DOCSLIB.ORG

Cohomological Descent for Logarithmic Differential Forms in the Log Etale

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Cohomological Descent for Logarithmic Differential Forms in the Log Etale Cohomological descent for logarithmic differential forms in the log etale topology Dissertation zur Erlangung des Doktorgrades vorgelegt von Vetere Elmiro an der Fakult¨atf¨urMathematik und Physik der Albert-Ludwigs-Universit¨atFreiburg Oktober 2018 Dekan: Prof. Dr. Gregor Herten Referenten: Prof. Dr. Annette Huber-Klawitter Prof. Dr. Stefan Kebekus Prof. Dr. Wies lawa Nizio l Datum der Promotion: 28.11.2018 Abstract (in German) In dieser Dissertation beweisen wir kohomologischen Abstieg zwischen dem (kleinen) ´etalen Situs und dem logarithmischen Situs f¨urdie Garbe der logarithmischen Differentialformen. Die existierenden Methoden in der Literatur erlauben uns, kohomologischen Abstieg f¨ureine gr¨oßereKlasse von Garben zu beweisen. Ins- besondere vergleichen wir die Kummer log ´etaleKohomologie mit der ´etalenKo- homologie f¨urquasi-koh¨arente Garben auf dem Kummer log ´etalenSitus. Damit beweisen wir, dass Morphismen von log Schemata, f¨urdie der zugrun- deliegende Morphismus von Schemata affin ist, triviale h¨oheredirekte Bilder f¨ur ket lokal klassische quasi-koh¨arente Garben auf dem Kummer log ´etalenSitus haben. In derselben Weise vergleichen wir die log ´etaleKohomologie mit der ´etalen Kohomologie f¨urklassische Vektorb¨undel auf dem log ´etalenSitus. Wir geben außerdem einen Beweis f¨urdie als bekannt angesehenen Tatsache, dass log regul¨arelog Schemata rational singul¨ar sind. Der Fall von Zariski log Schemata wurde bereits von K. Kato in [Kat94] behandelt. Wir geben einen Beweis f¨ur den Fall von ´etalenlog Schemata. Als letzte Anwendung definieren wir algebraische log de Rham Kohomologie f¨urfs log Schemata, mithilfe des log etalen Situs. Das liefert einen neuen Beweis f¨ur den Satz von A. Ogus, [Ogu18, Theorem V.4.2.5(1)], ¨uber die Verallgemeinerung algebraischer de Rham Kohomologie f¨urSchemata mit torischen Singularit¨aten. Abstract (in English) In this thesis we prove cohomological descent between the (small) etale site and the logarithmic sites for the sheaf of logarithmic differential forms. The existing techniques in the literature allow us to prove cohomological descent for a broader class of sheaves. Specifically, we compare the Kummer log etale cohomology with the etale coho- mology for quasi-coherent sheaves defined on the Kummer log etale site. By using this, we prove that morphisms of log schemes where the underlying morphism of schemes is affine have trivial higher direct images for ket locally classical quasi- coherent sheaves on the Kummer log etale site. In the same spirit, we compare the log etale cohomology with the etale cohomology for classical vector bundles on the log etale site. We also give a proof for the fact that log regular log schemes are rationally singular, something which is considered known. The case of Zariski log schemes has been already treated by K. Kato in [Kat94]. We provide a proof for the case of etale log schemes. As a final application we define algebraic log de Rham cohomology for fs log schemes using the log etale site. Along these lines, we prove with a different method the theorem of A. Ogus, [Ogu18, Theorem V.4.2.5(1)], regarding the gen- eralization of algebraic de Rham cohomology to schemes with toric singularities. QwrÐc mtaiec antarsÐec na deic kai na deqteÐc ta sÔnora tou anjr¸pinou nou, kai mèsa st' austhr toÔta sÔnora adiamartÔrhta, akatpauta na douleÔeic. Na poio eÐnai to pr¸to sou qrèoc. Me antreÐa, me sklhrìthta sterèwse pnw sto saleuìmeno qoc to katastrìg- gulo, to kataf¸tisto al¸ni tou nou, n' alwnÐseic, na liqnÐseic, sa noikokÔrhc, ta sÔmpanta. Kajar na xeqwrÐseic ki hrwik na deqteÐc tic pikrèc gìnimec toÔtec, anjr¸pinec, srka