foundations of perfectoid geometry, i

Transcription

1 foundations of perfectoid geometry, i ALESSANDRO MARIA MASULLO 1 Abstract We introduce Tate-perfectoid rings (the rings we ve been calling perfectoid, so far) and perfectoid rings, and relate the two notions. We discuss the tilting equivalence in the generality of both perfectoid rings and Tate-perfectoid rings, and describe the equivalence of the finite-étale sites over a Tate-perfectoid ring and its tilt. We globalize the results and introduce perfectoid spaces, establishing the basic foundational features. We discuss all results at the aforementioned level of generality, explaining the motivation for the choices made throughout, in light of applications we have in mind. These notes are organized as follows. In 1 we discuss a slightly nonstandard perspective on the Zariski Nagata purity theorem in [SGA2], explaining the meaning of Falting s Almost Purity and how to regard it as a purity result. We give motivation and describe the nature of this work, while announcing the results contained in the second part of it. In 2 we discuss the needed preliminaries on perfectoid rings. We ll explain how to relate the notion of perfectoid ring and Tate-perfectoid ring bearing in mind the geometric picture. In 3 we discuss the tilting equivalence in the generality of perfectoid rings stressing the fact that almost mathematics does not come into play in the proof, which, rather, relies on a core property of perfectoid rings we discuss and ensure the deformation theory machinery, reviewed in B, works. On the other hand, in this section we prove the tilting equivalence for Tate-perfectoid rings deducing the result from the equivalence for perfectoid rings, pointing out where almost mathematics (though mildly) comes into the picture and why. In 4 we discuss finite étale tilts, set up the necessary preliminaries for establishing the equivalence of the almost finite étale sites on perfectoid rings and finite étale sites on Tate-perfectoid rings, and on their tilts, respectively. We prove the almost purity theorem for Tate-perfectoid rings in positive characteristic, and draw consequences from it. We finally adress completely the equivalence for perfectoid fields, recovering the Fontaine- Wintenberger correspondence. In 5 we finally introduce perfectoid spaces, exploiting the full power of the generality adopted throughout. We globalize the tilting functor and prove that the category of perfectoid spaces (upon fixing a base perfectoid space) contains fibered products. In 6 we define the small étale site on a perfectoid space, and prove the global equivalence of the étale site of a perfectoid space and its tilt, achieving the geometric form of Faltings almost purity. In the first appendix A we discuss some loose ends, recollecting some needed tools from almost mathematics, and explaining a couple of different proofs to results discussed in previous sections. In the second appendix B, we recollect the basics of deformation theory via the cotangent complex, for convenience of the reader. The purpose of this handout is to make the setup and foundational results on perfectoid spaces uniform and uniformly treated at this level of generality, with the aim of providing a unified and consisent reference. 1 The purpose of these notes is to give a uniform treatment of the theory, given the various evolutionary steps the theory encountered so far. I hope it helps. Comments to a.m.masullo@stanford.edu. Lecture notes covering the prerequisites, available at conrad/perfseminar/. Notes explaining/expanding the proof of finiteness of p-adic étale cohomology for proper rigid-analytic varieties, coming soon under the title Foundations of Perfectoid Geometry, II. Proof of Poincaré duality for p-adic cohomology of proper rigid-analytic varieties, also included in Foundations, II.

2 Contents Contents 2 1 Introduction 3 2 Perfectoid rings 17 3 The tilting equivalence over perfectoid rings 34 4 Finite étale extensions and finite étale tilts 57 5 Perfectoid spaces 68 6 Small étale sites and tilting 83 A Loose ends on Almost Mathematics 101 B The cotangent complex 108 References 155

3 foundations of perfectoid geometry, i 3 1 Introduction If we let X be a locally noetherian scheme, and Y X a closed subscheme, in case we have a natural isomorphism πét 1 (Y ) πét 1 (X), we have an equivalence of categories between étale covers of Y and X respectively. Grothendieck s strategy to tackle situations along these lines and decide whether such equivalence holds or not, is to first compare the category of étale covers of X with that of a neighbourhood U of Y in X. Then compare the category of étale covers of U with that of the formal completion of X along Y. As a last step, compare the category of étale covers of the formal completion of X along Y, with that of Y. The very first step relies on the idea that if Y is large enough, then passing from étale covers of U to étale covers of X should involve addition of high codimension subvarieties, which will inaffect the category of étale covers. We review the Zariski - Nagata purity theorem in SGA2 from a slightly non-standard perspective. We shall discuss its proof in full detail, and then describe the nature of Faltings method of almost étale extensions as being, actually, an instance of purity in a wider sense. 1.1 Aims of this work and announcement This introduction will be used in later lecture notes for the purposes of producing significant examples, first, and motivate the introduction of what I call the Iwasawa topology on proper formal Spf(A inf )-schemes X, A inf being Fontaine s ring W (O ), O being a perfectoid ring, and O being its tilt. The site (X/A inf ) Iw will be closely related to Iwasawa theory, and has the feature of yielding a topos theoretic cohomology theory specializing via canonical integral comparison isomorphisms to: RΓ crys (X O/p /A crys ), RΓét (X O O Frac(O), Z p ), RΓ crys (X O O k/w (k)), RΓ dr (X O O k) and, notably: RΓét (X O O Frac(O ), Z p ). Bhatt, Morrow and Scholze already produce such cohomology theory in [BMS], by constructing a complex of A inf -modules AΩ on X Zar, whose hypercohomology does the job. Their approach is concrete and relies on very explicit local calculations which suggest indeed that one should be able to assign a good A inf -version of the infinitesimal site (this latter being bound not to work at all, though) recovering AΩ site theoretically. It turns out that, via the natural topos-theoretic projection ε : (X/A inf ) Iw X Zar, we do have AΩ Rε(O XIw ) canonically and naturally in X, and the specialization theorems in [BMS] can be reproved geometrically. Thanks to (X/A inf ) Iw, Faltings almost purity will be indeed a purity theorem, with no intervention of the theory of almost étale extensions, which, as we shall see at times throughout this volume and explicitly explain in the second part, is ill suited to work integrally. We choose to explain here the original theory of perfectoid spaces in the setup given by Peter Scholze and going back to Faltings, ie. making use of almost mathematics, but we ll limit the use of almost mathematics to the minimum. We shall address the integral theory of perfectoid spaces in the second part of these notes, where we shall prove an analogue of Theorem and Theorem , in the context of p-adic formal schemes and their perfectoid Iwasawa covers. There will be no ad hoc of almost mathematics, but rather we shall give a geometric interpretation to it, which will clarify what

