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New Approach to Arakelov Geometry Nikolai Durov 7 0 April 17, 2007 0 2 r p Introduction A 6 The principal aim of this work is to provide an alternative algebraic frame- 1 work for Arakelov geometry, and to demonstrate its usefulness by presenting ] several simple applications. This framework, called theory of generalized G rings and schemes, appears to be useful beyond the scope of Arakelov ge- A ometry, providing a uniform description of classical scheme-theoretical alge- . h braic geometry (“schemes over SpecZ”), Arakelov geometry (“schemes over t a m SpecZ andS\pecZ”),tropicalgeometry(“schemesoverSpecTandSpecN”) ∞ [ and the geometry over the so-called field with one element (“schemes over SpecF ”). Therefore, we develop this theory a bit further than it is strictly 1 1 v necessary for Arakelov geometry. 0 The approach to Arakelov geometry developed in this work is completely 3 0 algebraic, in the sense that it doesn’t require the combination of scheme- 2 theoretical algebraic geometry and complex differential geometry, tradition- . 4 ally used in Arakelov geometry since the works of Arakelov himself. 0 7 However, we show that our models X¯/S\pecZ of algebraic varieties X/Q 0 define both a model X /SpecZ in the usual sense and a (possibly singular) : v Banach (co)metric on (the smooth locus of) the complex analytic variety i X X(C). This metric cannot be chosen arbitrarily; however, some classical r metrics like the Fubini–Study metric on Pn do arise in this way. It is in- a teresting to note that “good” models from the algebraic point of view (e.g. finitely presented) usually give rise to not very nice metrics on X(C), and conversely, nice smooth metrics like Fubini–Study correspond to models with “bad” algebraic properties (e.g. not finitely presented). Our algebraic approach has some obvious advantages over the classical one. For example, we never need to require X to be smooth or proper, and we can deal with singular metrics. In order to achieve this goal we construct a theory of generalized rings, commutative or not, which include classical rings (always supposed to be 1 2 Introduction associativewithunity), thendefinespectra ofsuch(commutative) generalized rings, and construct generalized schemes by patching together spectra of generalized rings. Of course, these generalized schemes are generalized ringed spaces, i.e. topological spaces, endowed with a sheaf of generalized rings. Then the “compactified” SpecZ, denoted by S\pecZ, is constructed as a (pro-)generalized scheme, and our models X¯/S\pecZ are (pro-)generalized schemes as well. All the “generalized” notions we discuss are indeed generalizations of corresponding “classical” notions. More precisely, “classical” objects (e.g. commutative rings) always constitute a full subcategory of the category of corresponding “generalized” objects (e.g. commutative generalized rings). In this way we can always treat for example a classical scheme as a generalized scheme, since no new morphisms between classical schemes arise in the larger category of generalized schemes. In particular, the category of (commutative) generalized rings contains all classical (commutative) rings like Z, Q, R, C, Z , ..., as well as some p new objects, such as Z (the “archimedian valuation ring” of R, similar to p- ∞ adic integers Z Q ), Z (the “non-completed localization at ”, or the p p ( ) ⊂ ∞ ¯ ∞ “archimedian valuation ring” of Q), Z (“the integral closure of Z in C”). ∞ ∞ Furthermore, once these “archimedian valuation rings” are constructed, we can define some other generalized rings, such as F := Z Z, or the “field 1 ± ∞∩ with one element” F . Tropical numbers T and other semirings are also 1 generalized rings, thus fitting nicely into this picture as well. In this way we obtain not only a “compact model” S\pecZ of Q (called also “compactification of SpecZ”), and models X S\pecZ of algebraic → varieties X/Q, but a geometry over “the field with one element” as well. For example, S\pecZ itself is a pro-generalized scheme over F and F . 