DOCSLIB.ORG

Norms in Motivic Homotopy Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Norms in Motivic Homotopy Theory NORMS IN MOTIVIC HOMOTOPY THEORY TOM BACHMANN AND MARC HOYOIS Abstract. If f : S0 ! S is a finite locally free morphism of schemes, we construct a symmetric monoidal 0 \norm" functor f⊗ : H•(S ) ! H•(S), where H•(S) is the pointed unstable motivic homotopy category 0 over S. If f is finite ´etale,we show that it stabilizes to a functor f⊗ : SH(S ) ! SH(S), where SH(S) is 1 the P -stable motivic homotopy category over S. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic E1-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck{Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum HZ, the homotopy K-theory spectrum KGL, and the algebraic cobordism spectrum MGL. The normed spectrum structure on HZ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology. Contents 1. Introduction 3 1.1. Norm functors 3 1.2. Normed motivic spectra 4 1.3. Examples of normed spectra 5 1.4. Norms in other contexts 7 1.5. Norms vs. framed transfers 8 1.6. Summary of the construction 8 1.7. Summary of results 9 1.8. Guide for the reader 10 1.9. Remarks on 1-categories 10 1.10. Standing assumptions 11 1.11. Acknowledgments 11 2. Preliminaries 11 2.1. Nonabelian derived 1-categories 11 2.2. Unstable motivic homotopy theory 13 2.3. Weil restriction 14 3. Norms of pointed motivic spaces 15 3.1. The unstable norm functors 15 3.2. Norms of quotients 17 4. Norms of motivic spectra 19 4.1. Stable motivic homotopy theory 19 4.2. The stable norm functors 20 5. Properties of norms 21 5.1. Composition and base change 21 5.2. The distributivity laws 22 5.3. The purity equivalence 25 5.4. The ambidexterity equivalence 26 5.5. Polynomial functors 29 6. Coherence of norms 30 6.1. Functoriality on the category of spans 30 Date: May 28, 2020. 1 2 TOM BACHMANN AND MARC HOYOIS 6.2. Normed 1-categories 32 7. Normed motivic spectra 35 7.1. Categories of normed spectra 35 7.2. Cohomology theories represented by normed spectra 39 8. The norm{pullback{pushforward adjunctions 43 8.1. The norm{pullback adjunction 44 8.2. The pullback{pushforward adjunction 47 9. Spectra over profinite groupoids 48 9.1. Profinite groupoids 49 9.2. Norms in stable equivariant homotopy theory 51 10. Norms and Grothendieck's Galois theory 53 10.1. The profinite ´etalefundamental groupoid 54 10.2. Galois-equivariant spectra and motivic spectra 55 10.3. The Rost norm on Grothendieck{Witt groups 57 11. Norms and Betti realization 58 11.1. A topological model for equivariant homotopy theory 58 11.2. The real Betti realization functor 60 12. Norms and localization 61 12.1. Inverting Picard-graded elements 62 12.2. Inverting elements in normed spectra 63 12.3. Completion of normed spectra 66 13. Norms and the slice filtration 67 13.1. The zeroth slice of a normed spectrum 67 13.2. Applications to motivic cohomology 69 13.3. Graded normed spectra 71 13.4. The graded slices of a normed spectrum 74 14. Norms of cycles 75 14.1. Norms of presheaves with transfers 75 14.2. The Fulton{MacPherson norm on Chow groups 77 14.3. Comparison of norms 78 15. Norms of linear 1-categories 82 15.1. Linear 1-categories 83 15.2. Noncommutative motivic spectra and homotopy K-theory 85 15.3. Nonconnective K-theory 90 16. Motivic Thom spectra 92 16.1. The motivic Thom spectrum functor 92 16.2. Algebraic cobordism and the motivic J-homomorphism 95 16.3. Multiplicative properties 96 16.4. Free normed spectra 100 16.5. Thom isomorphisms 101 Appendix A. The Nisnevich topology 103 Appendix B. Detecting effectivity 104 Appendix C. Categories of spans 106 C.1. Spans in extensive 1-categories 106 C.2. Spans and descent 110 C.3. Functoriality of spans 113 Appendix D. Relative adjunctions 114 Table of notation 116 References 117 NORMS IN MOTIVIC HOMOTOPY THEORY 3 1. Introduction The goal of this paper is to develop the formalism of norm functors in motivic homotopy theory, to study the associated notion of normed motivic spectrum, and to construct many examples. 