apì th srka mac, al jeiec: a) O nouc tou anjr¸pou fainìmena monqa mporeÐ na sullbei, potè thn ousÐa. b) Ki ìqi ìla ta fainìmena, par monqa ta fainìmena thc Ôlhc. g) Ki akìma sten¸tera: ìqi kan ta fainìmena toÔta thc Ôlhc, par monqa touc metaxÔ touc suneirmoÔc. d) Ki oi suneirmoÐ toÔtoi den eÐnai pragmatikoÐ, anexrthtoi apì ton njrwpo. EÐnai ki autoÐ genn mata tou anjr¸pou. e) Kai den eÐnai oi mìnoi dunatoÐ anjr¸pinoi, par monqa oi pio bolikoÐ gia tic praqtikèc kai nohtikèc tou angkec. Mèsa sta sÔnora toÔta, o nouc eÐnai o nìmimoc apìlutoc monrqhc. Kami llh exousÐa sto basÐleio tou den uprqei. AnagnwrÐzw ta sÔnora toÔta, ta dèqoumai m' egkartèrhsh, gennaiìthta ki agph, ki agwnÐzoumai mèsa sthn perioq touc neta sa na 'moun eleÔteroc. Upotzw thn Ôlh, thn anagkzw na gÐnei kalìc agwgìc tou mualoÔ mou. Qa- Ðroumai ta fut, ta z¸a, touc anjr¸pouc, touc jeoÔc san paidi mou. 'Olo to SÔmpanto to ni¸jw na sofilizei apnw mou kai na me akoloujei sa s¸ma. Se xafnec foberèc stigmèc astrftei mèsa mou: -'Ola toÔta eÐnai paiqnÐdi sklhrì kai mtaio, dÐqwc arq , dÐqwc tèloc, dÐqwc nìhma. Ma xanazeÔoumai, pli, gorg ston troqì thc angkhc ki ìlo to SÔmpanto xanarqinei gÔra trogÔra mou thn peristrof tou. PeijarqÐa, na h an¸tath aret . 'Etsi monqa sozugizetai h dÔnamh me thn epijumÐa kai karpÐzei h prospjeia tou anjr¸pou. Na pwc me saf neia kai me sklhrìthta na kajorÐzeic thn pantodunamÐa tou nou mèsa sta fainìmena kai thn anikanìthta tou nou pèra apì ta fainìmena, prin na kin seic gia th lÔtrwsh. Alli¸c den mporeÐc na lutrwjeÐc. Askhtik , NÐkoc Kazantzkhc To see and accept the boundaries of the human mind without vain rebellion, and in these severe limitations to work ceaselessly without protest - this is where man's first duty lies. Build over the unsteady abyss, with manliness and austerity, the fully round and luminous arena of the mind where you may thresh and winnow the universe like a lord of the land. Distinguish clearly these bitter yet fertile human truths, flesh of our flesh, and admit them heroically: a) The mind of man can perceive appearances only, and never the essence of things. b) And not all appearances but only the appearances of matter. c) And more narrowly still: not even these appearances of matter, but only relationships between them. d) And these relationships are not real and independent of man, for even these are his creations. e) And they are not the only ones humanly possible, but simply the most convenient for his practical and perceptive needs. Within these limitations the mind is the legal and absolute monarch. No other power reigns within its kingdom. I recognize these limitations, I accept them with resignation, bravery, and love, and I struggle at ease in their closure, as though I were free. I subdue matter and force it to become my mind's good medium. I rejoice in plants, in animals, in man and in gods, as though they were my children. I feel all the universe nestling about me and following me as though it were my own body. In sudden dreadful moments a thought flashes through me: -"This is all a cruel and futile game, without beginning, without end, without meaning". But again I yoke myself swiftly to the wheels of necessity, and all the universe begins to revolve around me once more. Discipline is the highest of all virtues. Only so may strength and desire be counterbalanced and the endeavors of man bear fruit. This is how, with clarity and austerity, you may determine the omnipotence of the mind amid appearances and the incapacity of the mind beyond appearances - before you set out for salvation. You may not otherwise be saved. The Saviors of God (Salvatores dei), Nikos Kazantzakis Contents Introduction Acknowledgments 1 Monoids and Log schemes1 1.1 Monoids................................ 1 1.2 Kummer and exact homomorphisms of monoids.......... 6 1.3 Pushouts in the category of fs monoids............... 