INTERSECTION THEORY CLASSES 20 AND 21: BIVARIANT INTERSECTION THEORY
|
|
- Ashlyn Jackson
- 5 years ago
- Views:
Transcription
1 INTERSECTION THEORY CLASSES 20 AND 21: BIVARIANT INTERSECTION THEORY RAVI VAKIL CONTENTS 1. What we re doin this week 1 2. Precise statements Basic operations and properties 4 3. Provin thins The Chow ( cohomoloy ) rin 6 4. Thins you miht want to be true Poincare duality Chern classes commute with all bivariant classes Bivariant classes vanish in dimensions that you d expect them to 8 1. WHAT WE RE DOING THIS WEEK In this inal week o class, I ll describe bivariant intersection theory, coverin much o Chapter 20. Aain, you should notice that iven chapters 1 throuh 6, we can comortably jump into chapter 20. Suppose : Y is any morphism. Throuhout today and Wednesday s lectures, we ll use the ollowin notation. Suppose we are iven any Y Y. Deine = Y Y, so we have a iber square Y Y. Recall that the inal undamental intersection construction we came up with was the ollowin. Suppose is a local complete intersection o codimension d (or more enerally a local complete intersection morphism). Then we deined! : A k Y A k d Date: Monday, November 29 and Wednesday, December 1,
2 or all Y Y. These Gysin pullbacks were well-behaved in all ways, and in particular compatible with proper pushorward, lat pullback, and intersection products. An earlier example was that o a lat pullback; i i lat o relative dimension n, then is too, and we ot : A k Y A k n, which aain behaves well with respect to everythin else. We ll now eneralize this notion. Deine a bivariant class or any (not just lci) as ollows. It is a collection o homomorphisms A k Y A k p or all Y Y, all k, aain compatible with pushorward, pullback, and intersection products. We ll call the roup o such thins A p ( : Y). We ll see that the roup A k ( pt) will be (canonically) isomorphic to A k. We ll see that A k (id : ) is a rin, which Fulton calls the cohomoloy roup; I miht call the resultin rin the Chow rin. We ll denote this by A k. The rin structure is a product o the orm A p A q A p+q. We ll deine more enerally A p ( : Y) A q ( : Y Z) A p+q ( : Z). We ll prove Poincare duality when is smooth: A = A (as rins recall we deined a rin structure on the latter). We ll deine proper pushorward and pullback operations or bivariant roups. Basically, they ll behave the way you d expect rom homoloy and cohomoloy. This will ive a cap product A A A. Alarmin act: This rin is apparently not known to be commutative in eneral, because the arument requires resolution o sinularities. (It is known to be commutative in characteristic 0, and or smooth thins in positive characteristic, and a ew more thins.) I think it should be possible to show that the rin is commutative in eneral usin technoloy not available when this theory was irst developed, usin Johan de Jon s alteration theorem in positive characteristic. I you would like to patch this hole, then come talk to me. Okay, let s et started. Today I ll outline the results, and prove a ew thins; Wednesday I ll prove some more thins. 2. PRECISE STATEMENTS Let : Y be a morphism. For each : Y Y, orm the iber square Y Y A bivariant class c in A p ( : Y) is a collection o homomorphisms c (k) : A ky A k p 2.