1 1 ± In other words, we obtain rigorous definitions both of the “archimedian local ring” Z and of the “field with one element” F . They have been 1 ∞ discussed in mathematical folklore for quite a long time, but usually only in a very informal fashion. We would like to say a few words here about some applications of the theory of generalized rings and schemes presented in this work. Apart from defining generalized rings, their spectra, and generalized schemes, we discuss some basic properties of generalized schemes, essentially transferring some results of EGA I and II to our case. For example, we discuss projective (generalized) schemes and morphisms, study line and vector bundles, define Picard groups and so on. Afterwards we do some homological (actually homotopic) algebra over generalized rings and schemes, define perfect simplicial objects and cofibra- Introduction 3 tions (which replace perfect complexes in this theory), define K of per- 0 fect simplicial objects and vector bundles, briefly discuss higher algebraic K-theory (Waldhausen’s construction seems to be well-adapted to our sit- uation), and construct Chow rings and Chern classes using the γ-filtration on K , in the way essentially known since Grothendieck’s proof of Riemann– 0 Roch theorem. We apply the above notions to Arakelov geometry as well. For example, wecomputePicardgroup,ChowringandChernclassesofvectorbundlesover S\pecZ, and construct the moduli space of such vector bundles. In particular, we obtainthe notionof(arithmetic) degree degE logQ ofa vector bundle ∈ ×+ E over S\pecZ; it induces an isomorphism deg : Pic(S\pecZ) logQ . We → ×+ also prove that any affine or projective algebraic variety X over Q admits a finitely presented model X over S\pecZ, and show (under some natural conditions) that rational points P X(Q) extend to uniquely determined ∈ sections σ : S\pecZ X . We show that when X is a closed subvariety P → of the projective space Pn, and its model X is chosen accordingly (e.g. X Q is the “scheme-theoretical closure” of X in Pn ), then the (arithmetic) S\pecZ degree of the pullback σP∗(OX(1)) of the ample line bundle of X equals the logarithmic height of point P X(Q) Pn(Q). ∈ ⊂ There are several reasons to believe that our “algebraic” Arakelov geom- etry can be related to its more classical variants, based on K¨ahler metrics, differential forms and Green currents, as developed first by Arakelov himself, and then in the series of works of H. Gillet, C. Soul´e and their collobora- tors. The simplest reason is that our “algebraic” models give rise to some (co)metrics, andclassicalmetricsliketheFubini–Studydoappearinthisway. More sophisticated arguments involve comparison with the non-archimedian variant of classical Arakelov geometry, developed in [BGS] and [GS]. This non-archimedian Arakelov geometry is quite similar to (classical) archime- dianArakelovgeometry, andatthesametimeadmitsanaturalinterpretation in terms of models of algebraic varieties over discrete valuation rings. An- alytic torsion corresponds in this picture to torsion in the special fiber (i.e. lack of flatness). Therefore, one might hope to transfer eventually the results of these two works to archimedian context, using our theory of generalized schemes, thus establishing a direct connection between our “algebraic” and classical “ana- lytic” variant of Arakelov geometry. Acknowledgements. First of all, I would like to thank my scientific advisor, Gerd Faltings, for his constant attention to this work, as well as for teaching me a lot of arithmetic geometry during my graduate studies in Bonn. I would also like to thank Alexandr Smirnov for some very interesting 4 Introduction discussions and remarks, some of which have considerably influenced this work. I’d like to thank Christophe Soul´e for bringing very