4 4 alessandro maria masullo goes wrong in its usage on the integral level, and led Faltings to introduce almost mathematics in the first place in order to obtain rational comparison isomorphisms. This is also why we call these notes Foundations of perfectoid geometry : as they aim at not being limited to the world of rigid geometry, in terms of applicability. The ultimate results will be applicable to proper smooth schemes over Spec(Z[q ]), q being a parameter to be specialized from time to time, and will be made available by formal geometry. To guide the reader through the theory, we first address the theory of perfectoid spaces, for then moving to formal and algebraic geometry in the second part. The naive slogan these notes aim to convey is that (integral) perfectoid geometry enables one to relate geometry locally at primes p and p, with possibly p p. Along the way, we shall establish a number of foundational results for the p-adic étale cohomology of (proper) rigid analytic varieties over characteristic 0 complete algebraically closed nonarchimedean fields C, including Poincaré duality and the Künneth formula, both discussed in the second part together with a new proof of finiteness of p-adic étale cohomology, the only existing one being given by Peter Scholze in his p-adic Hodge Theory paper ([Sch2]). Let s go back to the introduction, and review purity a la Grothendieck. One can safely say that our interpretation of Faltings purity is modeled on what follows. 1.2 Comparison of Ét(Y ) and Ét( X) We fix our setup for the whole of the introduction as being that of a locally noetherian scheme X,I an ideal of definition of a closed immersion Y X. We shall denote by X the formal completion of X along Y. Let n be a positive integer. We set Y n to denote the scheme (Y, (O X /I n+1 ) Y ), where we endow the structure sheaf (O X /I n+1 ) Y with the discrete topology (I is of finite type). The Y n s form a directed system of schemes, and we denote by X its colimit formal scheme, ie. the formal completion of X along Y. By [SGA1, 1.8.3] we know that the assignement of a locally finite type X-formal scheme S, is equivalent to the assignement of an inductive system (S n ) of Y n -schemes of finite type, together with isomorphisms of schemes S n S n+1 Yn+1 Y n. Moreover, S is an étale cover of X if and only if Sn is an étale cover of Y n for all n 1. It s an easy exercise to check that the base change functor: Ét(Y n+1 ) Ét(Y n ) is an equivalence of categories for all n 1, and that we have the following: Proposition The natural functor: is an equivalence of categories. Ét( X) Ét(Y ) The proof of the above Proposition is an immediate application of the theory of the cotangent complex. Proof. For any étale cover a : Y Y, the cotangent complex L Y /Y vanishes in D qcoh (O Y ). It is, by definition, L OY /a 1 O Y, and its vanishing yields unique (up to unique isomorphism) unobstructed deformations of Y Y to some Y 1 Y 1, which can be arranged to be flat and of finite type. Flatness ensures vanishing of L Y 1 /Y 1 L O Y (O Y 1 /J O Y 1 ) in D qcoh (O Y ), which 1 yields vanishing of L Y 1 /Y 1. By induction, we conclude.

5 foundations of perfectoid geometry, i 5 The upshot is that, in our setup, the categories of étale covers of Y and of X are always equivalent. The reason of this is to be seen in the vanishing of the full cotangent complex of an étale morphism of schemes. Such vanishing, however, occurs in more general situations where no finiteness assumption is being given. We briefly digress on this. 1.3 Digression on the vanishing of the full cotangent complex For the convenience of the reader, we briefly digress on the relation between étaleness of a ring map, together with weaker notions, and vanishing of the full cotangent complex. This digression, as the whole of the introduction, is elementary and won t be used in the sequel, and the reader may want to skip it at a first reading. It will be nonetheless useful to familiarize with the usage of the cotangent complex, which will be deepened in Appendix B, and will be absolutely crucial in the second volume of these notes. We first discuss an example of a formally étale ring map whose cotangent complex is nontrivial (not isomorphic to a trivial complex in the derived category). We define R := Z[t r, r Q >0 ], and consider the ideal generated by the indeterminates indexed by all positive rational numbers, say I. Here we require t r t r = t rr, so our R is not actually a polynomial algebra over Z, but rather a quotient by a great number of relations. The restriction on r being always positive ensures that such relations are not too many and 1 0 in R, so R is nonzero. Then, we have I 2 = I and therefore R R/I is formally étale (check!). We now consider A := R/(t r t r ), where r r are two fixed positive rationals. Again A is nonzero. We call J := I mod (t r t r ). A base change of a formally étale morphism is formally étale, so A A/J is a formally étale ring map. J is not flat as an A-module, as the multiplication map J A J J sends t r t r to zero. It s easy to show we have the following exact injection of (A/J)-modules: 0 ker(j J J) H 2 (L (A/J)/A ) and we conclude A A/J is an example of a formally étale ring map with non-vanishing cotangent complex. Such non-vanishing had to appear in degree 2, as it is an easy task to show that a formally étale ring map has vanishing truncated cotangent complex. Proposition Let u : A B be a ring homomorphism. The following conditions are equivalent: (1) u is formally étale. (2) τ 1 L B/A = 0 in D(B). If we require u to be of finite presentation, then being étale is equivalent to the vanishing of the full cotangent complex. Proof. We show that if τ 1 L B/A = 0, then u is formally étale. Indeed, we must show that for any A-algebra C and nilpotent ideal I of C, given an A-algebra map v 0 : B C/I there exists a unique A-algebra map v : B C lifting v 0, that is, we have a bijection Hom A (B, C) = Hom A (B, C/I). We may and do assume I is square-zero, by an easy induction.