1.1. Norm functors. Norm functors in unstable motivic homotopy theory were first introduced by Voevod- sky in [Del09, x5.2], in order to extend the symmetric power functors to the category of motives [Voe10a, x2]. In this work, we will be mostly interested in norm functors in stable motivic homotopy theory. For every finite ´etalemorphism of schemes f : T ! S, we will construct a norm functor f⊗ : SH(T ) ! SH(S); where SH(S) is Voevodsky's 1-category of motivic spectra over S [Voe98, Definition 5.7]. Another functor associated with f is the pushforward f∗ : SH(T ) ! SH(S). For comparison, if r: S t S ! S is the fold map, then r∗ : SH(S t S) ' SH(S) × SH(S) ! SH(S) is the direct sum functor, whereas r⊗ : SH(S t S) ' SH(S) × SH(S) ! SH(S) is the smash product functor. This example already shows one of the main technical difficulties in the theory of norm functors: unlike f∗, the functor f⊗ preserves neither limits nor colimits. The norm functors are thus a new basic functoriality of stable motivic homotopy theory, to be added to ∗ ! the list f], f , f∗, f!, f . The three covariant functors f], f∗, and f⊗ should be viewed as parametrized versions of the sum, product, and tensor product in SH(S): categorical structure parametrized extension sum f] for f smooth product f∗ for f proper tensor product f⊗ for f finite ´etale Of course, the sum and the product coincide in SH(S), and much more generally the functors f] and f∗ coincide up to a \twist" when f is both smooth and proper. The basic interactions between sums, products, and tensor products extend to this parametrized setting. For example, the associativity and commutativity of these operations are encoded in the compatibility of the assignments f 7! f]; f∗; f⊗ with composition and base change. The distributivity of tensor products over sums and products has a parametrized extension as well. To express these properties in a coherent manner, we are led to consider 2-categories of spans1, which will be ubiquitous throughout the paper. The most fundamental example is the 2-category Span(Fin) of spans of finite sets: its objects are finite sets, its morphisms are spans X Y ! Z, and composition of morphisms is given by pullback. The one-point set in this 2-category happens to be the universal commutative monoid: if C is any 1-category with finite products, then a finite-product-preserving functor Span(Fin) ! C is precisely a commutative monoid in C (we refer to Appendix C for background on 1-categories of spans and for a proof of this fact). For example, the 1-category SH(S) is itself a commutative monoid in the 1-category Cat1 of 1-categories, under either the direct sum or the smash product. These commutative monoid structures correpond to functors ⊕ ⊗ I SH(S) ; SH(S) : Span(Fin) ! Cat1;I 7! SH(S) : I ` Using that SH(S) ' SH( I S), one can splice these functors together as S varies to obtain ⊕ ⊗ SH ; SH : Span(Sch; all; fold) ! Cat1;S 7! SH(S): Here, Span(Sch; all; fold) is a 2-category whose objects are schemes and whose morphisms are spans X Y ! Z where X Y is arbitrary and Y ! Z is a sum of fold maps (i.e., there is a finite coproduct ` tni decomposition Z = i Zi and isomorphisms Y ×Z Zi ' Zi over Zi). It turns out that the functors SH⊕ and SH⊗ encode exactly the same information as the presheaves of symmetric monoidal 1-categories 1All the 2-categories that appear in this introduction are (2; 1)-categories, i.e., their 2-morphisms are invertible. 4 TOM BACHMANN AND MARC HOYOIS S 7! SH(S)⊕ and S 7! SH(S)⊗. The basic properties of parametrized sums, products, and tensor products are then encoded by extensions of SH⊕ and SH⊗ to larger 2-categories of spans: t f p ∗ SH : Span(Sch; all; smooth) ! Cat1;S 7! SH(S); (X − Y −! Z) 7! p]f ; × f p ∗ SH : Span(Sch; all; proper) ! Cat1;S 7! SH(S); (X − Y −! Z) 7! p∗f ; ⊗ f p ∗ SH : Span(Sch; all; f´et) ! Cat1;S 7! SH(S); (X − Y −! Z) 7! p⊗f ; where \f´et"is the class of finite ´etalemorphisms. A crucial difference between the first two functors and the third one is that the former are completely determined by their restriction to Schop and the knowledge ∗ t × that the functors f] and f∗ are left and right adjoint to f . The existence of the extensions SH and SH is then a formal consequence of the smooth base change theorem (which holds in SH by design) and the proper base change theorem (formulated by Voevodsky [Del02] and proved by Ayoub [Ayo08]). By contrast, the functor SH⊗ must be constructed by hand.