3 or all : Y Y, and all k, compatible with proper pushorwards, lat pullbacks, and intersection products. I ll make that precise in a moment, by statin 3 conditions explicitly. But irst I want to show you that you ve seen this beore in several circumstances. Example 1. I is a local complete intersection, or more enerally an lci morphism, we ve deined!. This ives some inklin as to why we want to deal with maps Y. We could have just had a class on Y, but we have more reined inormation; we have a class on, that pushes orward to the more reined class on Y. Example 2. I : Y is the identity, and V is a vector bundle on Y, then the Chern classes are o this orm: α ( c k (V)) α. Example 3 (which eneralizes urther): pseudodivisors. Let L be a line bundle on Y, and the zero-scheme o a section s o L. (s miht cut out a Cartier divisor, i.e. will contain no associated points o Y; at the other extreme, s miht be 0 everywhere.) Then we deined cappin with a pseudo-divisor : A k Y A k 1. Because pseudodivisors pull back, is a pseudodivisor on Y (with correspondin line bundle L, and correspondin section s), so we et a map A k Y A k 1, and this behaves well respect to everythin else. So we re creatin a machine that in some sense incorporates most thins we ve done so ar. Here are the conditions. (C 1 ): I h : Y Y is proper, : Y Y is arbitrary, and one orms the iber diaram (1) Y, h h Y Y then or all α A k Y, c (k) (h α) = h c (k) h α in A k p. (C 2 ): I h : Y Y is lat o relative dimension n, and : Y Y is arbitrary, and one orms the iber diaram (1), then, or all α A k Y, in A k+n p. c (k+n) h (h α) = h c (k) α 3
4 (C 3 ): I : Y Y, h : Y Z are morphisms, and i : Z Z is a local complete intersection o codimension e, and one orms the diaram (2) then, or all α A k Y, in A k p e. Y i i Y Y h h Z i Z i! c (k) (α) = c(k e) i (i! α) 2.1. Basic operations and properties. Here are some basic operations on bivariant Chow roups A ( Y). (P 1 ) Product: For all : Y, : Y Z, we have : A p ( : Y) A q ( : Y Z) A p+q ( : Z). It is pretty immediate to show this: iven any Z Z, orm the iber diaram (3) Y Z Y Z. I α A k Z, then d(α) A k 1 Y and c(d(α)) A k q p, so we deine c d by c d(α) := c(d(α)). (P 2 ) Pushorward: Let : Y be proper, : Y Z any morphism. Then there is a homomorphism ( proper pushorward ): : A p ( : Z) A p ( : Y Z). Aain, it s straihtorward: iven Z Z, orm the iber diaram (3). I c A p ), and α A k (Z ), then c(α) A k p ( ). Since is proper, (c(α)) A k p(y ). Deine (c) by (c)(α) = (c(α)). (P 3 ): Pullback (not necessarily lat!!): Given : Y, : Y 1 Y, orm the iber square (4) 1 1 For each p there is a homomorphism Y 1 Y. : A p ( : Y) A p ( 1 : 1 Y 1 ). 4
5 Aain, we just ollow our nose. Given c A p (), Y Y 1, then composin with ives a morphism Y. Thereor c(α) A k p ( ), = Y Y = 1 Y1 Y. Set (c)(α) = c(α). Here are seven more axioms, which can also be easily veriied. (A pr ) Associativity o products. I c A( Y), d A(Y Z), e A(Z W), then (c d) e = c (d e) A( W). (A p ) Functoriality o proper pushorward. I : Y and : Y Z are proper, Z W arbitrary, and c A( W), then () (c) = ( c) A(Z W). (A pb ) Functoriality o pullbacks. I c A( Y), : Y 1 Y, h : Y 2 Y 1, then (h) (c) = h (c) A( Y Y 2 Y 2 ). (A prp ) Product and pushorward commute. I : Y is proper, Y Z and Z W are arbitrary and c A( Z), d A(Z W), then (c) d = (c d) A(Y W). (A prpb ) Product and pullback commute. I c A( : Y), d A(Y Z), and : Z 1 Z is a morphism, orm the iber diaram 1 Y 1 Z 1 Y Z. Then (c d) = (c) (d) A( 1 Z 1 ). (A ppb ) Proper pushorward and pullback commute. I : Y is proper, Y Z, : Z 1 Z, and c A( Z) are iven, then, with notation as in the precedin diaram (A? ): Projection ormula. Given a diaram c = ( c) A(Y 1 Z 1 ). Y Y h Z. with proper, the square a iber square, and c A( Y), d A(Y Z), then c (d) = ( (c) d) A( Z). 5
6 3. PROVING THINGS Let S = Spec K, where K is some base ield. There is a canonical homomorphism iven by c c([s]). Proposition. This is an isomorphism. φ : A p ( S) A p () Proo. We will deine the inverse morphism. Given a A p (), deine a bivariant class ψ(a) A p ( S) as ollows: or any Y S, and any α A k Y, deine ψ(a)(α) = a α A p+k ( S Y). (Here a α is the exterior product.) Since exterior products are compatible with proper pushorward, lat pullback, and intersections, ψ(a) is a bivariant class. Let s check that this really is an inverse to φ. ψ(a)([s]) = a immediately, so φ ψ is the identity. To show that ψ φ is the identity, we have to show that c(α) = φ(c) α A k+p ( S Y) or all α A k Y. By compatibility with pushorward, we can assume α = [V], and V = Y a variety o dimension k: S V S Y V Y cl. imm. S Then α = p [S], where p : V S is the morphism rom V to S. Since c commutes with lat pullback, c(α) = c(p [S]) = p c([s]) = φ(c) [V] as desired The Chow ( cohomoloy ) rin. Deine A p := A p (id : ). We have a cup product. We also have an element 1 A 0, which acts as the identity. We have a cap product determined by : A p A q A q p A p ( ) A q ( S) A (q p) ( S) which makes A into a let A -module. All o this ollows ormally rom our axioms. 6
