The Spectral Sequence Relating Algebraic K-Theory to Motivic Cohomology
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The Leray-Serre Spectral Sequence
THE LERAY-SERRE SPECTRAL SEQUENCE REUBEN STERN Abstract. Spectral sequences are a powerful bookkeeping tool, used to handle large amounts of information. As such, they have become nearly ubiquitous in algebraic topology and algebraic geometry. In this paper, we take a few results on faith (i.e., without proof, pointing to books in which proof may be found) in order to streamline and simplify the exposition. From the exact couple formulation of spectral sequences, we ∗ n introduce a special case of the Leray-Serre spectral sequence and use it to compute H (CP ; Z). Contents 1. Spectral Sequences 1 2. Fibrations and the Leray-Serre Spectral Sequence 4 3. The Gysin Sequence and the Cohomology Ring H∗(CPn; R) 6 References 8 1. Spectral Sequences The modus operandi of algebraic topology is that \algebra is easy; topology is hard." By associating to n a space X an algebraic invariant (the (co)homology groups Hn(X) or H (X), and the homotopy groups πn(X)), with which it is more straightforward to prove theorems and explore structure. For certain com- putations, often involving (co)homology, it is perhaps difficult to determine an invariant directly; one may side-step this computation by approximating it to increasing degrees of accuracy. This approximation is bundled into an object (or a series of objects) known as a spectral sequence. Although spectral sequences often appear formidable to the uninitiated, they provide an invaluable tool to the working topologist, and show their faces throughout algebraic geometry and beyond. ∗;∗ ∗;∗ Loosely speaking, a spectral sequence fEr ; drg is a collection of bigraded modules or vector spaces Er , equipped with a differential map dr (i.e., dr ◦ dr = 0), such that ∗;∗ ∗;∗ Er+1 = H(Er ; dr): That is, a bigraded module in the sequence comes from the previous one by taking homology. -
Spectral Sequences: Friend Or Foe?
SPECTRAL SEQUENCES: FRIEND OR FOE? RAVI VAKIL Spectral sequences are a powerful book-keeping tool for proving things involving com- plicated commutative diagrams. They were introduced by Leray in the 1940's at the same time as he introduced sheaves. They have a reputation for being abstruse and difficult. It has been suggested that the name `spectral' was given because, like spectres, spectral sequences are terrifying, evil, and dangerous. I have heard no one disagree with this interpretation, which is perhaps not surprising since I just made it up. Nonetheless, the goal of this note is to tell you enough that you can use spectral se- quences without hesitation or fear, and why you shouldn't be frightened when they come up in a seminar. What is different in this presentation is that we will use spectral sequence to prove things that you may have already seen, and that you can prove easily in other ways. This will allow you to get some hands-on experience for how to use them. We will also see them only in a “special case” of double complexes (which is the version by far the most often used in algebraic geometry), and not in the general form usually presented (filtered complexes, exact couples, etc.). See chapter 5 of Weibel's marvelous book for more detailed information if you wish. If you want to become comfortable with spectral sequences, you must try the exercises. For concreteness, we work in the category vector spaces over a given field. However, everything we say will apply in any abelian category, such as the category ModA of A- modules. -
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. -
What Is the Motivation Behind the Theory of Motives?
What is the motivation behind the Theory of Motives? Barry Mazur How much of the algebraic topology of a connected simplicial complex X is captured by its one-dimensional cohomology? Specifically, how much do you know about X when you know H1(X, Z) alone? For a (nearly tautological) answer put GX := the compact, connected abelian Lie group (i.e., product of circles) which is the Pontrjagin dual of the free abelian group H1(X, Z). Now H1(GX, Z) is canonically isomorphic to H1(X, Z) = Hom(GX, R/Z) and there is a canonical homotopy class of mappings X −→ GX which induces the identity mapping on H1. The answer: we know whatever information can be read off from GX; and are ignorant of anything that gets lost in the projection X → GX. The theory of Eilenberg-Maclane spaces offers us a somewhat analogous analysis of what we know and don’t know about X, when we equip ourselves with n-dimensional cohomology, for any specific n, with specific coefficients. If we repeat our rhetorical question in the context of algebraic geometry, where the structure is somewhat richer, can we hope for a similar discussion? In algebraic topology, the standard cohomology functor is uniquely characterized by the basic Eilenberg-Steenrod axioms in terms of a simple normalization (the value of the functor on a single point). In contrast, in algebraic geometry we have a more intricate set- up to deal with: for one thing, we don’t even have a cohomology theory with coefficients in Z for varieties over a field k unless we provide a homomorphism k → C, so that we can form the topological space of complex points on our variety, and compute the cohomology groups of that topological space. -
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. -
Some Extremely Brief Notes on the Leray Spectral Sequence Greg Friedman