6 6 alessandro maria masullo Via pullback, we reduce our consideration to the case B = C/I and I is a B-module. Recall (see B) that the obstruction o(v 0 ) to existence of v lies in Ext 1 B(L B/A, I), which is trivial, since τ 1 L B/A is trivial. Hence v exists. Any other v lifting v 0 is reached from v via action of Ext 1 B(L B/A, I) = Ext 1 B(τ 1 L B/A, I), which is again trivial, and hence v is unique up to automorphism. The automorphism group of v is Ext 0 B(τ 1 L B/A, I) = Ext 0 B(L B/A, I) = Hom B (Ω 1 B/A, I), which is again trivial, and we conclude v is unique (up to unique isomorphism). We show the converse. For every B-module M we have Ext 1 B(τ 1 L B/A, M) Exal A (B, M) the right side denoting the group of A-algebra extensions of B by M. We claim that if B is formally étale over A, the latter group is trivial. Indeed, take µ : J/J 2 M a B-module homomorphism such that, under the above bijection, a given extension: is isomorphic to the pushout of the extension: 0 M B B 0 0 J/J 2 A[B] B 0 along µ. Since B is formally étale over A, it is in particular formally smooth, so this latter extension is trivial in Exal A (B, J/J 2 ). Then, B is the trivial extension of B by M in Exal A (B, M), implying that Ext 1 B(τ 1 L B/A, M) = 0 for all B-modules M. The same way formal étaleness implies that Ext 0 B(L B/A, M) = 0 for all B-modules M. Hence we have that RHom B (τ 1 L B/A, M) = 0 for all B-modules M, implying that τ 1 L B/A = 0 in D(B), as desired. We now show that if, in addition, we require beforehand u to be locally of finite presentation, then we have that u is étale if and only if L B/A = 0 in D(B) and u is locally of finite presentation. We set up the following pushout diagram: A u B u B B A B We have B A B = B L A B, so by flat base change we have that: L B/A B (B A B) L B A B/B is an isomorphism in D(B). Consider the maps: B α B A B β B

7 foundations of perfectoid geometry, i 7 where α(b) = b 1 and β is the multiplication. In particular βα = id. Now we consider the corresponding fundamental distinguished triangle: L B A B/B B A B B L B/B L B/B A B L B A B/B[1] B A B B We now use the quasi-isomorphism: L B/A B (B A B) qi L B A B/B due to flatness of B A B over B, to get: L B/A B (B A B) B A B B L B/B L B/B A B L B/A B (B A B)[1] B A B B and now we use the fact that βα = id, to get that this is actually the following: L B/A L B/B L B/B A B L B/A [1] We are left to check that L B/B A B is trivial. So far we have just used formal unramifiedness of u, but now we observe that our assumption on u being étale ensures that the diagonal : Spec(B) Spec(B) A Spec(B) is an open immersion, thus yielding the desired vanishing. The converse is clear, as if we have L B/A = 0 then in particular τ 1 L B/A = 0, which, together with the finite presentation assumption on u, implies u is an étale ring map. In order to single out a class of morphisms of rings, and more in general sheaves of rings, hence schemes X Y, satisfying vanishing of the full cotangent complex, the key observation is that formation of the cotangent complex L X/Y is stable under flat localization. We claim that if a morphism of scheme f : X Y is flat with flat diagonal, then it has vanishing full cotangent complex. We call this class of morphisms weakly étale. This class of morphism plays an essential role in the introduction of the proétale topology for schemes, and we refer the interested reader to [BSch]. We will review the foundations of the pro-étale topology for adic spaces and schemes in the second part of these notes, discussing the étale and pro-étale topology on p-adic formal schemes as well. Proposition Let u : A B be a weakly étale ring map. Then L B/A 0. Proof. We re required to check L B/B A B 0 in D(B), though now we have no finite presentation assumption on A B(!) In fact a priori we do not even know τ 1 L B/A = 0 because we are not assuming A B is formally étale. We claim: B L B A B B B. If so, we conclude by flat base change. We know that B is (B A B)-flat, so we need to check: B B B A B B. Let J be the kernel of the surjective flat ring map B A B B, and upon applying ( ) B A B B to the following short exact sequence of (B A B)-modules: we obtain: 0 J B A B B 0 0 J B A B B B B B A B B 0 Now the multiplication map J B A B B JB is injective by flatness of B over B A B, and since JB = 0, we conclude.

8 8 alessandro maria masullo Remark We remark that in [BSch], the class of weakly étale ring maps generates the small pro-étale site on a scheme (coverings being given by fpqc coverings). The pro-étale topology can be defined for adic spaces possessing a well behaved étale site, as well as for perfectoid spaces, whose definition will be engaged later. We d like to stress here that weakly étale maps have the feature of having vanishing full cotangent complex, as shown above. Such vanishing is a property which, as we shall discuss, is enjoyed by any map between perfectoid rings in a suitable sense, and is at the core of the whole theory. We first need a couple of definitions. 1.4 Comparison of Ét(Y ) and Ét(U) Definition The setup being given, we consider the pair (X, Y ). (1) We write Lef(X, F ) and say (X, Y ) satisfies the Lefschetz condition if and only if for evey open U in X containing Y, and every coherent locally free sheaf F on U, the natural map: Γ(U, F ) Γ( X, F ) is an isomorphism. (2) We write Leff(X, Y ) and say (X, Y ) satisfies the effective Lefschetz condition, if and only if we have Lef(X, Y ) and, in addition, for every coherent locally free sheaf F on X there exists an open neighbourhood U of Y and a coherent locally free sheaf F 0 on U, coming with an isomorphism F 0 F. These two conditions are fulfilled in two important examples. Proposition Let A be a noetherian ring, and let ϖ rad(a) be an A-regular element in the radical of A. We assume A is a quotient of a regular local ring, and that A is ϖ-adically complete. Let X := Spec(A) and Y := Spec(A/ϖ). We set x := Spec(A/rad(A)), and define X := X {x}, Y := Y {x}. Then: (1) If for every prime ideal p of A with dim(a/p) = 1 we have depth(a p ) 2, then we have Lef(X, Y ). (2) If, in addition, for every prime ideal p of A containing ϖ and such that dim(a/p) = 1 we have depth(a p ) 3, then we have Leff(X, Y ). Sketch of proof. We won t discuss the full proof, but rather mention that part (1) goes through showing that if we let U to be an open neighbourhood of Y in X and E a locally free O U - module, we set Z := X Y and let j : U X the canonical open immersion, then j (E ) is a coherent O X -module. This is equivalent to showing that H i Z(E ) is coherent in degrees 0 and 1, where E is a coherent extension of E to X. To achieve this one uses the fact that if X is a locally noetherian scheme, Y a closed subset of X and F a coherent O X -module, and if X has locally a dualizing complex, then as soon as one has, for all x X Y : H i c(x) (F x ) = 0 for i Z and c(x) := codim({x} Y, {x}), then one has that the cohomology sheaf H i Y (F ) is coherent. This is [SGA2, Thm. 2.1, Exp. VIII]. To see that the above consitions are met by E, one first shows the complement U of Y in X is a union of finitely many closed points. Since U is open in X, one calls Z the closed complement, defined by an ideal I. Since Z Y = {x}, the ring A/(I + ϖa) is Artin, and

