Projective Geometry for Perfectoid Spaces Gabriel Dorfsman-Hopkins August 19, 2018 Abstract To understand the structure of an algebraic variety we often embed it in various projective spaces. This develops the notion of projective geometry which has been an invaluable tool in algebraic geometry. We develop a perfectoid analog of projective geometry, and explore how equipping a perfectoid space with a map to a certain analog of projective space can be a powerful tool to understand its geometric and arithmetic structure. In particular, we show that maps from a perfectoid space X to the perfectoid analog of projective space correspond to line bundles on X together with some extra data, reflecting the classical theory. Along the way we give a complete classification of vector bundles on the perfectoid unit disk, and compute the Picard group of the perfectoid analog of projective space. Contents 1 Introduction 3 2 Algebraic Preliminaries 5 2.1 The Fontaine-Wintenberger Isomorphism . .5 2.2 Topological Rings and Fields . .5 2.3 A Convergence Result . .9 2.4 Perfectoid Fields . 11 2.5 Perfectoid Algebras . 14 3 Adic Spaces 17 3.1 Valuation Spectra . 17 3.2 Huber Rings . 19 3.3 The Adic Spectrum of a Huber Pair . 20 3.4 The Adic Unit Disk . 21 3.5 The Structure Presheaf OX ..................................... 23 3.6 The Categories of Pre-Adic and Adic Spaces . 26 3.7 Examples of Adic Spaces . 27 3.8 Sheaves on Adic Spaces . 28 1 4 Perfectoid Spaces 30 4.1 Affinoid Perfectoid Spaces . 30 4.2 Globalization of the Tilting Functor . 31 4.3 The Etale´ Site . 32 4.4 Examples of Perfectoid Spaces and their Tilts . 33 5 The Perfectoid Tate Algebra 35 5.1 The Group of Units . 36 5.2 Krull Dimension in Characteristic p ................................ 37 5.3 Weierstrass Division . 38 5.4 Generating Regular Elements . 41 6 Vector Bundles on the Perfectoid Unit Disk 46 6.1 Finite Free Resolutions and Unimodular Extension . 46 6.2 Coherent Rings . 48 6.3 Finite Projective Modules on the Residue Ring . 49 6.4 Finite Projective Modules on the Ring of Integral Elements . 50 6.5 The Quillen-Suslin Theorem for the Perfectoid Tate Algebra . 51 7 Line Bundles and Cohomology on Projectivoid Space 57 7.1 Reductions Using Cechˇ Cohomology . 57 7.2 The Picard Group of Projectivoid Space . 60 7.3 Cohomology of Line Bundles . 62 7.3.1 Koszul-to-Cech:ˇ The Details . 64 8 Maps to Projectivoid Space 66 8.1 L -Distinguished Open Sets . 67 8.2 Construction of the Projectivoid Morphism . 68 8.3 The Positive Characteristic Case . 70 9 Untilting Line Bundles 71 9.1 Cohomological Untilting . 71 9.2 Untilting Via Maps to Projectivoid Space . 73 9.3 Injectivity of θ ............................................ 76 2 1 Introduction An important dichotomy in algebraic geometry is the distinction between characteristic 0 and prime character- istic p > 0. Algebraic geometry provides a framework to do geometry in positive characteristic, transporting our classical intuition to a more exotic algebraic world. But in positive characteristic we are also provided with extra tools, such as the Frobenius map, making many geometric results more accessible. Therefore transporting information from positive characteristic up to characteristic 0 proves very useful as well. In [9], Fontaine and Wintenberger produced an isomorphism which hinted at a deep correspondence between algebraic objects of each type. Theorem 1.1 (Fontaine-Wintenberger) 1=p1 There is a canonical isomorphism of topological groups between the absolute Galois groups of Qp p 1=p1 and Fp t . An algebraic geometer would perhaps ask if there is some deeper geometric correspondence, of which this is a manifestation on the level of points. Quite recently, in [27], Scholze introduced a class of geometric objects called perfectoid spaces, which exhibit this very correspondence. In particular, to a perfectoid space X of any characteristic, we can associate its tilt X[ which is a perfectoid space of characteristic p, and furthermore X and X[ have isomorphic ´etalesites. These spaces have proved useful far beyond giving a geometric framework in which to understand the Fontaine-Wintenberger isomorphism. Indeed, they have found applications in extending instances of Deligne's Weight-Monodromy Conjecture, classifying p-divisible groups, have been used in work on the geometric Langlands program, and even aid in the understanding of singularities in positive characteristic. For a survey, see [28]. This paper is inspired by the goal of understanding vector bundles on perfectoid spaces, and how they behave under the so called tilting correspondence of Scholze. To do so, we develop a perfectoid analog of projective geometry. We define a perfectoid analog of projective space, which we call projectivoid space and denote by Pn;perf , and show that maps from a perfectoid space X to Pn;perf correspond to line bundles on X together with some extra data, giving an analog to the classical theory of maps to projective space. To get to this point we must first understand the theory of line bundles on projectivoid space itself, and in particular, its Picard group. In his dissertation [7], Das worked toward computing Picard group of the projectivoid line, P1;perf . His proof relied on having certain local trivializations of line bundles, requiring a perfectoid analog of the Quillen-Suslin theorem. Therefore, in order to begin developing the theory of so called projectivoid geometry, we must prove this first. The Quillen-Suslin theorem says that finite dimensional vector bundles on affine n-space over a field are all trivial. Equivalently, all finite projective modules on a polynomial ring K[T ] are free, where T is an n-tuple of indeterminates. In rigid analytic geometry, we replace polynomial rings with rings of convergent power series called Tate algebras, denoted KhT i, and it can be shown that over such rings the Quillen-Suslin theorem still holds, that is, all finite projective KhT i-modules are free. The analog of these rings for perfectoid spaces is 1 the ring KhT 1=p i of convergent power series where the indeterminates have all their pth power roots. The difficulty in extending the theorem to this perfectoid Tate algebra is that the ring is no longer noetherian, and so the result cannot be easily reduced to the polynomial case. Sections 2 through 4 of this paper set up the theory of perfectoid spaces. In Section 2 we define our fundamental algebraic objects, perfectoid fields and algebras, and explore some of their algebraic properties in relating characteristic 0 and characteristic p. In Section 3 we review Huber's theory of adic spaces, which provide the geometric framework for globalizing perfectoid algebras into spaces (playing a role analogous to schemes in algebraic geometry). In Section 4 we apply Huber's theory of adic spaces to perfectoid rings and algebras, and explore the geometric properties of perfectoid spaces and their tilts. Sections 5 through 9 constitute the author's work on the subject. In Section 5 we explore the commutative ring theoretic properties of the perfectoid Tate algebra. We compute its unit group, and prove perfectoid 3 analogues of Weierstrass division and preparation. We also compute the Krull dimension of the perfectoid Tate algebra in positive characteristic. In Section 6 we have our first main theorem. Theorem 1.2 (The Quillen-Suslin Theorem for the Perfectoid Tate Algebra) D 1=p1 1=p1 E Finite projective modules on the perfectoid Tate algebra K T1 ; ··· ;Tn are all free. Equiva- lently, finite dimensional vector bundles on the perfectoid unit disk are all isomorphic to the trivial vector bundle. This completes Das' proof, and lays the groundwork to begin studying vector bundles on more general perfectoid spaces. In Section 7 we develop the theory of line bundles on projectivoid space, extending Das' result for n = 1. Theorem 1.3 (The Picard Group of Projectivoid Space) Pic Pn;perf =∼ Z[1=p]. We also compute the cohomology of all line bundles on projectivoid space. In Section 8 we compute the functor of points of projectivoid space, showing that (much like in the classical theory) it is deeply connected to the theory of line bundles on perfectoid spaces. Theorem 1.4 (The Functor of Points of Projectivoid Space) n;perf (i) Let X be a perfectoid space over a field K. Morphisms X ! P correspond to tuples Li; sj ;'i , n (i) (i)o ⊗p ∼ where Li 2 Pic X, s0 ; ··· ; sn are n+1 global sections of Li which generate Li, and 'i : Li+1 −! ⊗p (i+1) (i) Li are isomorphisms under which sj 7! sj . We also provide refinements of this theorem in characteristic p and see how it behaves under the tilting equivalence of Scholze. In Section 9 we test out this new theory, using it to compare the Picard groups of a perfectoid space X and [ n;perf its tilt X . In particular, since the tilting equivalence builds a correspondence between maps X ! PK [ n;perf and maps X ! PK[ , we can chain this together with the correspondence of line bundles and maps to projectivoid space to compare line bundles on X and X[. The main result follows. Theorem 1.5 Suppose X is a perfectoid space over K. Suppose that X has an ample line bundle and that H0(X ; ) = K. Then there is a natural injection K OXK θ : Pic X[ ,! lim Pic X: − L 7!L p In particular, if Pic X has no p torsion, then composing with projection onto the first coordinate gives an injection [ θ0 : Pic X ,! Pic X: 4 2 Algebraic Preliminaries Algebraic geometry is locally commutative algebra, that is, the spaces we study are locally a `model space,' which is the prime spectrum of a commutative ring.