8 1.4 Monoid algebras and Toric varieties................. 10 1.5 Log structures and log schemes ................... 16 1.6 Fs log schemes and charts ...................... 20 1.7 Strict, Kummer morphisms of log schemes............. 24 1.8 Fiber products in the category of coherent log schemes . 25 2 Differential forms on Log schemes 28 2.1 Logarithmic differential forms.................... 28 2.2 Log deformation theory........................ 34 3 Log regularity and Log blow-ups 38 3.1 Log regularity............................. 38 3.2 Divisorial log structures ....................... 43 3.3 Log blow-ups ............................. 45 4 Log sites 51 4.1 The Kummer log etale topology................... 52 4.2 The log etale topology ........................ 57 5 Sheaves on Log sites 60 5.1 Intoduction .............................. 60 5.2 ket descent for the extension presheaf................ 62 5.3 let descent for the extension presheaf................ 69 5.4 Classical and locally classical sheaves................ 73 6 Cohomological descent for Log sites 77 6.1 Comparison of ket cohomology and etale cohomology . 77 6.2 Comparison of let cohomology and etale cohomology . 80 7 Applications and concluding remarks 84 7.1 Applications: Algebraic log de Rham cohomology for fs log schemes 84 7.2 Concluding remarks.......................... 90 8 Appendix 92 Notation (X; triv) The log scheme with underlying space X and the trivial log structure. Xtriv The trivial locus of the log structure of X: X The underlying scheme of a log scheme X: f The underlying morphism of schemes of a log morphism f: X´et The etale site for the underlying scheme of a log scheme (X; M): Xket The Kummer log etale site for a log scheme (X; M): Xlet The log etale site for a log scheme (X; M): Fτ The extension presheaf of a module F for the topology τ: OXτ The sheaf of rings on the site X equiped with the τ topology. Sh(Xτ ) The category of sheaves for the topology τ over the scheme/log scheme (X; M): Mod The category of modules over a scheme/log scheme. Qcoh The category of quasicoherent modules over a scheme/log scheme. Vect The category of locally free sheaves of finite rank over a scheme/log scheme.
Load more

Recommended publications
  • Rational Curves on Hypersurfaces
    Rational curves on hypersurfaces ||||||{ Lecture notes for the II Latin American School of Algebraic Geometry and Applications 1{12 June 2015 Cabo Frio, Brazil ||||||{ Olivier Debarre January 4, 2016 Contents 1 Projective spaces and Grassmannians 3 1.1 Projective spaces . 3 1.2 The Euler sequence . 4 1.3 Grassmannians . 4 1.4 Linear spaces contained in a subscheme of P(V )................ 5 2 Projective lines contained in a hypersurface 9 2.1 Local study of F (X) ............................... 9 2.2 Schubert calculus . 10 2.3 Projective lines contained in a hypersurface . 12 2.3.1 Existence of lines in a hypersurface . 12 2.3.2 Lines through a point . 16 2.3.3 Free lines . 17 2.4 Projective lines contained in a quadric hypersurface . 18 2.5 Projective lines contained in a cubic hypersurface . 19 2.5.1 Lines on a smooth cubic surface . 20 2.5.2 Lines on a smooth cubic fourfold . 22 2.6 Cubic hypersurfaces over finite fields . 25 3 Conics and curves of higher degrees 29 3.1 Conics in the projective space . 29 3.2 Conics contained in a hypersurface . 30 3.2.1 Conics contained in a quadric hypersurface . 32 3.2.2 Conics contained in a cubic hypersurface . 33 3.3 Rational curves of higher degrees . 35 4 Varieties covered by rational curves 39 4.1 Uniruled and separably uniruled varieties . 39 4.2 Free rational curves and separably uniruled varieties . 41 4.3 Minimal free rational curves . 44 Abstract In the past few decades, it has become more and more obvious that rational curves are the primary player in the study of the birational geometry of projective varieties.