9 foundations of perfectoid geometry, i 9 we re done. Thanks to this and the fact that A is a quotient of a regular ring, as soon as p U, given by the ideal p A, is such that c(p) = 1, then dim(a/p) = 2. Since E is locally free, then one has that for all points p in the support of E : depth(e p ) = depth(o U,p ). One puts everything together by observing that if p U satisfies c(p) = 1, we have: depth(e p) = depth(e p ) = depth(o U,p ) = depth(a p ) 3 2 = 1. We have that the natural map Γ(X, j (E )) Γ(U, E ) is an isomorphism. Now set F := j (E ), which is coherent and of depth 2. Then if we denote by f : X X the canonical immersion, R i f (E ) are coherent for i = 0, 1. It follows the natural map: Γ(X, F ) Γ( X, F ) is also an isomorphism, and since it factors through the natural isomorphism Γ(X, F ) Γ(U, E ), it follows the natural map: is an isomorphism, whence (1) is proved. Γ(U, E ) Γ(X, Ê ) To show (2), we take E to be a coherent locally free sheaf on X, and we use formal GAGA to check that it s algebrizable, ie. it is isomorphic to the formal completion of a coherent O X - module E. Then it s easy to see E is locally free on a neighbourhood of Y, which achieves (2). Some details omitted. The above basic cases are the key to show that if one has Lef(X, Y ), then for any open neighbourhood U of Y in X the functor: Ét(U) Ét(Y ) is fully faithful, and if Leff(X, Y ) holds, then for every étale cover T Y, there exists an open neighbourhood U of Y and an étale cover T U such that T U Y T. As an immediate consequence, if Lef(X, Y ) holds and the notation being the same as above, we have that the natural map: πét 1 (Y ) πét 1 (U) is surjective for all such U, and we can recover πét 1 (Y ) as the inverse limit of the πét 1 (U) s over all such U s (upon fixing base points on Y and X). We begin with the following definition. 1.5 Comparison of πét 1 (U) and πét 1 (X) Definition Let X be a scheme, and Z a closed subset of X. We set U := X Z. We say the pair (X, Z) is pure if and only if for every open V X, the functor: assigned by: is an equivalence of categories. Ét(V ) Ét(V U) V V V (V U) If X = Spec(A) is the spectrum of a local noetherian ring A, and rad(a) is the radical of A, with x = rad(a) the closed point of X, we say A is pure if the couple (X, {x}) is.

10 10 alessandro maria masullo We finally get to the statement of the purity theorem. Theorem ([SGA2, Thm. 3.4, Exp. X]) Every noetherian regular local ring of dimension 2 is pure. Every noetherian local ring of dimension 3 which is a complete intersection is pure. The first part of the statement is known as the Zariski-Nagata purity theorem. The first key input which goes in the proof is the following Lemma. Lemma Let X be a locally noetherian scheme, and U an open subscheme in X, j : U X being the canonical open immersion. The following conditions are equivalent: (1) For all open subschemes V X, if we set V := V U, the functor F F V from the category of locally free coherent O V -modules to the category of locally free coherent O V -modules, is fully faithful. (2) The natural map O X j (O U ) is an isomorphism. (3) For all z Z, we have depth(o X,z ) 2. The proof is easy, and left to the reader. Upon establishing the fact that purity is insensitive of faithfully flat extensions, and by means of the above Lemma 1.5.3, we can reduce the proof of Theorem to the case the noetherian regular local ring A is complete. At this point one has to show the following. Lemma Let A be a noetherian regular local ring, and let ϖ rad(a) be an A-regular element. We assume A is complete with respect to the ϖ-adic topology, and that A is a quotient of a regular local ring. We set B := A/ϖA. (1) If for all prime ideals p of A such that dim(a/p) = 1, we have depth(a p ) 2, then if B is pure so is A. (2) If for all prime ideals p of A such that dim(a/p) = 1, we have depth(a p ) 2, then if A p is pure whenever ϖ is not contained in p, and if depth(a p ) 3 if ϖ p, then if A is pure so is B. The proof is a straightforward combination of the results in 1.4, and is left to the reader. The above Lemma is [SGA2, Lemma 3.9, Exp. X]. We ll comment on this Lemma soon, but first let s finally achieve Theorem Proof of Theorem We show (1) by induction on the dimension. Let A be a noetherian regular local ring of dimension 2. Set X := Spec(A), x := rad(a) and X := X {x}. We have depth(a) = 2. We apply Lemma to the pair (X, {x}), and hence the functor Ét(X ) Ét(X) is fully faithful. We let f : W X an étale cover defined by a coherent locally free étale O X -algebra A := f (O W ). Calling j : X X the canonical immersion, we claim B := j (A ) is a coherent O X -algebra. We now have that the depth of B at x is 2, as it s the direct image of an O X -module supported on X {x}. Since A is a regular ring of dimension 2, we have: pd(b) + depth(b) = dim(a) = 2. It follows the projective dimension of B is zero, hence B is projective, hence free. As a result, B defines a finite flat cover of X. The non-étale locus of such cover in X is closed and defined