interesting works [GS] and [BGS] on non-archimedian Arakelov geometry to my attention. This work hasbeen writtenduring my graduatestudies inBonn, asa part of the IMPRS (International Max Planck Research School for Moduli Spaces and their applications), a joint programof the Bonn University and the Max- Planck-Institut fu¨r Mathematik. I’d like to thank all people involved from these organizations for their help and support. Special thanks go to Chris- tian Kaiser, the coordinator of the IMPRS, whose help was indispensable throughout all my stay in Bonn. Introduction 5 Contents Overview 9 0.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 0.2. Z -structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 ∞ 0.3. -categories, algebras and monads . . . . . . . . . . . . . . . . 15 ⊗ 0.4. Algebraic monads as non-commutative generalized rings . . . . 17 0.5. Commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0.6. Localization and generalized schemes . . . . . . . . . . . . . . 37 0.7. Applications to Arakelov geometry . . . . . . . . . . . . . . . . 45 0.8. Homological and homotopic algebra . . . . . . . . . . . . . . . 50 0.9. Homotopic algebra over topoi . . . . . . . . . . . . . . . . . . . 57 0.10. Perfect cofibrations and intersection theory . . . . . . . . . . 61 1 Motivation: Looking for a compactification of SpecZ 65 1.1. Original motivation . . . . . . . . . . . . . . . . . . . . . . . . 65 1.5. Vector bundles over SpecZ . . . . . . . . . . . . . . . . . . . . 70 1.6. Usual description of Arakelov varieties . . . . . . . . . . . . . . 71 d 2 Z -Lattices and flat Z -modules 73 ∞ ∞ 2.1. Lattices stable under multiplication . . . . . . . . . . . . . . . 73 2.3. Maximal compact submonoids of End(E) . . . . . . . . . . . . 75 2.4. Category of Z -lattices . . . . . . . . . . . . . . . . . . . . . . 78 ∞ 2.7. Torsion-free Z -modules . . . . . . . . . . . . . . . . . . . . . 84 ∞ 2.11. Category of torsion-free algebras and modules . . . . . . . . . 92 2.12. Arakelov affine line . . . . . . . . . . . . . . . . . . . . . . . . 95 2.13. Spectra of flat Z -algebras . . . . . . . . . . . . . . . . . . . 102 ∞ 2.14. Abstract Z -modules . . . . . . . . . . . . . . . . . . . . . . 109 ∞ 3 Generalities on monads 119 3.1. AU -categories . . . . . . . . . . . . . . . . . . . . . . . . . . 119 ⊗ 3.2. Categories of functors . . . . . . . . . . . . . . . . . . . . . . . 124 3.3. Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.4. Examples of monads . . . . . . . . . . . . . . . . . . . . . . . . 137 3.5. Inner functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4 Algebraic monads and algebraic systems 155 4.1. Algebraic endofunctors on Sets . . . . . . . . . . . . . . . . . . 155 4.3. Algebraic monads . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.4. Algebraic submonads and strict quotients . . . . . . . . . . . . 168 4.5. Free algebraic monads . . . . . . . . . . . . . . . . . . . . . . . 173 6 Introduction 4.6. Modules over an algebraic monad . . . . . . . . . . . . . . . . 181 4.7. Categories of algebraic modules . . . . . . . . . . . . . . . . . . 190 4.8. Addition. Hypoadditivity and hyperadditivity . . . . . . . . . 197 4.9. Algebraic monads over a topos . . . . . . . . . . . . . . . . . . 203 5 Commutative monads 215 5.1. Definition of commutativity . . . . . . . . . . . . . . . . . . . . 215 5.2. Topos case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 5.3. Modules over generalized rings . . . . . . . . . . . . . . . . . . 228 5.4. Flatness and unarity . . . . . . . . . . . . . . . . . . . . . . . . 239 5.5. Alternating monads and exterior powers . . . . . . . . . . . . . 244 5.6. Matrices with invertible determinant . . . . . . . . . . . . . . . 253 5.7. Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 6 Localization, spectra and schemes 269 6.1. Unary localization . . . . . . . . . . . . . . . . . . . . . . . . . 269 6.2. Prime spectrum of a generalized ring . . . . . . . . . . . . . . . 