Load more

Recommended publications
  • Partial Mal'tsevness and Partial Protomodularity
    PARTIAL MAL’TSEVNESS AND PARTIAL PROTOMODULARITY DOMINIQUE BOURN Abstract. We introduce the notion of Mal’tsev reflection which allows us to set up a partial notion of Mal’tsevness with respect to a class Σ of split epimorphisms stable under pullback and containing the isomorphisms, and we investigate what is remaining of the properties of the global Mal’tsev context. We introduce also the notion of partial protomodularity in the non-pointed context. Introduction A Mal’tsev category is a category in which any reflexive relation is an equivalence relation, see [9] and [10]. The categories Gp of groups and K-Lie of Lie K-algebras are major examples of Mal’tsev categories. The terminology comes from the pi- oneering work of Mal’tsev in the varietal context [16] which was later on widely developped in [18]. In [3], Mal’tsev categories were characterized in terms of split epimorphisms: a finitely complete category D is a Mal’tsev one if and only if any pullback of split epimorphisms in D: t¯ X′ o X O g¯ / O ′ ′ f s f s t Y ′ o Y g / is such that the pair (s′, t¯) is jointly extremally epic. More recently the same kind of property was observed, but only for a certain class Σ of split epimorphisms (f,s) which is stable under pullback and contains arXiv:1507.02886v1 [math.CT] 10 Jul 2015 isomorphisms. By the classes of Schreier or homogeneous split epimorphisms in the categories Mon of monoids and SRng of semi-rings [8], by the classes of puncturing or acupuncturing split epimorphims in the category of quandles [6], by the class of split epimorphic functors with fibrant splittings in the category CatY of categories with a fixed set of objects Y [5].
    [Show full text]
  • Arxiv:0906.3146V1 [Math.NT] 17 Jun 2009
    Λ-RINGS AND THE FIELD WITH ONE ELEMENT JAMES BORGER Abstract. The theory of Λ-rings, in the sense of Grothendieck’s Riemann– Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce Λ-algebraic geometry. We show that Λ-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry. Introduction Many writers have mused about algebraic geometry over deeper bases than the ring Z of integers. Although there are several, possibly unrelated reasons for this, here I will mention just two. The first is that the combinatorial nature of enumer- ation formulas in linear algebra over finite fields Fq as q tends to 1 suggests that, just as one can work over all finite fields simultaneously by using algebraic geome- try over Z, perhaps one could bring in the combinatorics of finite sets by working over an even deeper base, one which somehow allows q = 1. It is common, follow- ing Tits [60], to call this mythical base F1, the field with one element. (See also Steinberg [58], p. 279.) The second purpose is to prove the Riemann hypothesis. With the analogy between integers and polynomials in mind, we might hope that Spec Z would be a kind of curve over Spec F1, that Spec Z ⊗F1 Z would not only make sense but be a surface bearing some kind of intersection theory, and that we could then mimic over Z Weil’s proof [64] of the Riemann hypothesis over function fields.1 Of course, since Z is the initial object in the category of rings, any theory of algebraic geometry over a deeper base would have to leave the usual world of rings and schemes.