11 foundations of perfectoid geometry, i 11 by a principal ideal: the discriminant ideal of B/A. By design, it s contained in x and hence it s empty, since dim(a) = 2. Suppose, now, A is a noetherian regular local ring of dimension n 3. We assume (1) is settled for dimension less than n. Without loss of generality, we can assume A is complete. Let ϖ rad(a) such that its image in rad(a)/rad(a) 2 is nonzero. Then, B := A/ϖA is a noetherian regular local ring of dimension n 1, and hence (1) holds for B, as n 1 2. We conclude by Lemma 1.5.4, which we can apply since A is complete. We now show (2). Let s assume there exists a noetherian regular local ring B and a B-sequence (ϖ 1,..., ϖ k ) such that A B/(ϖ 1,..., ϖ k ). We proceed by induction on k. If k = 0, we conclude by (1). We assume the result is settled for all k < k, k 1. Let C := B/(ϖ 1,..., ϖ k 1 ) so A C/ϖ k C, and ϖ k is C-regular. By inductive hypothesis, C is pure, and it s enough to check Lemma applies. Since dim(c) 4, for all prime ideals p of C such that dim(c/p) = 1 we have depth(c p ) 3. Moreover, C p is a complete intersection for k k 1. By inductive hypothesis, C is pure, as desired. As a direct consequence, we easily deduce the following: Theorem Let X be a connected locally noetherian scheme, and Y closed in X. We assume we have Leff(X, Y ), and that for every open neighbourhood U of Y and every x X U, the local ring O X,x is regular of dimension 2 or a complete intersection of dimension 3. Then: πét 1 (X) πét 1 (Y ). In order to re-gain the above results in the context of p-adic formal schemes, a new notion of fundamental group will be needed, and will be offered by the introduction of the Iwasawa site. Faltings almost purity theorem will take the form of an actual purity theorem along the above lines. We, for the moment, stick to Faltings original approach and make use of his theory of almost étale extensions. We therby clarify the sense in which we can regard Faltings almost purity as a purity result, in the following section. 1.6 Presque pureté In this section we explain how to regard Faltings almost purity theorem as, indeed, a purity theorem, in light of the preceding sections of the introduction. We fix a nonarchimedean field k of mixed characteristic (0, p), equipped with a nontrivial rank 1 valuation. We call O k the valuation ring of k, and let ϖ be a nonzero topologically nilpotent element in O such that ϖ p p in O. Call, for each integer n 1, k n := k(ϖ 1/pn ). Set k 0 := k, and we call O n the ring of integers of k n, for all n 0. We call: R n + := O n [T ±p n 1,..., T r ±p n ] and R n := R n + [1/ϖ]. Since R n is O n -smooth, it s regular, and the transition maps R n R n+1 are all finite étale. Let S 0 + be a finite and normal R+ 0 -algebra, and assume that S 0 := S 0 + [1/ϖ] is finite étale as an R 0 -algebra. Essentially, we re given a directed system (R n ) n 0 of finite étale R 0 -algebras, together with a direct system (R + n ) n 0 of integral structures which are, by design, not finite étale as R algebras, as their reductions modulo ϖ are ramified, but they are upon inverting ϖ.

12 12 alessandro maria masullo S 0 is given as being finite étale away from the locus {ϖ = 0}, and if we call S n + the R n - normalization of S 0 + R + R n +, we have that S n := S n + [1/ϖ] is R n -finite étale, and the map R n + 0 S n + is, instead, ramified along the locus {ϖ = 0}. Note that this is exactly the ramification locus of S 0 + over R+ 0, and we re considering its pullback to S+ n for all n 0. Let s give a measure of such ramification. We can consider the cotangent complex L S + n /R + n L Sn/R n 0 in D (S n ) for all n 0, and observe that: since R n S n is étale, for all n 0. On the other hand, formation of the cotangent complex is stable under flat localization, and hence: L Sn/R n L S + n /R + n R + n R n for all n 0. Since each one of the S + n is R + n -finite, then L S + n /R + n is represented in D (S + n ) by a complex of coherent S + n -modules (note that all the rings R + n are coherent), and in particular has S + n -finitely generated cohomology, implying that sufficiently large powers of ϖ kill the cohomology groups of L S + n /R + n in D (S + n ), for each n 0. More precisely, fix n 0. S + n S n is flat, and since each cohomology group of L S + n /R + n is S + n -finitely generated, then Ann S + n (H i (L S + n /R + n ))[1/ϖ] = Ann S n (H i (L S + n /R + n [1/ϖ]) = S n, i 0. It follows that, calling m n the radical of ϖ in S n +, all cohomology groups of L S + n /R n + are supported on V (m n ). Let s call X n := Spec(R n + ), j n : U n := Spec(R n ) X n, and likewise X n := Spec(S n + ) and j n : U n := Spec(S n ) X n. We eventually call X the affine scheme given by lim X n, X := lim X n, and likewise for U and U. We call i n, i n the closed immersions of the complements of U n in X n and U n in X n respectively. Likewise, we denote with j, j, i, i the obvious remaining morphisms. We form the triangles in D qcoh (X n): Rj n! j nl X n /X n L X n /X n i n Li nl X n /X n Rj n! j nl X n /X n [1] thus deducing a quasi-isomorphism: L X n /X n i n Li nl X n /X n i n i nl X n /X n in D qcoh (X n) (note that L X/Y is, by very design using its simplicial representative in the category of complexes by the Dold-Kan correspondence, represented by a complex of flat O X - modules!). Likewise, L X /X i i L X /X in D qcoh (X ). Formation of the cotangent complex of a morphism of sheaves of rings commutes with direct limits (of sheaves of rings), and hence we obtain: L X /X lim i n i nl X n /X n in D qcoh (X ). Let s study L X /X by first specializing to a simple case.

