟨pn

g

⟩)→ Spec(A∞

⟨pn

g

⟩) is proper and finite etale after inverting

p because g divides pn on the base Spec(A∞

⟨pn

g

⟩), and the special case discussed in lemma 6.5.2 gives

us that

A∞

⟨pn

g

⟩→ RΓ (Z,OZ) , Z := X ×Spec(A) Spec

(A∞

⟨pn

g

⟩)is almost split. We finish by noting that we have the almost isomorphism

RΓ (Z,OZ) ∼=a RΓ(X,OX)⊗LA A∞⟨pn

g

⟩(mod pm)

as A→ A∞ is an almost faithfully flat extension modulo p.

Lemma 6.5.2. Let A be an integral perfectoid K◦-algebra, S := Spec(A). Let f : Y → S be a propermorphism that is finite etale after inverting p. Then A→ RΓ(Y,OY ) is almost split.

Proof. Let B := H0(Y,OY ). B is an integral extension of A that is finite etale after inverting p,and Y is naturally a B-scheme. Almost purity 4.5.3 gives us a map B → C which is an isomorphismafter inverting p, such that A → C is an almost finite etale cover. This means C is integral perfectoidand A → C is almost split (see corollary 4.5.4). We can make a base change of f : Y → Spec(B)to Y ⊗B C → Spec(C) by 6.1.1 and assume that f is an isomorphism after inverting p. This means

the p-adic completion f can be dominated by an admissible blowup of S. Set Sη := Spa(A[p−1]) tobe the associated affinoid perfectoid space. The natural map (Sη,O+

Sη)→ S factors then through every

admissible blowup of S. In particular, we get (Sη,O+Sη

)→ Y → S and taking cohomology of the structure

sheaf gives Ag−→ RΓ(Y,OY )

h−→ RΓ(Sη,O+Sη

).

Scholze’s vanishing theorem 4.4.7 gives us that RΓ(Sη,O+Sη

) is almost zero in all degrees i 6= 0 and thath ◦ g is an almost isomorphism. Hence g is almost split.

36

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[20] E. Witt. Zyklische Korper und Algebren der Charakteristik p vom Grad pn. Journal fur reine undangewandte Mathematik, 176:126–140, 1937.

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