7 4. THINGS YOU MIGHT WANT TO BE TRUE 4.1. Poincare duality. Theorem. Let Y be a smooth, purely n-dimensional scheme (variety) not necessarily proper (compact). (a) The canonical homomorphism [Y] : A p Y A n p Y is an isomorphism. (b) The rin structure on A Y is compatible with that deined on A Y earlier. More enerally, i : Y is a morphism, β A Y, α A, then the class (β) α A coincides with that constructed earlier. We ll show somethin more eneral. Theorem. Let : Y Z be a smooth morphism o relative dimension n, and let [] A n ( : Y Z) be the bivariant class correspondin to lat pullback. Then or any morphism : Y and any p, is an isomorphism. [] : A p ( : Y) A p n ( : Z) In eneral, i : Y is a lat morphism, or a local complete intersection, or a local complete intersection morphism, the (lat or Gysin) pullback we ve deined earlier deines a bivariant class, which we ll denote []. ([ ] miht be better.) Fulton calls this bivariant class a canonical orientation. I m not sure o the motivation or this terminoloy, so I ll avoid it. Proo. We ll deine the inverse homomorphism Consider the iber diaram A p n ( : Z) A p ( : Y). Y γ Z Y δ Y Z Y q Y p Y p Z where δ is the diaonal map, and p and q are the irst and second projections. Here γ is the raph o the morphism Y over Z. Deine L : A p n ( : Z) A P ( Y) by L(c) = [γ] (c). Notice that γ and δ is a local complete intersection o codimension n, with [δ] = [γ]. (This requires a check in the case o γ.) Notice that p γ : and q δ : Y Y are both the identity morphisms (on and Y respectively). 7
8 Let s veriy that L and multiplication by [] are inverse homomorphisms. (This is easier to understand i you see someone pointin at diarams!) First, Second, L(c) [] = [γ] ( [c] []) (axiom (A pr )) = [γ] [p ] c (axiom (C 2 )) = [p γ] c = 1 c = c (axiom (A pr )) L(c []) = [δ] p (c) [] (axioms (A prpb ), (A pr )) = (p δ) (c) [δ][q] (axiom (C 2 )) = c [δ q] = c 1 = c (axiom (A pr )) Chern classes commute with all bivariant classes. Put another way, any operation which commutes with proper pushorward, pullback, and intersections, automatically commutes with Chern classes. Precisely: Proposition. Let c A q ( : Y), Y Y, α A k (Y ), E a vector bundle on Y. Then where : = Y Y Y. c(c p (E) α) = c p ( E) c(α) A k q p Proo. Recall our deinition o Chern classes. They are certain polynomials in Sere classes. Sere classes are deined usin operations o the orm α p (c 1 (O(1)) i p α), and since c commutes with p and p, we just have to show that c commutes with c 1 (L), where L is a line bundle on Y. We may assume α = [V]. Because c commutes with proper pushorward, we may assume V = Y. Let L = O(D), D a Cartier divisor on V. We can replace V by V, where V V is proper and birational, so we may assume D = D 1 D 2, where D 1 and D 2 are eective. (Recall our trick in chapter 2: it isn t true that a Cartier divisor is a dierence o two eective Cartier divisors, but we can do a clever blow-up and turn it into a dierence o two eective Cartier divisors.) Let i : D V be the inclusion. Then we ve shown that c 1 (L) α = i i! α, and since c commutes with i and i!, c commutes with c(l) Bivariant classes vanish in dimensions that you d expect them to. Proposition. Let : Y be a morphism. Let m = dim Y, and let n be the larest dimension o any iber 1 y, y Y. A p ( : Y) = 0 i p < n or p > m. (Think about why this is what you d expect!) In particular, or any, A p = 0 unless 0 p dim. (Proo omitted.) address: vakil@math.stanord.edu 8
INTERSECTION THEORY CLASS 17
INTERSECTION THEORY CLASS 17 RAVI VAKIL CONTENTS 1. Were we are 1 1.1. Reined Gysin omomorpisms i! 2 1.2. Excess intersection ormula 4 2. Local complete intersection morpisms 6 Were we re oin, by popular
More informationINTERSECTION THEORY CLASS 7
INTERSECTION THEORY CLASS 7 RAVI VAKIL CONTENTS 1. Intersecting with a pseudodivisor 1 2. The first Chern class of a line bundle 3 3. Gysin pullback 4 4. Towards the proof of the big theorem 4 4.1. Crash
More informationINTERSECTION THEORY CLASS 12
INTERSECTION THEORY CLASS 12 RAVI VAKIL CONTENTS 1. Rational equivalence on bundles 1 1.1. Intersecting with the zero-section of a vector bundle 2 2. Cones and Segre classes of subvarieties 3 2.1. Introduction
More informationINTERSECTION THEORY CLASS 16
INTERSECTION THEORY CLASS 16 RAVI VAKIL CONTENTS 1. Where we are 1 1.1. Deformation to the normal cone 1 1.2. Gysin pullback for local complete intersections 2 1.3. Intersection products on smooth varieties
More informationINTERSECTION THEORY CLASS 6
INTERSECTION THEORY CLASS 6 RAVI VAKIL CONTENTS 1. Divisors 2 1.1. Crash course in Cartier divisors and invertible sheaves (aka line bundles) 3 1.2. Pseudo-divisors 3 2. Intersecting with divisors 4 2.1.