    [Show full text]
  • MAPPING STACKS and CATEGORICAL NOTIONS of PROPERNESS Contents 1. Introduction 2 1.1. Introduction to the Introduction 2 1.2
    MAPPING STACKS AND CATEGORICAL NOTIONS OF PROPERNESS DANIEL HALPERN-LEISTNER AND ANATOLY PREYGEL Abstract. One fundamental consequence of a scheme being proper is that there is an algebraic space classifying maps from it to any other finite type scheme, and this result has been extended to proper stacks. We observe, however, that it also holds for many examples where the source is a geometric stack, such as a global quotient. In our investigation, we are lead naturally to certain properties of the derived category of a stack which guarantee that the mapping stack from it to any geometric finite type stack is algebraic. We develop methods for establishing these properties in a large class of examples. Along the way, we introduce a notion of projective morphism of algebraic stacks, and prove strong h-descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. Contents 1. Introduction 2 1.1. Introduction to the introduction2 1.2. Mapping out of stacks which are \proper enough"3 1.3. Techniques for establishing (GE) and (L)5 1.4. A long list of examples6 1.5. Comparison with previous results7 1.6. Notation and conventions8 1.7. Author's note 9 2. Artin's criteria for mapping stacks 10 2.1. Weil restriction of affine stacks 11 2.2. Deformation theory of the mapping stack 12 2.3. Integrability via the Tannakian formalism 16 2.4. Derived representability from classical representability 19 2.5. Application: (pGE) and the moduli of perfect complexes 21 3. Perfect Grothendieck existence 23 3.1.
    [Show full text]
  • Universality of Multiplicative Infinite Loop Space Machines
    UNIVERSALITY OF MULTIPLICATIVE INFINITE LOOP SPACE MACHINES DAVID GEPNER, MORITZ GROTH AND THOMAS NIKOLAUS Abstract. We establish a canonical and unique tensor product for commutative monoids and groups in an ∞-category C which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that En-(semi)ring objects in C give rise to En-ring spectrum objects in C. In the case that C is the ∞-category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic K-theory of rings and ring spectra. The main tool we use to establish these results is the theory of smashing localizations of presentable ∞-categories. In particular, we identify preadditive and additive ∞-categories as the local objects for certain smashing localizations. A central theme is the stability of algebraic structures under basechange; for example, we show Ring(D ⊗ C) ≃ Ring(D) ⊗ C. Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in ∞-categories. Contents 0. Introduction 1 1. ∞-categories of commutative monoids and groups 4 2. Preadditive and additive ∞-categories 6 3. Smashing localizations 8 4. Commutative monoids and groups as smashing localizations 11 5. Canonical symmetric monoidal structures 13 6. More functoriality 15 7. ∞-categories of semirings and rings 17 8. Multiplicative infinite loop space theory 19 Appendix A. Comonoids 23 Appendix B. Algebraic theories and monadic functors 23 References 26 0. Introduction The Grothendieck group K0(M) of a commutative monoid M, also known as the group completion, is the universal abelian group which receives a monoid map from M.
    [Show full text]
  • Monads of Effective Descent Type and Comonadicity
    Theory and Applications of Categories, Vol. 16, No. 1, 2006, pp. 1–45. MONADS OF EFFECTIVE DESCENT TYPE AND COMONADICITY BACHUKI MESABLISHVILI Abstract. We show, for an arbitrary adjunction F U : B→Awith B Cauchy complete, that the functor F is comonadic if and only if the monad T on A induced by the adjunction is of effective descent type, meaning that the free T-algebra functor F T : A→AT is comonadic. This result is applied to several situations: In Section 4 to give a sufficient condition for an exponential functor on a cartesian closed category to be monadic, in Sections 5 and 6 to settle the question of the comonadicity of those functors whose domain is Set,orSet, or the category of modules over a semisimple ring, in Section 7 to study the effectiveness of (co)monads on module categories. Our final application is a descent theorem for noncommutative rings from which we deduce an important result of A. Joyal and M. Tierney and of J.-P. Olivier, asserting that the effective descent morphisms in the opposite of the category of commutative unital rings are precisely the pure monomorphisms. 1. Introduction Let A and B be two (not necessarily commutative) rings related by a ring homomorphism i : B → A. The problem of Grothendieck’s descent theory for modules with respect to i : B → A is concerned with the characterization of those (right) A-modules Y for which there is X ∈ ModB andanisomorphismY X⊗BA of right A-modules. Because of a fun- damental connection between descent and monads discovered by Beck (unpublished) and B´enabou and Roubaud [6], this problem is equivalent to the problem of the comonadicity of the extension-of-scalars functor −⊗BA :ModB → ModA.