13 foundations of perfectoid geometry, i 13 Let s assume we were in the good situation in which for some n, the ramification at the generic point of the locus V (m), with m = lim m n = (ϖ, ϖ 1/p,, ϖ 1/pn, ), is actually zero. Then the Zariski-Nagata purity theorem tells us that the ramification locus of X n over X n has to be pure of codimension 1. Since there is no ramification at the codimension 1 points (!) we deduce that there is no ramification, that is, X n X n is étale. Hence L X n /X n 0 in D qcoh (X n). In particular, L X /X 0 in D qcoh (X ). The almost purity theorem says that this result extends to the limit, that is, X X is étale in the sense of almost mathematics, hence L X /X has almost zero cohomology. Since ϖ is S + n -regular for all n 0, then m n S + n L X n /X n 0, which implies m S + L X /X 0, meaning exactly that L X /X has almost zero cohomology. We make the above ideas more concrete on an example, which will also shed light on a prototypical version of a tool which will be quite crucial all over the theory: a lemma of Gabber and Ramero on henselian approximation. Example We call R n := k n = Q p (p 1/pn ), and S n := k n = k n (p 1/2p ), for all n. We set R + n := O n, and S + n := O n, meaning of the notation being as before. We see that S + n = R + n [x]/(x 2 p 1/pn+1 ), n 0 and since p 1/pn+1 (x 2 p 1/pn+1, 2x), then p 1/pn+1 annihilates Ω 1 S + n /R + n = S+ n dx 2S n + x dx R n + [x] (x 2 p 1/pn+1, 2x). We have ring maps: yielding a triangle: R + n R + n [x] S + n L R + n [x]/r + n R + n [x] S+ n L S + n /R + n L S + n /R + n [x] L R + n [x]/r + n [1] R + n S+ n which is, in D(R + n ): Ω 1 R + n [x]/r + n R + n [x] S+ n [0] L S + n /R + n L S + n /R + n [x] I n/i 2 n[ 1] where I n := (x 2 p 1/pn+1 ) R + n [x]. It follows that if we let n grow arbotrarily large, we obtain, notation as before, a quasi-isomorphism in D(S + ): Ω 1 R + [x]/r + R + [x] S+ [0] L S + /R + which, in particular, reads: L S + /R + Ω1 S + /R + [0]. The ideal m := (p 1/pN ), which singles out the support of the cohomology of L S + /R + on Spec(S ), + kills Ω 1 as predicted. S /R, + + Remark We remark an important fact. In this example, it turns out that S is a perfectoid R -algebra. However, the cotangent complex L S + /R + does not vanish! We re lucky, and such cotangent complex is concentrated in degree 0 in this case. We observe that: Ω 1 S + /R + R + (R + /p) Ω 1 (S + /p)/(r + /p)

14 14 alessandro maria masullo and the right side is zero. Indeed, if R S is a map of F p -algebras such that on both R and S the pth power map is surjective, then Ω 1 S/R 0. Indeed, you can write every s S as s p, and Ω 1 S/R is R-linearly spanned by the derivations ds, s S, ie. by ds p = 0, s S. This is a toy case in which we observe that, calling S := S + and R := R +, we have: L S/R R (R/p) 0. This implies, in particular, that the (derived) p-adic completion of L S/R is trivial (ie. analytic cotangent complex of R S vanishes). In our case this just reads Ω 1 S/R 0. Remark In Example we noted how the cotangent complex captured the ramification at the generic point of the locus {ϖ = 0} in X over X. In fact, we can develop the following little idea: define the (generalized) notion of discriminant of a finite locally free morphism of schemes f : X Y of constant rank n in terms of L X/Y, guided by the following fact: disc X/Y Fitt 0 (H 0 (L X/Y )) = Fitt 0 (Ω 1 X/Y ), the left side being the discriminant of X Y, and the right being the 0th fitting ideal of Ω 1 X/Y. For a geometric construction of disc X/Y, recall that under the above assumptions we may define a trace form (which can be defined more in general for quasi-finite flat separated morphisms): Tr f : f (O X ) O Y and using the O Y -structure on f (O X ) we get a bilinear pairing: inducing a bilinear pairing: f (O X ) OY f (O X ) O Y, n f (O X ) OY n f (O X ) O Y, whose image is a locally principal ideal sheaf on Y, disc X/Y. For example, if f is étale, then, by computing disc X/Y fiberwise we see disc X/Y O Y. If Y is a Dedekind scheme, then we recover the classical notion of discriminant in algebraic number theory. We claim disc X/Y := det L X/Y recovers the above definition whenever f : X Y is finite locally free of constant rank (the reader is encouraged to check this on his own: one needs to make sure L X/Y is a perfect complex, which is the case if f is locally a complete intersection, or if f is smooth, syntomic, étale. In all these cases L X/Y is a perfect complex of finite locally free O X -modules concentrated in degrees 1, 0). This gives motivation to consider, in any event, the full cotangent complex when studying ramification, as in Example 1.6.1, and so we ll do. In these notes, we explain the intervention of two key tools which are at the very core of the classical theory of perfectoid spaces. One is the theory of almost étale extensions, and particularly Faltings almost purity in the positive characteristic case. As the reader will progressively realize, the almost purity in the zero characteristic case is proved by means of an henselian approximation process we shall explain at the appropriate moment in 6, which Gabber and Ramero prove as a generalization of Elkik s [Elkik, Thm. 7] to the non-noetherian situation. Its usefulness trascends the applications to almost mathematics. One can regard such principle as a vast generalization of Krasner s Lemma, and we now discuss an easy example to explain how, already in the following basic case, Krasner s Lemma intervenes substantially. the

