Lecture notes for a class on perfectoid spaces Bhargav Bhatt April 23, 2017 Contents Preface 3 1 Conventions on non-archimedean fields4 2 Perfections and tilting6 3 Perfectoid fields9 3.1 Definition and basic properties............................9 3.2 Tilting......................................... 10 4 Almost mathematics 17 4.1 Constructing the category of almost modules.................... 17 4.2 Almost commutative algebra............................. 22 4.3 Almost etale´ extensions................................ 25 4.4 Some more almost commutative algebra....................... 30 5 Non-archimedean Banach algebras via commutative algebra 33 5.1 Commutative algebras: completions and closures.................. 33 5.2 The dictionary..................................... 36 6 Perfectoid algebras 40 6.1 Reminders on the cotangent complex......................... 40 6.2 Perfectoid algebras.................................. 44 7 Adic spaces 53 7.1 Tate rings....................................... 53 7.2 Affinoid Tate rings.................................. 56 7.3 Affinoid adic spaces: definition and basic properties................. 61 7.4 Spectrality of affinoid adic spaces.......................... 68 7.5 The structure presheaf and adic spaces........................ 76 8 The adic spectrum via algebraic geometry 86 8.1 Topological spaces.................................. 86 8.2 Comparing sheaves.................................. 90 1 8.3 A fully faithful embedding of affinoid Tate rings into a geometric category..... 93 9 Perfectoid spaces 95 9.1 Perfectoid affinoid algebras.............................. 95 9.2 Tilting rational subsets................................ 97 9.3 Tate acyclicity and other sheaf-theoretic properties................. 105 9.4 Zariski closed subsets................................. 112 10 The almost purity theorem 118 11 The direct summand conjecture 124 11.1 DSC for maps which are unramified in characteristic 0 ............... 125 11.2 Quantitative Riemann Hebbarkeitssatz........................ 127 11.3 Almost-pro-isomorphisms.............................. 129 11.4 Proof of DSC..................................... 131 2 Preface These are notes for a class on perfectoid spaces taught in Winter 2017. The goal of the class was to develop the theory of perfectoid spaces up through a proof of the almost purity theorem, and then explain the proof of the direct summand conjecture. I have tried to make these notes self-contained, and hopefully accessible to anyone with a back- ground algebraic geometry. In particular, I have not assumed any familiarity with rigid geometry, so the relevant theory of adic spaces is developed from scratch. Likewise, I have not assumed any familiarity with Hochster’s network of “homological conjectures”, so the direct summand conjecture is proven directly, and not via (a non-trivial) reduction to some other statement. Two exceptions: (a) I have used the language of derived categories in a couple of spots where I think it brings out the essence of the argument faster, and (b) I have used some results in point set topology (of spectral spaces) without proof. Disclaimers. There are surely many errors, so please use at your own risk. The notes are unstable, and being constantly revised. Also, essentially all references and attributions are missing, and will be added later. 3 Chapter 1 Conventions on non-archimedean fields We establish some standard notation about non-archimedean fields. Definition 1.0.1. A (complete) non-archimedean or NA field is a field K equipped with a multi- ∗ plicative valuation j · j : K ! R>0 such that K is complete for the valuation topology. The group ∗ jK j ⊂ R>0 is called the value group of K. Remark 1.0.2. Some comments are in order: 1. It is convenient to extend the valuation to a map j · j : K ! R≥0 by setting j0j = 0. With this extension, the valuation topology on K is the unique group topology with a basis of open −1 subgroups given by j · j ((0; γ)) for γ 2 R>0. This topology defined by a metric on K: d(x; y) = jx − yj. 2. Most authors do not impose a completeness hypothesis. However, our later constructions with adic spaces work best for complete rings, so we impose completeness right away. 3. We shall typically work in a mixed/positive characteristic setting, i.e., jpj < 1 for some prime p. 4. We shall always assume that K is nontrivially valued, i.e., there exists a nonzero element t 2 K with 0 < jtj < 1. A NA field comes naturally with some associated rings and ideals: Definition 