More informationUMS 7/2/14. Nawaz John Sultani. July 12, Abstract
UMS 7/2/14 Nawaz John Sultani July 12, 2014 Notes or July, 2 2014 UMS lecture Abstract 1 Quick Review o Universals Deinition 1.1. I S : D C is a unctor and c an object o C, a universal arrow rom c to S
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.
More informationVALUATIVE CRITERIA BRIAN OSSERMAN
VALUATIVE CRITERIA BRIAN OSSERMAN Intuitively, one can think o separatedness as (a relative version o) uniqueness o limits, and properness as (a relative version o) existence o (unique) limits. It is not
More informationOriented Bivariant Theories, I
ESI The Erwin Schrödiner International Boltzmannasse 9 Institute or Mathematical Physics A-1090 Wien, Austria Oriented Bivariant Theories, I Shoji Yokura Vienna, Preprint ESI 1911 2007 April 27, 2007 Supported
More informationAPPENDIX 3: AN OVERVIEW OF CHOW GROUPS
APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss
More informationDescent on the étale site Wouter Zomervrucht, October 14, 2014
Descent on the étale site Wouter Zomervrucht, October 14, 2014 We treat two eatures o the étale site: descent o morphisms and descent o quasi-coherent sheaves. All will also be true on the larger pp and
More informationChern classes à la Grothendieck
Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces
More informationCategorical Background (Lecture 2)
Cateorical Backround (Lecture 2) February 2, 2011 In the last lecture, we stated the main theorem o simply-connected surery (at least or maniolds o dimension 4m), which hihlihts the importance o the sinature
More informationVALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS
VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Recall that or prevarieties, we had criteria or being a variety or or being complete in terms o existence and uniqueness o limits, where
More informationTHE GORENSTEIN DEFECT CATEGORY
THE GORENSTEIN DEFECT CATEGORY PETTER ANDREAS BERGH, DAVID A. JORGENSEN & STEFFEN OPPERMANN Dedicated to Ranar-Ola Buchweitz on the occasion o his sixtieth birthday Abstract. We consider the homotopy cateory
More informationThe excess intersection formula for Grothendieck-Witt groups
manuscripta mathematica manuscript No. (will be inserted by the editor) Jean Fasel The excess intersection ormula or Grothendieck-Witt roups Received: date / Revised version: date Abstract. We prove the
More informationCategory Theory 2. Eugenia Cheng Department of Mathematics, University of Sheffield Draft: 31st October 2007
Cateory Theory 2 Euenia Chen Department o Mathematics, niversity o Sheield E-mail: echen@sheieldacuk Drat: 31st October 2007 Contents 2 Some basic universal properties 1 21 Isomorphisms 2 22 Terminal objects
More informationMath 248B. Base change morphisms
Math 248B. Base change morphisms 1. Motivation A basic operation with shea cohomology is pullback. For a continuous map o topological spaces : X X and an abelian shea F on X with (topological) pullback
More informationINTERSECTION THEORY CLASS 19
INTERSECTION THEORY CLASS 19 RAVI VAKIL CONTENTS 1. Recap of Last day 1 1.1. New facts 2 2. Statement of the theorem 3 2.1. GRR for a special case of closed immersions f : X Y = P(N 1) 4 2.2. GRR for closed
More informationCOMPLEX ALGEBRAIC SURFACES CLASS 9
COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution
More informationCategories and Natural Transformations
Categories and Natural Transormations Ethan Jerzak 17 August 2007 1 Introduction The motivation or studying Category Theory is to ormalise the underlying similarities between a broad range o mathematical
More informationA Peter May Picture Book, Part 1
A Peter May Picture Book, Part 1 Steve Balady Auust 17, 2007 This is the beinnin o a larer project, a notebook o sorts intended to clariy, elucidate, and/or illustrate the principal ideas in A Concise
More informationCLASSIFICATION OF GROUP EXTENSIONS AND H 2
CLASSIFICATION OF GROUP EXTENSIONS AND H 2 RAPHAEL HO Abstract. In this paper we will outline the oundations o homoloical alebra, startin with the theory o chain complexes which arose in alebraic topoloy.