15 foundations of perfectoid geometry, i 15 Remark The setup being as in Example 1.6.1, the tilt C of C := lim k n, comes together with its integral structure C 0. Recall that C 0 lim C 0 /p x x p as multiplicative monoids. We shall soon review the definition of the multiplicative map, and of the ring structure on both sides, whose underlying monoid structure is preserved by such map. The natural ring map: C 0 C 0 /p given by the projection onto the first component, allows us to lift p to an element t of its kernel, so that: C 0 /t C 0 /p Z p [p p ]/p. Following the philosophy of replacing p with t, we have: C 0 /t F p [t][t p ]/t. As a consequence, C = F p ((t))(t p ). Let us focus on the tilting operation ( ) in this case, by attempting a hands-on definition of the tilt of any finite extension C of C, which we denote C. Again the philosophy is replacing p with t. Let us assume C is given by adjoining the roots of some polynomial, say x 2 p 1/p, assuming p 2, as in our Example Then C should be given by adjoining the roots of x 2 t 1/p. Clearly, such assignment is manifestly not well defined. C is far from being uniquely determined by C, as C is given as well by adjoining the roots of x 2 + (p 1)t 1/p, whereas by no means the extension of C given by adjoining the roots of x 2 + (p 1)p 1/p should be C. Instead, let s consider the polynomials x 2 (p 1/p ) 1/pn = x 2 p 1/pn+1, for n 0. For each integer n 0, we have that the splitting field C n of x 2 p 1/pn+1 and C n+1 of x 2 p 1/pn+2 will be closer and closer, and eventually stabilize by Krasner s Lemma. We can eventually assign: ) C := (lim C 0 n/p [1/p] C (t 1/2p ). Remark We conclude the introduction with one last comment about purity. Let s step back to Lemma If we consider a perfectoid ring A, to use the notation of the Lemma (the notion of perfectoid ring to be introduced in Definition and the discussion in 2) then A basically satisfies none of the assumptions of the Lemma. However, we ll assume ϖ is A-regular, for reasons coming from geometry (ϖ will play the role of the parameter q in Z[q ] mentioned in the introduction), so let s focus on the basic case A is the valuation ring of a perfectoid field. A is complete with respect to the ϖ-adic topology, ϖ being in rad(a) and A-regular. Upon allowing almost étale covers in Definition 1.5.1, then we ll see that the pair (Spec(A/ϖ), {m}), m being the maximal ideal of A and m the reduction modulo ϖ, is almost pure (this is Proposition 4.3.2). The whole of 6 explains how this implies that the pair (Spf(A), {m}) is almost pure (this is Theorem ), which implies the pair (Spa(A[1/ϖ], A), {v m }) is pure. Roughly, perfectoidness of A is, upon allowing for the moment almost mathematics into the picture, a good substitute to the assumptions in Lemma We ll expand on this.

16 16 alessandro maria masullo 1.7 Prerequisites The first part of these notes is largely expository, and intended to be as self contained as possible (and reasonable), and as elementary as possible. Basic notions from rigid geometry will be reviewed when necessary. The theory of the cotangent complex is reviewed and mostly re-developed from scratch, and its application to deformation theory explained. Familiarity with the basic theory of derived categories is necessary. The second part of these notes, under the name Foundations of perfectoid geometry, II, builds on the first but is original in nature. It requires good familiarity with classical rigid geometry, for motivational reasons and for gaining a good understanding of the materials. Particularly ideas borrowed from Raynaud s approach via formal models. Familiarity with Raynaud and Gruson s paper on flattening techniques is auspicable. A very good familiarity with étale cohomology of schemes, good familiarity with étale cohomology of p-adic analytic spaces, and étale cohomology of adic spaces, will be rewarding, though everything will be reviewed when necessary, and/or referenced. For the final chapter, a very good familiarity with the theory of algebraic spaces is required. Moreover, familiarity with Voevodsky s h-topology and h-descent, together with de Jong s theorem on alterations, its proof, and Gabber s improvement, is highly recommended. We ll try to avoid the use of higher algebra as much as possible (completely successfully in this first part, slightly unsuccessfully in the second due to descent theory) to make the materials more immediately accessible. Note, however, that a number of the main results in the second part of these notes will actually find their most natural form in the language of higher algebra.

17 foundations of perfectoid geometry, i 17 2 Perfectoid rings For later purposes, we discuss all the results in this handouts and in subsequent lectures in the wider generality of perfectoid rings, a class of rings we re about to introduce. To this extent, we discuss a new notion of perfectoidness for rings following Gabber and Ramero, and make our new notion consistent with those already present in the literature. In 2.1 we discuss Tate-perfectoid rings, and in the subsequent 2.3 we define perfectoid rings and relate the two notions. A careful study of Fontaine s functor A inf ( ) will be needed, and discussed partially starting from the treatment in [BMS]. 2.1 Tate-perfectoid rings In Scholze s original paper, he introduced perfectoid fields and perfectoid algebras over a perfectoid field. Fontaine s Bourbaki report [Fontaine], removed the assumption of having ground field, which turned out to be very useful for the purposes of taking into consideration interesting examples coming from the theory of formal schemes. We re about to give a uniform framework to deal with Fontaine s notion of perfectoidness, which so far takes into account Tate Huber rings only. The reader should nevertheless keep in mind the initial case of perfectoid algebras over a perfectoid field, and more to the point, of perfectoid fields, as this basic case will turn out to be conceptually fundamental in sight of a sofisticated generalization of Krasner s Lemma, which lies at the heart of the whole theory. To give an outline of what the field case is about, take any arithmetically profinite extension K of Q p with residue field κ. One can attach to it its field of norms K, which, non-canonically, is isomorphic to κ((t)). Fontaine and Wintenberger show that the Galois theory of K is identified with that of K. The completion E of K is a perfectoid field, and the completion of the purely inseparable closure of K is what we shall call its tilt E. Another way to say that the Galois theories of these fields coincide is that the small étale sites of K, K, E, E are (canonically) identified (!). This phenomenon will be likewise true for Tate-perfectoid rings and their tilt, and in fact for perfectoid spaces and their tilt, as we shall discuss in later lectures. It will hold likewise true for perfectoid rings, upon allowing a relaxed notion of étaleness coming from almost mathematics. Notation and conventions As usual for Huber rings, we denote the Huber rings involved usually with letters A, B, C, D, until we set up the theory of Tate-perfectoid rings completely, in which case we shall denote Tate-perfectoid rings with the letters R, R, S etc. For a Huber ring A, we always write A 0 for the subring of power-bounded elements of A, and A + shall denote any open and integrally closed subring of A 0. In this section, all Huber rings will be Tate with respect to a regular pseudo-uniformizer ϖ in A 0, a ring of definition for A. ϖ will induce the ϖ-adic topology on A 0, which will in turn induce the respective Huber ring topology on A. Definition A Tate-perfectoid ring is a complete Tate ring A (Banach with topologically nilpotent unit ϖ A) satisfying the following properties: (1) A 0 is bounded. (2) There exists a topologically nilpotent unit ϖ with ϖ p p in A 0. (3) The pth power map ϕ on A 0 /ϖ is surjective.