    [Show full text]
  • The Jouanolou-Thomason Homotopy Lemma
    The Jouanolou-Thomason homotopy lemma Aravind Asok February 9, 2009 1 Introduction The goal of this note is to prove what is now known as the Jouanolou-Thomason homotopy lemma or simply \Jouanolou's trick." Our main reason for discussing this here is that i) most statements (that I have seen) assume unncessary quasi-projectivity hypotheses, and ii) most applications of the result that I know (e.g., in homotopy K-theory) appeal to the result as merely a \black box," while the proof indicates that the construction is quite geometric and relatively explicit. For simplicity, throughout the word scheme means separated Noetherian scheme. Theorem 1.1 (Jouanolou-Thomason homotopy lemma). Given a smooth scheme X over a regular Noetherian base ring k, there exists a pair (X;~ π), where X~ is an affine scheme, smooth over k, and π : X~ ! X is a Zariski locally trivial smooth morphism with fibers isomorphic to affine spaces. 1 Remark 1.2. In terms of an A -homotopy category of smooth schemes over k (e.g., H(k) or H´et(k); see [MV99, x3]), the map π is an A1-weak equivalence (use [MV99, x3 Example 2.4]. Thus, up to A1-weak equivalence, any smooth k-scheme is an affine scheme smooth over k. 2 An explicit algebraic form Let An denote affine space over Spec Z. Let An n 0 denote the scheme quasi-affine and smooth over 2m Spec Z obtained by removing the fiber over 0. Let Q2m−1 denote the closed subscheme of A (with coordinates x1; : : : ; x2m) defined by the equation X xixm+i = 1: i Consider the following simple situation.
    [Show full text]
  • Vector Bundles As Generators on Schemes and Stacks
    Vector Bundles as Generators on Schemes and Stacks Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨atD¨usseldorf vorgelegt von Philipp Gross aus D¨usseldorf D¨usseldorf,Mai 2010 Aus dem Mathematischen Institut der Heinrich-Heine-Universit¨atD¨usseldorf Gedruckt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨atder Heinrich-Heine-Universit¨atD¨usseldorf Referent: Prof. Dr. Stefan Schr¨oer Koreferent: Prof. Dr. Holger Reich Acknowledgments The work on this dissertation has been one of the most significant academic challenges I have ever had to face. This study would not have been completed without the support, patience and guidance of the following people. It is to them that I owe my deepest gratitude. I am indebted to my advisor Stefan Schr¨oerfor his encouragement to pursue this project. He taught me algebraic geometry and how to write academic papers, made me a better mathematician, brought out the good ideas in me, and gave me the opportunity to attend many conferences and schools in Europe. I also thank Holger Reich, not only for agreeing to review the dissertation and to sit on my committee, but also for showing an interest in my research. Next, I thank the members of the local algebraic geometry research group for their time, energy and for the many inspiring discussions: Christian Liedtke, Sasa Novakovic, Holger Partsch and Felix Sch¨uller.I have had the pleasure of learning from them in many other ways as well. A special thanks goes to Holger for being a friend, helping me complete the writing of this dissertation as well as the challenging research that lies behind it.
    [Show full text]
  • NOTES on CARTIER and WEIL DIVISORS Recall: Definition 0.1. A
    NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil's book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil's book, and the originality of these notes lies in the mistakes. I learned some of this from [1] though. Recall: Definition 0.1. A line bundle on a ringed space X (e.g. a scheme) is a locally free sheaf of rank one. The group of isomorphism classes of line bundles is called the Picard group and is denoted Pic(X). Here is a standard source of line bundles. 1. The twisting sheaf 1.1. Twisting in general. Let R be a graded ring, R = R0 ⊕ R1 ⊕ ::: . We have discussed the construction of the scheme ProjR. Let us now briefly explain the following additional construction (which will be covered in more detail tomorrow). L Let M = Mn be a graded R-module. Definition 1.1. We define the sheaf Mf on ProjR as follows. On the basic open set D(f) = SpecR(f) ⊂ ProjR, we consider the sheaf associated to the R(f)-module M(f). It can be checked easily that these sheaves glue on D(f) \ D(g) = D(fg) and become a quasi-coherent sheaf Mf on ProjR. Clearly, the association M ! Mf is a functor from graded R-modules to quasi- coherent sheaves on ProjR. (For R reasonable, it is in fact essentially an equiva- lence, though we shall not need this.) We now set a bit of notation.