More information( ) x y z. 3 Functions 36. SECTION D Composite Functions
3 Functions 36 SECTION D Composite Functions By the end o this section you will be able to understand what is meant by a composite unction ind composition o unctions combine unctions by addition, subtraction,
More informationINTERSECTION THEORY CLASS 2
INTERSECTION THEORY CLASS 2 RAVI VAKIL CONTENTS 1. Last day 1 2. Zeros and poles 2 3. The Chow group 4 4. Proper pushforwards 4 The webpage http://math.stanford.edu/ vakil/245/ is up, and has last day
More informationSPLITTING OF SHORT EXACT SEQUENCES FOR MODULES. 1. Introduction Let R be a commutative ring. A sequence of R-modules and R-linear maps.
SPLITTING OF SHORT EXACT SEQUENCES FOR MODULES KEITH CONRAD 1. Introduction Let R be a commutative rin. A sequence o R-modules and R-linear maps N M P is called exact at M i im = ker. For example, to say
More informationAlgebraic Geometry I Lectures 22 and 23
Algebraic Geometry I Lectures 22 and 23 Amod Agashe December 4, 2008 1 Fibered Products (contd..) Recall the deinition o ibered products g!θ X S Y X Y π 2 π 1 S By the universal mapping property o ibered
More informationThe Category of Sets
The Cateory o Sets Hans Halvorson October 16, 2016 [[Note to students: this is a irst drat. Please report typos. A revised version will be posted within the next couple o weeks.]] 1 Introduction The aim
More informationAlgebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism
Algebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism UWO January 25, 2005 Marc Levine Prelude: From homotopy theory to A 1 -homotopy theory A basic object in homotopy theory is a generalized
More informationCLASS NOTES MATH 527 (SPRING 2011) WEEK 6
CLASS NOTES MATH 527 (SPRING 2011) WEEK 6 BERTRAND GUILLOU 1. Mon, Feb. 21 Note that since we have C() = X A C (A) and the inclusion A C (A) at time 0 is a coibration, it ollows that the pushout map i
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Proper morphisms 1 2. Scheme-theoretic closure, and scheme-theoretic image 2 3. Rational maps 3 4. Examples of rational maps 5 Last day:
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44 RAVI VAKIL CONTENTS 1. Flat implies constant Euler characteristic 1 2. Proof of Important Theorem on constancy of Euler characteristic in flat families
More informationINTERSECTION THEORY CLASS 1
INTERSECTION THEORY CLASS 1 RAVI VAKIL CONTENTS 1. Welcome! 1 2. Examples 2 3. Strategy 5 1. WELCOME! Hi everyone welcome to Math 245, Introduction to intersection theory in algebraic geometry. Today I
More informationSTRONGLY GORENSTEIN PROJECTIVE MODULES OVER UPPER TRIANGULAR MATRIX ARTIN ALGEBRAS. Shanghai , P. R. China. Shanghai , P. R.
STRONGLY GORENSTEIN PROJETIVE MODULES OVER UPPER TRINGULR MTRIX RTIN LGERS NN GO PU ZHNG Department o Mathematics Shanhai University Shanhai 2444 P R hina Department o Mathematics Shanhai Jiao Ton University
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24
FOUNDATIONS OF ALGEBRAIC GEOMETR CLASS 24 RAVI VAKIL CONTENTS 1. Normalization, continued 1 2. Sheaf Spec 3 3. Sheaf Proj 4 Last day: Fibers of morphisms. Properties preserved by base change: open immersions,
More informationCS 361 Meeting 28 11/14/18
CS 361 Meeting 28 11/14/18 Announcements 1. Homework 9 due Friday Computation Histories 1. Some very interesting proos o undecidability rely on the technique o constructing a language that describes the
More informationRELATIVE GOURSAT CATEGORIES
RELTIVE GOURST CTEGORIES JULI GOEDECKE ND TMR JNELIDZE bstract. We deine relative Goursat cateories and prove relative versions o the equivalent conditions deinin reular Goursat cateories. These include
More informationThe Clifford algebra and the Chevalley map - a computational approach (detailed version 1 ) Darij Grinberg Version 0.6 (3 June 2016). Not proofread!