    [Show full text]
  • FINITE WEIL RESTRICTION of CURVES Introduction Let L Be A
    FINITE WEIL RESTRICTION OF CURVES E.V. FLYNN AND D. TESTA Abstract. Given number fields L ⊃ K, smooth projective curves C defined over L and B defined over K, and a non-constant L-morphism h: C ! BL, we denote by Ch the curve defined over K whose K-rational points parametrize the L-rational points on C whose images under h are defined over K. We compute the geometric genus of the curve Ch and give a criterion for the applicability of the Chabauty method to find the points of the curve Ch. We provide a framework which includes as a special case that used in Elliptic Curve Chabauty techniques and their higher genus versions. The set Ch(K) can be infinite only when C has genus at most 1; we analyze completely the case when C has genus 1. Introduction Let L be a number field of degree d over Q, let f 2 L[x] be a polynomial with coefficients in L, and define Af := x 2 L j f(x) 2 Q : Choosing a basis of L over Q and writing explicitly the conditions for an element of L to lie in Af , it is easy to see that the set Af is the set of rational solutions of d − 1 polynomials in d variables. Thus we expect the set Af to be the set of rational points of a (possibly reducible) curve Cf ; indeed, this is always true when f is non-constant. A basic question that we would like to answer is to find conditions on L and f that guarantee that the set Af is finite, and ideally to decide when standard techniques can be applied to explicitly determine this set.
    [Show full text]
  • Protomodular Aspect of the Dual of a Topos
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Advances in Mathematics 187 (2004) 240–255 http://www.elsevier.com/locate/aim Protomodular aspect of the dual of a topos Dominique Bourn Centre Universitaire de la Mi-voix, Lab. d’Analyse, Ge´ome´trie et Alge`bre, Universite´ du Littoral, 50 rue Ferdinand Buisson, BP699, 62228 Calais Cedex, France Received 3 March 2003; accepted 15 September 2003 Communicated by Ross Street Abstract The structure of the dual of a topos is investigated under its aspect of a Barr exact and protomodular category. In particular the normal monomorphisms in the fibres of the fibration of pointed objects are characterized, and the change of base functors with respect to this same fibration are shown to reflect those normal monomorphisms. r 2003 Elsevier Inc. All rights reserved. MSC: 18B25; 18D30; 08B10 Keywords: Topos; Fibrations; Mal’cev and protomodular categories 0. Introduction It is well known that, given a topos E; the subobject classifier O is endowed with an internal structure of Heyting Algebra, and that consequently the contravariant ‘‘power-set’’ functor OðÞ : Eop-E takes its values in the category HeytE of internal Heyting algebras in E: But the category Heyt of Heyting Algebras shares with the categories Gp of groups or Rg of rings the property of being protomodular. This property [3] says that the fibration of pointed objects p : PtC-C; whose fibre at X is the category PtX C of split epimorphisms with codomain X; has its change of base functors conservative (i.e.
    [Show full text]
  • A Cryptographic Application of Weil Descent
    A Cryptographic Application of Weil Descent Nigel Smart, Steve Galbraith† Extended Enterprise Laboratory HP Laboratories Bristol HPL-1999-70 26 May, 1999* function fields, This paper gives some details about how Weil descent divisor class group, can be used to solve the discrete logarithm problem on cryptography, elliptic curves which are defined over finite fields of elliptic curves small characteristic. The original ideas were first introduced into cryptography by Frey. We discuss whether these ideas are a threat to existing public key systems based on elliptic curves. † Royal Holloway College, University of London, Egham, UK *Internal Accession Date Only Ó Copyright Hewlett-Packard Company 1999 A CRYPTOGRAPHIC APPLICATION OF WEIL DESCENT S.D. GALBRAITH AND N.P. SMART Abstract. This pap er gives some details ab out howWeil descent can b e used to solve the discrete logarithm problem on elliptic curves which are de ned over nite elds of small characteristic. The original ideas were rst intro duced into cryptographybyFrey.We discuss whether these ideas are a threat to existing public key systems based on elliptic curves. 1. Introduction Frey [4] intro duced the cryptographic community to the notion of \Weil descent", n which applies to elliptic curves de ned over nite elds of the form F with n>1. q This pap er gives further details on how these ideas might be applied to give an attack on the elliptic curve discrete logarithm problem. We also discuss which curves are most likely to b e vulnerable to such an attack. The basic strategy consists of the following four stages which will b e explained in detail in the remainder of the pap er.