    [Show full text]
  • Bertini's Theorem on Generic Smoothness
    U.F.R. Mathematiques´ et Informatique Universite´ Bordeaux 1 351, Cours de la Liberation´ Master Thesis in Mathematics BERTINI1S THEOREM ON GENERIC SMOOTHNESS Academic year 2011/2012 Supervisor: Candidate: Prof.Qing Liu Andrea Ricolfi ii Introduction Bertini was an Italian mathematician, who lived and worked in the second half of the nineteenth century. The present disser- tation concerns his most celebrated theorem, which appeared for the first time in 1882 in the paper [5], and whose proof can also be found in Introduzione alla Geometria Proiettiva degli Iperspazi (E. Bertini, 1907, or 1923 for the latest edition). The present introduction aims to informally introduce Bertini’s Theorem on generic smoothness, with special attention to its re- cent improvements and its relationships with other kind of re- sults. Just to set the following discussion in an historical perspec- tive, recall that at Bertini’s time the situation was more or less the following: ¥ there were no schemes, ¥ almost all varieties were defined over the complex numbers, ¥ all varieties were embedded in some projective space, that is, they were not intrinsic. On the contrary, this dissertation will cope with Bertini’s the- orem by exploiting the powerful tools of modern algebraic ge- ometry, by working with schemes defined over any field (mostly, but not necessarily, algebraically closed). In addition, our vari- eties will be thought of as abstract varieties (at least when over a field of characteristic zero). This fact does not mean that we are neglecting Bertini’s original work, containing already all the rele- vant ideas: the proof we shall present in this exposition, over the complex numbers, is quite close to the one he gave.
    [Show full text]
  • On the De Rham Cohomology of Algebraic Varieties
    PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. ROBIN HARTSHORNE On the de Rham cohomology of algebraic varieties Publications mathématiques de l’I.H.É.S., tome 45 (1975), p. 5-99 <http://www.numdam.org/item?id=PMIHES_1975__45__5_0> © Publications mathématiques de l’I.H.É.S., 1975, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES by ROBIN HARTSHORNE CONTENTS INTRODUCTION .............................................................................. 6 CHAPTER I. — Preliminaries .............................................................. n § i. Cohomology theories ............................................................... 9 § 2. Direct and inverse images .......................................................... 10 § 3. Hypercohomology ................................................................. 11 § 4. Inverse limits...................................................................... 12 § 5. Completions ...................................................................... 18 §
    [Show full text]
  • The Spectral Sequence Relating Algebraic K-Theory to Motivic Cohomology
    Ann. Scient. Éc. Norm. Sup., 4e série, t. 35, 2002, p. 773 à 875. THE SPECTRAL SEQUENCE RELATING ALGEBRAIC K-THEORY TO MOTIVIC COHOMOLOGY BY ERIC M. FRIEDLANDER 1 AND ANDREI SUSLIN 2 ABSTRACT. – Beginning with the Bloch–Lichtenbaum exact couple relating the motivic cohomology of a field F to the algebraic K-theory of F , the authors construct a spectral sequence for any smooth scheme X over F whose E2 term is the motivic cohomology of X and whose abutment is the Quillen K-theory of X. A multiplicative structure is exhibited on this spectral sequence. The spectral sequence is that associated to a tower of spectra determined by consideration of the filtration of coherent sheaves on X by codimension of support. 2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. – Partant du couple exact de Bloch–Lichtenbaum, reliant la cohomologie motivique d’un corps F àsaK-théorie algébrique, on construit, pour tout schéma lisse X sur F , une suite spectrale dont le terme E2 est la cohomologie motivique de X et dont l’aboutissement est la K-théorie de Quillen de X. Cette suite spectrale, qui possède une structure multiplicative, est associée à une tour de spectres déterminée par la considération de la filtration des faisceaux cohérents sur X par la codimension du support. 2002 Éditions scientifiques et médicales Elsevier SAS The purpose of this paper is to establish in Theorem 13.6 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic K-theory of X: p,q p−q Z − −q − − ⇒ (13.6.1) E2 = H X, ( q) = CH (X, p q) K−p−q(X).