282 6.3. Localization theories . . . . . . . . . . . . . . . . . . . . . . . . 285 6.4. Weak topology and quasicoherent sheaves . . . . . . . . . . . . 296 6.5. Generalized schemes . . . . . . . . . . . . . . . . . . . . . . . . 302 6.6. Projective generalized schemes and morphisms . . . . . . . . . 313 7 Arakelov geometry 329 7.1. Construction of SpecZ . . . . . . . . . . . . . . . . . . . . . . 329 7.2. Models over SpecZ, Z and Z . . . . . . . . . . . . . . . . 350 ( ) ∞ ∞ 7.3. Z -models and mdetrics . . . . . . . . . . . . . . . . . . . . . . 356 ∞ 7.4. Heights of ratidonal points . . . . . . . . . . . . . . . . . . . . . 357 8 Homological and homotopical algebra 361 8.1. Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . 368 8.2. Simplicial and cosimplicial objects . . . . . . . . . . . . . . . . 376 8.3. Simplicial categories . . . . . . . . . . . . . . . . . . . . . . . . 381 8.4. Simplicial model categories . . . . . . . . . . . . . . . . . . . . 383 8.5. Chain complexes and simplicial objects over abelian categories 388 8.6. Simplicial Σ-modules . . . . . . . . . . . . . . . . . . . . . . . 393 8.7. Derived tensor product . . . . . . . . . . . . . . . . . . . . . . 398 9 Homotopic algebra over topoi 405 9.1. Generalities on stacks . . . . . . . . . . . . . . . . . . . . . . . 405 9.2. Kripke–Joyal semantics . . . . . . . . . . . . . . . . . . . . . . 411 9.3. Model stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Introduction 7 9.4. Homotopies in a model stack . . . . . . . . . . . . . . . . . . . 430 9.5. Pseudomodel stack structure on simplicial sheaves . . . . . . . 441 9.6. Pseudomodel stacks . . . . . . . . . . . . . . . . . . . . . . . . 453 9.7. Pseudomodel structure on sO-MOD and sO-Mod . . . . . . 459 E 9.8. Derived local tensor products . . . . . . . . . . . . . . . . . . . 464 9.9. Derived symmetric powers . . . . . . . . . . . . . . . . . . . . 468 10 Perfect cofibrations and Chow rings 487 10.1. Simplicial dimension theory . . . . . . . . . . . . . . . . . . . 487 10.2. Finitary closures and perfect cofibrations . . . . . . . . . . . . 496 10.3. K of perfect morphisms and objects . . . . . . . . . . . . . . 506 0 10.4. Projective modules over Z . . . . . . . . . . . . . . . . . . . 524 ∞ 10.5. Vector bundles over SpecZ . . . . . . . . . . . . . . . . . . . 532 10.6. Chow rings, Chern classes and intersection theory . . . . . . . 544 10.7. Vector bundles over SpdecZ: further properties . . . . . . . . . 557 d 8 Introduction Overview 9 Overview Wewouldliketostartwithabriefoverview oftherestofthiswork, discussing chapter after chapter. Purely technical definitions and statements will be omitted or just briefly mentioned, while those notions and results, which we consider crucial for the understanding of the remainder of this work, will be explained at some length. 0.1. (Motivation.) Chapter 1 is purely motivational. Here we discuss proper smooth models both of functional and number fields, and indicate why non- proper models (e.g. the affine line A1 as a model of k(t), or SpecZ as a k model of Q) are not sufficient for some interesting applications. We also in- troduce some notations. For example, S\pecZ denotes the “compactification” of SpecZ. Its closed points must correspond to all valuations of Q, archime- dian or not, i.e. we expect S\pecZ = SpecZ as a set, where denotes ∪{∞} ∞ a new “archimedian point”. We denote by Z Q := R and Z Q ( ) ∞ ⊂ ∞ ∞ ⊂ the completed and non-completed local rings of S\pecZ at , analogous to ∞ classical notations Z Q and Z Q. p p (p) ⊂ ⊂ 0.1.1. ItisimportanttonoticeherethatS\pecZ, Z andZ arenotdefined ( ) ∞ ∞ in this chapter. Instead, they are used in an informal way to describe the properties we would expect these objects to have. In this way we are even able to explain the classical approach to Arakelov geometry, which insists on defining an Arakelov model X¯ of a smooth projective algebraic variety X/Q as a flat proper model X SpecZ together with a metric on X(C) subject → to certain restrictions (e.g. being a K¨ahler metric). 