    [Show full text]
  • Invertible Objects in Motivic Homotopy Theory
    Invertible Objects in Motivic Homotopy Theory Tom Bachmann M¨unchen2016 Invertible Objects in Motivic Homotopy Theory Tom Bachmann Dissertation an der Fakult¨atf¨urMathematik, Informatik und Statistik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Tom Bachmann aus Chemnitz M¨unchen, den 19. Juli 2016 Erstgutachter: Prof. Dr. Fabien Morel Zweitgutachter: Prof. Dr. Marc Levine Drittgutachter: Prof. Dr. math. Oliver R¨ondigs Tag der m¨undlichen Pr¨ufung:18.11.2016 Eidesstattliche Versicherung Bachmann, Tom Name, Vorname Ich erkläre hiermit an Eides statt, dass ich die vorliegende Dissertation mit dem Thema Invertible Objects in Motivic Homotopy Theory selbständig verfasst, mich außer der angegebenen keiner weiteren Hilfsmittel bedient und alle Erkenntnisse, die aus dem Schrifttum ganz oder annähernd übernommen sind, als solche kenntlich gemacht und nach ihrer Herkunft unter Bezeichnung der Fundstelle einzeln nachgewiesen habe. Ich erkläre des Weiteren, dass die hier vorgelegte Dissertation nicht in gleicher oder in ähnlicher Form bei einer anderen Stelle zur Erlangung eines akademischen Grades eingereicht wurde. München, 1.12.2016 Ort, Datum Unterschrift Doktorandin/Doktorand Eidesstattliche Versicherung Stand: 31.01.2013 vi Abstract If X is a (reasonable) base scheme then there are the categories of interest in stable motivic homotopy theory SH(X) and DM(X), constructed by Morel-Voevodsky and others. These should be thought of as generalisations respectively of the stable homotopy category SH and the derived category of abelian groups D(Ab), which are studied in classical topology, to the \world of smooth schemes over X". Just like in topology, the categories SH(X); DM(X) are symmetric monoidal: there is a bifunctor (E; F ) 7! E ⊗ F satisfying certain properties; in particular there is a unit 1 satisfying E ⊗ 1 ' 1 ⊗ E ' E for all E.
    [Show full text]
  • Arxiv:Math/0504359V2 [Math.AG] 7 Aug 2005
    On the Structure of Weil Restrictions of Abelian Varieties Claus Diem Niko Naumann June 10, 2003 Abstract We give a description of endomorphism rings of Weil restrictions of abelian varieties with respect to finite Galois extensions of fields. The results are applied to study the isogeny decompositions of Weil restrictions. 2000 Mathematics Subject Classification Primary: 14K15, Secondary: 11G10. Introduction For the use of Weil restrictions of abelian varieties in various fields of math- ematics but also because of genuine interest in Weil restrictions themselves, it is important to determine the endomorphism rings and the isogeny de- compositions. This is what this article provides – at least in two important special cases. After giving a brief expos´eof general facts about Weil restrictions of abelian varieties in the first section, we study Weil restrictions with respect to extensions of finite fields in the second section. Here we determine the endomorphism algebra of a Weil restriction (see Theorem 1) and then show arXiv:math/0504359v2 [math.AG] 7 Aug 2005 that under rather general assumptions, the Weil restriction is simple over the base-field (see Theorem 2). In the third section, we deal with the following situation: K k is an ar- | bitrary finite Galois extension of fields, A an abelian variety over k, W the Weil restriction of AK with respect to K k. We describe the endomorphism | ring of W as a skew group ring over End(AK ) (see Theorem 3) and apply this result to study the isogeny decomposition of W over k. In the last sub- section the results are applied to give an explicit description of the isogeny decomposition of W in the case of a cyclic field extension; see Theorem 4.