The Cliord algebra and the Chevalley map - a computational approach detailed version 1 Darij Grinberg Version 0.6 3 June 2016. Not prooread! 1. Introduction: the Cliord algebra The theory o the Cliord
More informationGENERAL ABSTRACT NONSENSE
GENERAL ABSTRACT NONSENSE MARCELLO DELGADO Abstract. In this paper, we seek to understand limits, a uniying notion that brings together the ideas o pullbacks, products, and equalizers. To do this, we will
More informationare well-formed, provided Φ ( X, x)
(October 27) 1 We deine an axiomatic system, called the First-Order Theory o Abstract Sets (FOTAS) Its syntax will be completely speciied Certain axioms will be iven; but these may be extended by additional
More informationRepresentation Theory of Hopf Algebroids. Atsushi Yamaguchi
Representation Theory o H Algebroids Atsushi Yamaguchi Contents o this slide 1. Internal categories and H algebroids (7p) 2. Fibered category o modules (6p) 3. Representations o H algebroids (7p) 4. Restrictions
More informationCategories and Quantum Informatics: Monoidal categories
Cateories and Quantum Inormatics: Monoidal cateories Chris Heunen Sprin 2018 A monoidal cateory is a cateory equipped with extra data, describin how objects and morphisms can be combined in parallel. This
More informationCHEVALLEY S THEOREM AND COMPLETE VARIETIES
CHEVALLEY S THEOREM AND COMPLETE VARIETIES BRIAN OSSERMAN In this note, we introduce the concept which plays the role of compactness for varieties completeness. We prove that completeness can be characterized
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26 RAVI VAKIL CONTENTS 1. Proper morphisms 1 Last day: separatedness, definition of variety. Today: proper morphisms. I said a little more about separatedness of
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 23
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 23 RAVI VAKIL Contents 1. More background on invertible sheaves 1 1.1. Operations on invertible sheaves 1 1.2. Maps to projective space correspond to a vector
More informationPorteous s Formula for Maps between Coherent Sheaves
Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for
More informationLECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL
LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion
More information( x) f = where P and Q are polynomials.
9.8 Graphing Rational Functions Lets begin with a deinition. Deinition: Rational Function A rational unction is a unction o the orm ( ) ( ) ( ) P where P and Q are polynomials. Q An eample o a simple rational
More informationSTEENROD OPERATIONS IN ALGEBRAIC GEOMETRY
STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY ALEXANDER MERKURJEV 1. Introduction Let p be a prime integer. For a pair of topological spaces A X we write H i (X, A; Z/pZ) for the i-th singular cohomology group
More informationMath 754 Chapter III: Fiber bundles. Classifying spaces. Applications
Math 754 Chapter III: Fiber bundles. Classiying spaces. Applications Laurențiu Maxim Department o Mathematics University o Wisconsin maxim@math.wisc.edu April 18, 2018 Contents 1 Fiber bundles 2 2 Principle
More informationDefinition: Let f(x) be a function of one variable with continuous derivatives of all orders at a the point x 0, then the series.
2.4 Local properties o unctions o several variables In this section we will learn how to address three kinds o problems which are o great importance in the ield o applied mathematics: how to obtain the
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41 RAVI VAKIL CONTENTS 1. Normalization 1 2. Extending maps to projective schemes over smooth codimension one points: the clear denominators theorem 5 Welcome back!
More information2. ETA EVALUATIONS USING WEBER FUNCTIONS. Introduction
. ETA EVALUATIONS USING WEBER FUNCTIONS Introduction So ar we have seen some o the methods or providing eta evaluations that appear in the literature and we have seen some o the interesting properties
More information(C) The rationals and the reals as linearly ordered sets. Contents. 1 The characterizing results
(C) The rationals and the reals as linearly ordered sets We know that both Q and R are something special. When we think about about either o these we usually view it as a ield, or at least some kind o
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about
More informationA crash course in homological algebra
Chapter 2 A crash course in homoloical alebra By the 194 s techniques from alebraic topoloy bean to be applied to pure alebra, ivin rise to a new subject. To bein with, recall that a cateory C consists
More informationAlgebraic Cobordism. 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006.