    [Show full text]
  • Cyclic Homology, Cdh-Cohomology and Negative K-Theory
    Annals of Mathematics, 167 (2008), 549–573 Cyclic homology, cdh-cohomology and negative K-theory By G. Cortinas,˜ C. Haesemeyer, M. Schlichting, and C. Weibel* Abstract We prove a blow-up formula for cyclic homology which we use to show that infinitesimal K-theory satisfies cdh-descent. Combining that result with some computations of the cdh-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic K-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero. Introduction The negative algebraic K-theory of a singular variety is related to its ge- ometry. This observation goes back to the classic study by Bass and Murthy [1], which implicitly calculated the negative K-theory of a curve X. By def- inition, the group K−n(X) describes a subgroup of the Grothendieck group 1 n K0(Y ) of vector bundles on Y = X × (A −{0}) . The following conjecture was made in 1980, based upon the Bass-Murthy calculations, and appeared in [38, 2.9]. Recall that if F is any contravariant functor on schemes, a scheme X is called F -regular if F (X) → F (X × Ar)is an isomorphism for all r ≥ 0. K-dimension Conjecture 0.1. Let X be a Noetherian scheme of di- mension d. Then Km(X)=0for m<−d and X is K−d-regular. In this paper we give a proof of this conjecture for X essentially of finite type over a field F of characteristic 0; see Theorem 6.2.
    [Show full text]
  • V4A2 - Algebraic Geometry II (Sommersemester 2017)
    V4A2 - Algebraic Geometry II (Sommersemester 2017) Taught by Prof. Dr. Peter Scholze Typed by Jack Davies Contents 1 Overview of the Course 20/04/2017 1 2 Flatness 24/04/2017 4 3 Faithfully Flat Descent 27/04/2017 9 4 Smoothness 04/05/2017 13 5 Kähler Differentials 08/05/2017 18 6 Smoothness and the Jacobi Criterion 11/05/2017 24 7 Smoothness is Local on Source 15/05/2017 30 8 The First Sheaf Cohomology Group 18/05/2017 34 9 Homological Algebra I (Derived Functors) 22/05/2017 38 10 Homological Algebra II (Homotopy Categories) 29/05/2017 43 11 Homological Algebra III (Derived Categories) 01/06/2017 48 12 Cechˇ Cohomology 12/06/2017 54 13 Finiteness in Cohomology I 19/06/2017 58 14 Finiteness in Cohomology II 22/06/2017 62 15 Affine Criterion and Base Change 26/06/2017 66 16 Generalised Base Change and ⊗L 03/07/2017 71 17 Finiteness of RΓ(X; M) and Riemann-Roch 06/07/2017 75 18 Ext Functor and Serre Duality 10/07/2017 79 19 Ext Functor and the Proof of Serre Duality 13/07/2017 84 20 Formal Functions 17/07/2017 89 21 Zariski’s Main Theorem and Consequences 20/07/2017 94 22 Relatively Ample Line Bundles 24/07/2017 98 23 Left Derived Functors and T or 27/07/2017 103 Introduction This course was taught in Bonn, Germany over the Sommersemester 2017, by Prof. Dr. Peter Scholze. We started by looking at properties of flat maps between schemes, as a way of parametrising nice families of schemes over the base scheme.