0.1.2. Another thing discussed in this chapter is that the problem of con- structing models over S\pecZ can be essentially reduced to the problem of constructing Z -models of algebraic varieties X/Q, or Z -models of alge- ( ) ∞ ∞ braic varieties X/R. In other words, we need a notion of a Z -structure on ∞ an algebraic variety X over R; if X = SpecA is affine, this is the same thing as a Z -structure on an R-algebra A. So we see that a proper understand- ∞ ing of “compactified” models of algebraic varieties over Q must include an understanding of Z -structures on R-algebras and vector spaces. ∞ 0.2. (Z -structures.) Chapter 2 is dedicated to a detailed study of Z - ∞ ∞ structures on real vector spaces and algebras. We start from the simplest cases and extend our definitions step by step, arriving at the end to the “correct” definition of Z -Mod, the category of Z -modules. In this way ∞ ∞ we learn what the Z -modules are, without still having a definition of Z ∞ ∞ itself. 10 Overview The main method employed here to obtain “correct” definitions is the comparison with the p-adic case. 0.2.1. (Z -lattices: classical description.) The first step is to describe Z - ∞ ∞ structures onafinite-dimensional realvector spaceE, i.e.Z -lattices A E. ∞ ⊂ The classical solution is this. In the p-adic case a Z -lattice A in a p finite-dimensional Q -vector space E defines a maximal compact subgroup p K := Aut (A) = GL(n,Z ) in locally compact group G := Aut (E) = A Zp ∼ p Qp ∼ GL(n,Q ), all maximal compact subgroups of G arise in this way, and p K = K iff A and A are similar, i.e. A = λA for some λ Q . A A′ ′ ′ ∈ ×p Therefore, itisreasonabletoexpectsimilarity classesofZ -latticesinside ∞ realvectorspaceE tobeinone-to-onecorrespondencewithmaximalcompact subgroups K of locally compact group G := Aut (E) = GL(n,R). Such R ∼ maximal compact subgroups are exactly the orthogonal subgroups K = Q ∼ O(n,R), defined by positive definite quadratic forms Q on E, and K = K Q Q′ iff Q and Q are proportional, i.e. Q = λQ for some λ > 0. ′ ′ In this way the classical answer is that a Z -structure on a finite di- ∞ mensional real space E is just a positive definite quadratic form on E, and similarly, a Z¯ -structure on a finite dimensional complex vector space is a ∞ positive definite hermitian form. This point of view, if developed further, ex- plains why classical Arakelov geometry insists on equipping (complex points of) all varieties and vector bundles involved with hermitian metrics. 0.2.2. (Z -structures on finite R-algebras.) Now suppose that E is a finite ∞ R-algebra. We would like to describe Z -structures on this algebra, i.e. Z - ∞ ∞ lattices A E, compatible with the multiplication and unit of E. In the ⊂ p-adic case this would actually mean 1 A and A A A, but if we want ∈ · ⊂ to obtain “correct” definitions in the archimedian case, we must re-write these conditions for A in terms of corresponding maximal compact subgroup K G. A ⊂ And here a certain problem arises. These compatibility conditions cannot beeasily expressed in terms of maximal compact subgroups ofautomorphism groups even in the p-adic case. However, if we consider maximal compact submonoids of endomorphism monoids instead, this problem disappears. 0.2.3. (Maximal compact submonoids: p-adic case.) Thus we are induced to describe Z -lattices A in a finite-dimensional real space E in terms of max- ∞ imal compact submonoids M of locally compact monoid M := End (E) = A R ∼ M(n,R). When we study the corresponding p-adic problem, we see that all maximal compact submonoids of End (E) are of form M := ϕ : ϕ(A) Qp A { ⊂ A = End (A) = M(n,Z ) for a Z -lattice A E, and that M = M iff } ∼ Zp ∼ p p ⊂ A A′ A and A are similar, i.e. in the p-adic case maximal compact submonoids of ′

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