18 18 alessandro maria masullo Note that A 0 /ϖ is indeed an F p -algebra, thanks to condition (2), so the pth power map is a ring homomorphism. As a first important remark, we note that condition (3) is actually independent of ϖ. We prove, more in general, the following: Proposition For any complete Tate ring A and nonzero pseudo-uniformizer ϖ satisfying ϖ p p in A 0, the Frobenius map Φ : A 0 /ϖ A 0 /ϖ p is necessarily injective. The surjectivity condition is independent of the choice of such ϖ. Proof. Let x A 0 be satisfying x p = ϖ p a for some a A 0. Then the element x/ϖ A lies in A 0 because its pth power does. Hence ϕ is injective as claimed. We claim the surjectivity of ϕ is equivalent to surjectivity of the (necessarily injective) pth power map: A 0 /(p, ϖ n ) A 0 /(p, ϖ np ) for any n 1. For n = 1 this coincides with surjectivity of ϕ. Suppose such map is surjective for some n. Then it is surjective for all 1 m < n. We only need to check surjectivity for all m > n. Let x A 0 /p. We can write: x = a p + ϖ np b, a, b A 0 /p. Likewise, b = c p + ϖ p u, for some c, u A 0 /p. Hence: x = (a + ϖ n c) p + ϖ (n+1)p u. Surjectivity for m = n + 1 follows, and the claim is proved. We are now ready to conclude the proof. Suppose ϖ is another pseudo-uniformizer satisfying ϖ p p in A 0. If we take n large enough, then ϖ n ϖ A 0, and surjectivity of A 0 /(p, ϖ n ) A 0 /(p, ϖ np ) implies surjectivity of A 0 /ϖ A 0 /ϖ p. It follows that Φ is an isomorphism for all ϖ satisfying ϖ p p in A 0, as soon as it holds for one such ϖ. The proof is complete. Notation Given Proposition and its proof, and in the context of the same Proposition, we shall always denote by ϕ the pth power map on A 0 /ϖ, and likewise on A 0 /p, unless otherwise specified. Example Here is an example of a Tate-perfectoid ring which doesn t arise as an algebra over a (perfectoid) field: A = Z cyc p (p/x) 1/p [1/x]. One can take ϖ := x 1/p, as ϖ p = T divides p in A 0. It is a good exercise to figure out the rational subset U of X := Spa(Z cyc p [x 1/p ], Z cyc p [x 1/p ] ) which realizes A as O X (U). We note here that the ring Z cyc p [x 1/p ], the (p, x)-adic completion of Z cyc p [x 1/p ], wants to be a perfectoid ring, but it s not a Tate Huber ring, hence doesn t lie in the class of Tate-perfectoid rings. The notion we shall introduce in 2.3 will include this kind of examples. We are now ready to (re)introduce the notion of perfectoid fields. Naively, one could assign such notion by saying that a perfectoid field is a perfectoid ring which, in addition, is a field. However, it is not clear at all, a priori, if such a field is nonarchimedean with respect to a rank 1 valuation, and this latter property is quite essential to our purposes, for example when dealing with a crucial Henselian approximation process aiming at spreading finite étale algebras over perfectoid rings out of a certain perfectoid field. We, therefore, build this condition into the definition, for now. In fact, it turns out that a Tate-perfectoid ring which is a field, is always a perfectoid field, as we shall discuss later, for completeness. The key observation is that any

19 foundations of perfectoid geometry, i 19 Huber ring is Banach with respect to a submultiplicative real-valued norm, and we ll prove that for Banach rings whose underlying ring is a field, the uniformity condition does the job and makes them into nonarchimedean fields. 2 Definition A perfectoid field is a perfectoid ring C that is a field and its topology is defined by a rank 1 valuation : C R 0. Example Some examples and a number of non-examples: (1) An infinite family of non-examples is given by any discretely-valued nonarchimedean field k of residue characteristic p. Indeed, let O k be the valuation ring and ϖ be any nonzero element of its maximal ideal. It follows the quotients O k /ϖ and O k /ϖ p are Artin local rings of different lengths, hence they can never be isomorphic. (2) We consider Q p (p 1/p ) (with ϖ = p 1/p ) and Q cyc p, with ϖ coming from the Z/pZ piece of the (Z/p 2 Z) -field extension Q p (ζ p 2)/Q p. Their ϖ-adic completions are both perfectoid fields. (3) We consider: Q p x 1/p = lim n 1 Q p x 1/pn = lim Z p [x 1/pn ] [1/p]. n This is not a Tate-perfectoid ring. However, Q cyc p x 1/p = Q p x 1/p Zp Z cyc p is. This is also obtained as A[1/p], A being the p-adic completion of Z cyc p [x 1/p ]. This class of examples lies in the class of rings known as Tate-preperfectoid, and we shall discuss their theory in general in a later version of these notes. A few important properties of Tate-perfectoid rings. Lemma Any Tate-perfectoid ring A is reduced. Proof. Since A = A 0 [ 1 ϖ ], it s enough to check that A0 is reduced. Choose a A 0 such that a N = 0 for N 1. Then (a/ϖ n ) N = 0 for any n 1 and N 1, and so a/ϖ n is powerbounded for all n 1. Then a ϖ n A 0, which is 0, since ϖ is topologically nilpotent. Some preliminary remarks, before moving on. Remark Why doesn t this prove that any affinoid algebra is reduced? We are using that A is endowed with the (unique) Huber ring topology restricting to the ϖ-adic topology on A 0. Note also that if A was a Tate-perfectoid algebra over a perfectoid field C, reducedness of A was a consequence of the following simple observation. Suppose ε A 0 was a nilpotent element. Then Cε A 0 is a power-bounded subset which is not bounded, a contradiction. We refer the reader to the paper [BV] for a wider explanation of the importance of the boundedness condition on A 0 to hope the Huber pair (A, A 0 ) to be sheafy. Particularly, [BV, Prop. 13 ], and [BV, Prop. 17], which explains requiring boundedness of A 0 is not enough to ensure sheafness of (A, A 0 ), but we should rather require such boundedness to be stable under rational localization. Remark In Sholze s first paper [Sch], he defined a perfectoid field C of residue characteristic p > 0, by considering, rather, the pth power map on C 0 /pc 0. This notion turns 2 Complete proof being included.