Some extremely brief notes on the Leray spectral sequence Greg Friedman Intro. As a motivating example, consider the long exact homology sequence. We know that if we have a short exact sequence of chain complexes 0 ! C∗ ! D∗ ! E∗ ! 0; then this gives rise to a long exact sequence of homology groups @∗ ! Hi(C∗) ! Hi(D∗) ! Hi(E∗) ! Hi−1(C∗) ! : While we may learn the proof of this, including the definition of the boundary map @∗ in beginning algebraic topology and while this may be useful to know from time to time, we are often happy just to know that there is an exact homology sequence. Also, in many cases the long exact sequence might not be very useful because the interactions from one group to the next might be fairly complicated. However, in especially nice circumstances, we will have reason to know that certain homology groups vanish, or we might be able to compute some of the maps here or there, and then we can derive some consequences. For example, if ∼ we have reason to know that H∗(D∗) = 0, then we learn that H∗(E∗) = H∗−1(C∗). Similarly, to preserve sanity, it is usually worth treating spectral sequences as \black boxes" without looking much under the hood. While the exact details do sometimes come in handy to specialists, spectral sequences can be remarkably useful even without knowing much about how they work. What spectral sequences look like. In long exact sequence above, we have homology groups Hi(C∗) (of course there are other ways to get exact sequences). -
The Grothendieck Spectral Sequence (Minicourse on Spectral Sequences, UT Austin, May 2017)
The Grothendieck Spectral Sequence (Minicourse on Spectral Sequences, UT Austin, May 2017) Richard Hughes May 12, 2017 1 Preliminaries on derived functors. 1.1 A computational definition of right derived functors. We begin by recalling that a functor between abelian categories F : A!B is called left exact if it takes short exact sequences (SES) in A 0 ! A ! B ! C ! 0 to exact sequences 0 ! FA ! FB ! FC in B. If in fact F takes SES in A to SES in B, we say that F is exact. Question. Can we measure the \failure of exactness" of a left exact functor? The answer to such an obviously leading question is, of course, yes: the right derived functors RpF , which we will define below, are in a precise sense the unique extension of F to an exact functor. Recall that an object I 2 A is called injective if the functor op HomA(−;I): A ! Ab is exact. An injective resolution of A 2 A is a quasi-isomorphism in Ch(A) A ! I• = (I0 ! I1 ! I2 !··· ) where all of the Ii are injective, and where we think of A as a complex concentrated in degree zero. If every A 2 A embeds into some injective object, we say that A has enough injectives { in this situation it is a theorem that every object admits an injective resolution. So, for A 2 A choose an injective resolution A ! I• and define the pth right derived functor of F applied to A by RpF (A) := Hp(F (I•)): Remark • You might worry about whether or not this depends upon our choice of injective resolution for A { it does not, up to canonical isomorphism. -
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. -
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. -
A Users Guide to K-Theory Spectral Sequences
A users guide to K-theory Spectral sequences Alexander Kahle [email protected] Mathematics Department, Ruhr-Universt¨at Bochum Bonn-Cologne Intensive Week: Tools of Topology for Quantum Matter, July 2014 A. Kahle A users guide to K-theory The Atiyah-Hirzebruch spectral sequence In the last lecture we have learnt that there is a wonderful connection between topology and analysis on manifolds: K-theory. This begs the question: how does one calculate the K-groups? A. Kahle A users guide to K-theory From CW-complexes to K-theory We saw that a CW-decomposition makes calculating cohomology easy(er). It's natural to wonder whether one can somehow use a CW-decomposition to help calculate K-theory. We begin with a simple observation. A simple observation Let X have a CW-decomposition: ? ⊆ Σ0 ⊆ Σ1 · · · ⊆ Σn = X : Define K • (X ) = ker i ∗ : K •(X ) ! K •(Σ ): ZPk k This gives a filtration of K •(X ): K •(X ) ⊇ K • (X ) ⊇ · · · ⊇ K • (X ) = 0: ZP0 ZPn A. Kahle A users guide to K-theory One might hope that one can calculate the K-theory of a space inductively, with each step moving up one stage in the filtration. This is what the Atiyah-Hirzebruch spectral sequence does: it computes the groups K • (X ): ZPq=q+1 What then remains is to solve the extension problem: given that one knows K • (X ) and K • (X ), one must ZPq=q+1 q+1 somehow determine K • (X ), which fits into ZPq 1 ! K • (X ) ! K • (X ) ! K • (X ) ! 1: ZPq=q+1 ZPq ZPq+1 A. -
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. -
Finite Order Vinogradov's C-Spectral Sequence
Finite order formulation of Vinogradov’s C-spectral sequence Raffaele Vitolo Dept. of Mathematics “E. De Giorgi”, University of Lecce via per Arnesano, 73100 Lecce, Italy and Diffiety Institute, Russia email: Raff[email protected] Abstract The C-spectral sequence was introduced by Vinogradov in the late Seventies as a fundamental tool for the study of algebro-geometric properties of jet spaces and differential equations. A spectral sequence arise from the contact filtration of the modules of forms on jet spaces of a fibring (or on a differential equation). In order to avoid serious technical difficulties, the order of the jet space is not fixed, i.e., computations are performed on spaces containing forms on jet spaces of any order. In this paper we show that there exists a formulation of Vinogradov’s C- spectral sequence in the case of finite order jet spaces of a fibred manifold. We compute all cohomology groups of the finite order C-spectral sequence. We obtain a finite order variational sequence which is shown to be naturally isomorphic with Krupka’s finite order variational sequence. Key words: Fibred manifold, jet space, variational bicomplex, variational se- quence, spectral sequence. 2000 MSC: 58A12, 58A20, 58E99, 58J10. Introduction arXiv:math/0111161v1 [math.DG] 13 Nov 2001 The framework of this paper is that of the algebro-geometric formalism for differential equations. This branch of mathematics started with early works by Ehresmann and Spencer (mainly inspired by Lie), and was carried out by several people. One funda- mental aspect is that a natural setting for dealing with global properties of differential equations (such as symmetries, integrability and calculus of variations) is that of jet spaces (see [17, 18, 21] for jets of fibrings and [13, 26, 27] for jets of submanifolds).