    [Show full text]
  • On the K-Theory and Topological Cyclic Homology of Smooth Schemes Over a Discrete Valuation Ring
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 1, Pages 131–145 S 0002-9947(04)03599-8 Article electronically published on December 28, 2004 ON THE K-THEORY AND TOPOLOGICAL CYCLIC HOMOLOGY OF SMOOTH SCHEMES OVER A DISCRETE VALUATION RING THOMAS GEISSER AND LARS HESSELHOLT Abstract. We show that for a smooth and proper scheme over a henselian discrete valuation ring of mixed characteristic (0,p), the p-adic ´etale K-theory and p-adic topological cyclic homology agree. Introduction In this paper, we consider a smooth scheme X over a henselian discrete valuation ring V of mixed characteristic (0,p). Let i: Y→ X denote the inclusion of the special fiber. We have previously showed that the cyclotomic trace et Z v −∼→ Z v Kq (Y, /p ) TCq(Y ; p, /p ) defines an isomorphism of the Dwyer-Friedlander ´etale K-theory of Y with Z/pv- coefficients and the topological cyclic homology of Y with Z/pv-coefficients [2]. Starting from this result, we show: Theorem A. Suppose that X is smooth and proper over V . Then for all integers q and all v ≥ 1, the cyclotomic trace et Z v −∼→ Z v Kq (X, /p ) TCq(X; p, /p ) is an isomorphism. We briefly outline the proof. The left-hand term in the statement of Theorem A is the abutment of a strongly convergent spectral sequence s,t s − Z v ˜ ⇒ et Z v E2 = H (Xet,K−t( , /p ) ) K−s−t(X, /p ) from the cohomology of the sheaf on the small ´etale site Xet associated with the v presheaf which to U → X assigns K−t(U, Z/p ) [12].
    [Show full text]
  • Notes for Topics in AG: Deformation Theory Instructor: Daniel Litt
    Notes for Topics in AG: Deformation Theory Instructor: Daniel Litt Henry Liu December 11, 2017 Abstract These are my live-texed notes for the Fall 2017 offering of MATH GR6263 Topics in Algebraic Ge- ometry. Let me know when you find errors or typos. I'm sure there are plenty. 1 Introduction and motivation 1 2 Deformation functors 1 2.1 Tangent-obstruction theories . .2 2.2 Application: Lefschetz theorem for Pic . .3 2.3 Pro-representability and Schlessinger's criterion . .5 2.4 Application: smoothness of Hilbert scheme . .8 3 Tian{Todorov theorem 11 3.1 Deformations of a smooth variety . 12 3.2 T1-lifting theorem . 14 3.3 Proof of Tian{Todorov . 15 4 Generic vanishing 16 i 4.1 Analytic structure on S (X) ....................................... 17 4.2 The derivative complex via Hodge theory . 20 4.3 Proof of generic vanishing . 22 5 Unobstructedness of abelian varieties 23 5.1 Deformations of polarized AVs . 26 6 Formal GAGA 28 7 Bend and break 30 7.1 Moving curves in varieties . 31 7.2 Bend and break . 32 8 Isomonodromic deformations 33 9 Cotangent complex 36 9.1 Naive cotangent complex . 36 9.2 Using the cotangent complex . 38 9.3 Construction and properties . 38 9.4 More applications . 40 10 Gauss{Manin connection 40 a 1 Introduction and motivation Deformation theory is the study of the local structure of a space, e.g. a representable functor, near a point. Later on, a functor will not be sufficiently general, so we will work with categories co-fibered in groupoids.
    [Show full text]
  • Modules Over Algebraic Cobordism
    MODULES OVER ALGEBRAIC COBORDISM ELDEN ELMANTO, MARC HOYOIS, ADEEL A. KHAN, VLADIMIR SOSNILO, AND MARIA YAKERSON Abstract. We prove that the 1-category of MGL-modules over any scheme is equivalent to the 1-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite P1-loop spaces, we deduce that very effective MGL-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that Ω1 MGL is the A1-homotopy type of the moduli stack of virtual finite flat P1 local complete intersections, and that for n > 0, Ω1 Σn MGL is the A1-homotopy type of P1 P1 the moduli stack of finite quasi-smooth derived schemes of virtual dimension −n. Contents 1. Introduction 2 1.1. Modules over algebraic cobordism 2 1.2. Framed correspondences and the motivic recognition principle 2 1.3. Geometric models of motivic Thom spectra 3 1.4. Analogies with topology 4 1.5. Related work 5 1.6. Conventions and notation 5 1.7. Acknowledgments 6 2. Twisted framed correspondences 6 2.1. Stable vector bundles and K-theory 6 2.2. Presheaves of twisted framed correspondences 7 2.3. Descent and additivity 9 2.4. Comparison theorems 10 2.5. Base change 15 3. Geometric models of motivic Thom spectra 16 3.1.
    [Show full text]