1.0.3. Let K be a NA field. The subset K◦ := fx 2 K j jxj ≤ 1g is called the valuation ring of K; this is an open valuation subring of K with maximal ideal K◦◦ := fx 2 K j jxj < 1g. The quotient k := K◦=K◦◦ is called the residue field of K. Any nonzero element t 2 K◦◦ is called a pseudo-uniformizer. The next exercise shows that given the field K, specifying the topology on K is equivalent to specifying the valuation. In particular, it is meaningful to ask if a topological field K is NA. Exercise 1.0.4. Fix a NA field K. 4 1. A subset S ⊂ K is bounded if there exists a nonzero t 2 K such that t · K ⊂ K◦; equiva- lently, S is bounded for the metric topology on K. An element t 2 K is power bounded if the set tN := ftn j n ≥ 0g ⊂ K is bounded. Check that K◦ ⊂ K is exactly the set of power bounded elements. 2. Check that K◦◦ ⊂ K is exactly the topologically nilpotent elements of K, i.e., those t 2 K such that tn ! 0 as n ! 1. 3. Fix a pseudouniformizer t 2 K◦◦. Show that the t-adic topology on K◦ coincides with the valuation topology. 4. Show that the given NA valuation j · j on K can be reconstructed from the valuation ring K◦. The next table records some examples the concepts introduced above. table:NAmy-label K◦ K pu Value group Z Zp Qp p jpj Z OK K=Qp finite π jπj if π is a uniformizer Z Fp t Fp((t)) t jtj J K 1 1 1 \1 \1 Z[ ] Zp[p p ] Qp(p p ) p jpj p Q Zcp Cp := Qcp p jpj K◦ perfect K perfect t p-divisible Table 1.1: NA fields 5 Chapter 2 Perfections and tilting Recall that a characteristic p ring R is perfect is the Frobenius φ : R ! R is an isomorphism; if instead φ is merely assumed to be surjective, we say that R is semiperfect. In this chapter, we introduce and study Fontaine’s tilting functor: it attaches a perfect ring of characteristic p any commutative ring (and is typically of interest when the latter has mixed/positive characteristic). def:Tilt Definition 2.0.1 (The tilting functor). Let R be a ring. 1. If R has characteristic p, set R := lim R and Rperf := lim R, where φ : R ! R perf −!φ −φ denotes the Frobenius. [ perf 2. (Fontaine) For any ring R, set R := (R=p) := limφ R=p. Unless otherwise specified, this ring is endowed with the inverse limit topology, with each R=p being given the discrete topology. Remark 2.0.2 (Universal properties of perfections). When R has characteristic p, both Rperf and perf perf R are perfect. The canonical map R ! Rperf (resp. R ! R) is universal for maps into (resp. from) perfect rings. Moreover, the projection Rperf ! R is surjective exactly when R is semiperfect. ex:TiltingRings Example 2.0.3. We record some examples of these concepts. 1 1 perf 1. Fp[t]perf = Fp[t p ] and Fp[t] = Fp. [ 2. Fp[t] ' Fp. 3. Say R is a finite type algebra over an algebraically closed field k of characteristic p. Then R[ ' kπ0(Spec(R)) is the algebra of k-valued continuous functions on Spec(R). To see this, we may assume Spec(R) is connected and reduced (see Exercise 2.0.4). We must show k ' Rperf . Assume first that Spec(R) is irreducible. Picking a closed point x 2 Spec(R) gives a map R ! Rcx, where Rcx is the completion of the local ring at x; this map is injective as R is perf n a domain. It is therefore enough to show that Rcx ' k. We have Rcx = limn Rx=mx, and perf n perf n perf thus Rcx = limn(Rx=mx) . Using Exercise 2.0.4, it is easy to see that (Rx=mx) ' 6 perf perf (Rx=mx) ' k for all n, which gives the claim. The generalization to the case where R is not a domain is left to the reader. 1 1 1 perf \1 4. (Fp[t p ]=(t)) ' Fp[t p ]. More generally, if R is a perfect ring of characteristic p and f 2 R is a nonzerodivisor, then (R=f)perf is the f-adic completion of R. [ 5. (Zp) ' Fp. 1 1 perf \1 [ \1 6. (Zp[p p ]) ' Fp[t p ] ' F\p[t]perf ' Fp[t]perf =(t) . The perfection functors kill nilextensions. exer:PerfectionNilp Exercise 2.0.4. Let f : R ! S be a map of characteristic p rings that is surjective with nilpotent perf perf kernel. Then R ' S and Rperf ' Sperf . More generally, the same holds if f factors a power of Frobenius on either ring. We repeatedly use the following elementary lemma. lem:Binomial Lemma 2.0.5. Let R be a ring, and let t 2 R be an element such that p 2 (t). Given a; b 2 R with a = b mod t, we have apn = bpn mod tn+1 for all n.. Proof. We prove this by induction on n. If n = 0, there is nothing to show. Assume inductively pn pn n+1 p that a = b + t c for some c 2 R. Raising both sides to the p-th power, and using that p j i for 1 ≤ i ≤ p − 1, we get apn+1 = bpn+1 + p · tn+1 · d + tp·(n+1)cp for some d 2 R.