Algebraic Cobordism 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006 Marc Levine Outline: Describe the setting of oriented cohomology over a
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves of ideals, and closed subschemes 1 2. Invertible sheaves (line bundles) and divisors 2 3. Some line bundles on projective
More informationMath 216A. A gluing construction of Proj(S)
Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an N-graded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does
More informationCHOW S LEMMA. Matthew Emerton
CHOW LEMMA Matthew Emerton The aim o this note is to prove the ollowing orm o Chow s Lemma: uppose that : is a separated inite type morphism o Noetherian schemes. Then (or some suiciently large n) there
More informationLIMITS AND COLIMITS. m : M X. in a category G of structured sets of some sort call them gadgets the image subset
5 LIMITS ND COLIMITS In this chapter we irst briely discuss some topics namely subobjects and pullbacks relating to the deinitions that we already have. This is partly in order to see how these are used,
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 47 AND 48
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 47 AND 48 RAVI VAKIL CONTENTS 1. The local criterion for flatness 1 2. Base-point-free, ample, very ample 2 3. Every ample on a proper has a tensor power that
More informationCOARSE-GRAINING OPEN MARKOV PROCESSES. John C. Baez. Kenny Courser
COARE-GRAINING OPEN MARKOV PROCEE John C. Baez Department o Mathematics University o Caliornia Riverside CA, UA 95 and Centre or Quantum echnoloies National University o inapore inapore 7543 Kenny Courser
More informationDESCENT THEORY (JOE RABINOFF S EXPOSITION)
DESCENT THEORY (JOE RABINOFF S EXPOSITION) RAVI VAKIL 1. FEBRUARY 21 Background: EGA IV.2. Descent theory = notions that are local in the fpqc topology. (Remark: we aren t assuming finite presentation,
More informationAlgebraic Geometry (Intersection Theory) Seminar
Algebraic Geometry (Intersection Theory) Seminar Lecture 1 (January 20, 2009) We ill discuss cycles, rational equivalence, and Theorem 1.4 in Fulton, as ell as push-forard of rational equivalence. Notationise,
More informationAPPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY
APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY In this appendix we begin with a brief review of some basic facts about singular homology and cohomology. For details and proofs, we refer to [Mun84]. We then
More informationRoberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 1. Extreme points
Roberto s Notes on Dierential Calculus Chapter 8: Graphical analysis Section 1 Extreme points What you need to know already: How to solve basic algebraic and trigonometric equations. All basic techniques
More informationTangent Categories. David M. Roberts, Urs Schreiber and Todd Trimble. September 5, 2007
Tangent Categories David M Roberts, Urs Schreiber and Todd Trimble September 5, 2007 Abstract For any n-category C we consider the sub-n-category T C C 2 o squares in C with pinned let boundary This resolves
More informationCATEGORIES. 1.1 Introduction
1 CATEGORIES 1.1 Introduction What is category theory? As a irst approximation, one could say that category theory is the mathematical study o (abstract) algebras o unctions. Just as group theory is the
More informationCentral Limit Theorems and Proofs
Math/Stat 394, Winter 0 F.W. Scholz Central Limit Theorems and Proos The ollowin ives a sel-contained treatment o the central limit theorem (CLT). It is based on Lindeber s (9) method. To state the CLT
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Application of cohomology: Hilbert polynomials and functions, Riemann- Roch, degrees, and arithmetic genus 1 1. APPLICATION OF COHOMOLOGY:
More informationCOMPLEX ALGEBRAIC SURFACES CLASS 15
COMPLEX ALGEBRAIC SURFACES CLASS 15 RAVI VAKIL CONTENTS 1. Every smooth cubic has 27 lines, and is the blow-up of P 2 at 6 points 1 1.1. An alternate approach 4 2. Castelnuovo s Theorem 5 We now know that
More informationVector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle
Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,
More informationDEFINITION OF ABELIAN VARIETIES AND THE THEOREM OF THE CUBE
DEFINITION OF ABELIAN VARIETIES AND THE THEOREM OF THE CUBE ANGELA ORTEGA (NOTES BY B. BAKKER) Throughout k is a field (not necessarily closed), and all varieties are over k. For a variety X/k, by a basepoint
More informationHopf cyclic cohomology and transverse characteristic classes
Hopf cyclic cohomoloy and transverse characteristic classes Lecture 2: Hopf cyclic complexes Bahram Ranipour, UNB Based on joint works with Henri Moscovici and with Serkan Sütlü Vanderbilt University May
More information2 Coherent D-Modules. 2.1 Good filtrations
2 Coherent D-Modules As described in the introduction, any system o linear partial dierential equations can be considered as a coherent D-module. In this chapter we ocus our attention on coherent D-modules
More informationHodge Theory of Maps
Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar
More informationRecall for an n n matrix A = (a ij ), its trace is defined by. a jj. It has properties: In particular, if B is non-singular n n matrix,
Chern characters Recall for an n n matrix A = (a ij ), its trace is defined by tr(a) = n a jj. j=1 It has properties: tr(a + B) = tr(a) + tr(b), tr(ab) = tr(ba). In particular, if B is non-singular n n
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar
More informationMONDAY - TALK 4 ALGEBRAIC STRUCTURE ON COHOMOLOGY
MONDAY - TALK 4 ALGEBRAIC STRUCTURE ON COHOMOLOGY Contents 1. Cohomology 1 2. The ring structure and cup product 2 2.1. Idea and example 2 3. Tensor product of Chain complexes 2 4. Kunneth formula and
More informationTHE SNAIL LEMMA ENRICO M. VITALE
THE SNIL LEMM ENRICO M. VITLE STRCT. The classical snake lemma produces a six terms exact sequence starting rom a commutative square with one o the edge being a regular epimorphism. We establish a new
More informationA SHORT PROOF OF ROST NILPOTENCE VIA REFINED CORRESPONDENCES
A SHORT PROOF OF ROST NILPOTENCE VIA REFINED CORRESPONDENCES PATRICK BROSNAN Abstract. I generalize the standard notion of the composition g f of correspondences f : X Y and g : Y Z to the case that X
More informationEpimorphisms and maximal covers in categories of compact spaces
@ Applied General Topoloy c Universidad Politécnica de Valencia Volume 14, no. 1, 2013 pp. 41-52 Epimorphisms and maximal covers in cateories o compact spaces B. Banaschewski and A. W. Haer Abstract The
More information1 Relative degree and local normal forms
THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a
More informationMath 231b Lecture 16. G. Quick
Math 231b Lecture 16 G. Quick 16. Lecture 16: Chern classes for complex vector bundles 16.1. Orientations. From now on we will shift our focus to complex vector bundles. Much of the theory for real vector
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 14
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 14 RAVI VAKIL Contents 1. Dimension 1 1.1. Last time 1 1.2. An algebraic definition of dimension. 3 1.3. Other facts that are not hard to prove 4 2. Non-singularity:
More informationThe Gysin map is compatible with mixed Hodge Structures arxiv:math/ v1 [math.ag] 27 Apr 2005
The Gysin map is compatible with mixed Hodge Structures arxiv:math/0504560v1 [math.ag] 27 Apr 2005 1 Introduction Mark Andrea A. de Cataldo and Luca Migliorini The language of Bivariant Theory was developed
More informationHochschild homology and Grothendieck Duality
Hochschild homology and Grothendieck Duality Leovigildo Alonso Tarrío Universidade de Santiago de Compostela Purdue University July, 1, 2009 Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality
More information9.3 Graphing Functions by Plotting Points, The Domain and Range of Functions
9. Graphing Functions by Plotting Points, The Domain and Range o Functions Now that we have a basic idea o what unctions are and how to deal with them, we would like to start talking about the graph o
More informationFactorization of birational maps for qe schemes in characteristic 0
Factorization of birational maps for qe schemes in characteristic 0 AMS special session on Algebraic Geometry joint work with M. Temkin (Hebrew University) Dan Abramovich Brown University October 24, 2014
More informationThese slides available at joyce/talks.html
Kuranishi (co)homology: a new tool in symplectic geometry. II. Kuranishi (co)homology Dominic Joyce Oxford University, UK work in progress based on arxiv:0707.3572 v5, 10/08 summarized in arxiv:0710.5634
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves 1 2. Quasicoherent sheaves form an abelian category 5 We began by recalling the distinguished affine base. Definition.
More informationA NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 11, November 1997, Pages 3185 3189 S 0002-9939(97)04112-9 A NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES BENJI FISHER (Communicated by
More informationMILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES
MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.
More informationAlgebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra
Algebraic Varieties Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic varieties represent solutions of a system of polynomial
More informationHSP SUBCATEGORIES OF EILENBERG-MOORE ALGEBRAS
HSP SUBCATEGORIES OF EILENBERG-MOORE ALGEBRAS MICHAEL BARR Abstract. Given a triple T on a complete category C and a actorization system E /M on the category o algebras, we show there is a 1-1 correspondence
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationIntersection Theory course notes
Intersection Theory course notes Valentina Kiritchenko Fall 2013, Faculty of Mathematics, NRU HSE 1. Lectures 1-2: examples and tools 1.1. Motivation. Intersection theory had been developed in order to
More informationNotes on p-divisible Groups
Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More information