    [Show full text]
  • Arxiv:1311.0051V4 [Math.NT] 8 Nov 2016 3 Elrsrcinadtegenegfntr75 65 44 27 49 5 Functor Greenberg the and Restriction Weil Theorem 25 Functor Structure 13
    THE GREENBERG FUNCTOR REVISITED ALESSANDRA BERTAPELLE AND CRISTIAN D. GONZALEZ-AVIL´ ES´ Abstract. We extend Greenberg’s original construction to arbitrary (in partic- ular, non-reduced) schemes over (certain types of) local artinian rings. We then establish a number of basic properties of the extended functor and determine, for example, its behavior under Weil restriction. We also discuss a formal analog of the functor. Contents 1. Introduction 2 2. Preliminaries 4 2.1. Generalities 4 2.2. Vector bundles 7 2.3. Witt vectors 8 2.4. Groups of components 10 2.5. The connected-´etale sequence 11 2.6. Formal schemes 12 2.7. Weil restriction 16 2.8. The fpqc topology 20 3. Greenberg algebras 25 3.1. Finitely generated modules over arbitrary fields 25 3.2. Modules over rings of Witt vectors 27 4. The Greenberg algebra of a truncated discrete valuation ring 37 5. Greenbergalgebrasandramification 44 6. The Greenberg algebra of a discrete valuation ring 49 7. The Greenberg functor 50 8. The Greenberg functor of a truncated discrete valuation ring 56 9. The change of rings morphism 57 arXiv:1311.0051v4 [math.NT] 8 Nov 2016 10. The change of level morphism 64 11. Basic properties of the Greenberg functor 65 12. Greenberg’s structure theorem 69 13. Weil restriction and the Greenberg functor 75 Date: June 5, 2018. 2010 Mathematics Subject Classification. Primary 11G25, 14G20. Key words and phrases. Greenberg realization, formal schemes, Weil restriction. C.G.-A. was partially supported by Fondecyt grants 1120003 and 1160004. 1 2 ALESSANDRABERTAPELLEANDCRISTIAND.GONZALEZ-AVIL´ ES´ 14.
    [Show full text]
  • Weil Restriction for Schemes and Beyond
    WEIL RESTRICTION FOR SCHEMES AND BEYOND LENA JI, SHIZHANG LI, PATRICK MCFADDIN, DREW MOORE, & MATTHEW STEVENSON Abstract. This note was produced as part of the Stacks Project workshop in August 2017, under the guidance of Brian Conrad. Briefly, the process of Weil restriction is the algebro-geometric analogue of viewing a complex- analytic manifold as a real-analytic manifold of twice the dimension. The aim of this note is to discuss the Weil restriction of schemes and algebraic spaces, highlighting pathological phenoma that appear in the theory and are not widely-known. 1. Weil Restriction of Schemes Definition 1.1. Let S0 ! S be a morphism of schemes. Given an S0-scheme X0, 0 opp consider the contravariant functor RS0=S(X ): (Sch=S) ! (Sets) given by 0 0 T 7! X (T ×S S ): 0 If the functor RS0=S(X ) is representable by an S-scheme X, then we say that X is 0 0 0 the Weil restriction of X along S ! S, and we denote it by RS0=S(X ). Remark 1.2. Given any morphism of schemes f : S0 ! S and any contravari- ant functor F : (Sch=S0)opp ! (Sets), one can define the pushforward functor opp 0 f∗F : (Sch=S) ! (Sets) given on objects by T 7! F (T ×S S ). If F is repre- 0 0 0 sentable by an S -scheme X , then f∗F = RS0=S(X ). This approach is later used to define the Weil restriction of algebraic spaces. Theorem 1.3. Let S0 ! S be a finite and locally free morphism.
    [Show full text]