LECTURE NOTES IN EQUIVARIANT ALGEBRAIC GEOMETRY. Spec k = (G G) G G (G G) G G G G i 1 G e
|
|
- Ashlie Reynolds
- 5 years ago
- Views:
Transcription
1 LECTURE NOTES IN EQUIVARIANT ALEBRAIC EOMETRY 8/4/5 Let k be field, not necessaril algebraicall closed. Definition: An algebraic group is a k-scheme together with morphisms (µ, i, e), k µ, i, Spec k, which are subject to the following relations.. Associativit propert: the following diagram commutes. ( ) ( ) µ µ () () µ µ. Inverse propert: the following diagram commutes. i µ i e Spec k 3. Identit propert: the following diagrams commutes. e µ Spec k
2 e µ Spec k The above definition can be generalized to define an S-group, where S is a scheme. An S-scheme is called an S-group if we replace Spec k b the scheme S in the above definition. Suppose that X T are S-schemes. We let X(T ) Hom S (T, X) we refer to the elements of X(T ) as T -valued points. Observe that if X(T ), then the two morphisms i T : T T : T X induce the following commutative diagram. X S T X T If T is an S-scheme an S-group, then there is a natural multiplication map µ(t ) : (T ) (T ) (T ) defined b if µ(t )(g, g ) µ (g, g ) where (g, g ) : T S is the map induced b the S-morphisms T g T g. We can define an identit element in (T ) b letting e(t ) be the morphism T S i. There is a natural wa to define inverse elements in (T ) b i(t ) : (T ) (T ) given b g i g. It can be checked that (T ) together with µ(t ), i(t ), e(t ) make (T ) a group in the categor of sets. iven a morphism T T of S-schemes, there is an induced morphism of groups t : (T ) ( T ). So algebraic groups over a scheme S defines a contravariant functor from the categor of S-schemes to the categor of groups b sending an S-scheme T to the group (T ). A morphism of algebraic groups over S is a morphism φ : H of S-schemes such that the multiplication map on H is compatible with the multiplication map on, i.e., the following diagram commutes. H H φ φ S µ H µ H φ If φ is an immersion, i.e. locall the map on stalks is surjective, we sa that H is a subgroup of. If φ is a closed immersion then we sa that H is a closed subgroup of. Basic Eamples of Algebraic roups:. Suppose that is a finite group, S is an base scheme. Then we can define a group scheme structure on b Spec( g O S) in the obvious wa. Then (S).
3 ( ). Take Spec C[], a four point scheme over Spec C. Let µ : be the ( 4 ) morphism corresponding to the ring map of global sections µ # C[] : C[] C[] ( 4 ) ( 4 ) ( 4 ) sending ( )( ). Let e : Spec C be the morphism induced b e # : C[]/( 4 ) C sending, i.e. the morphism e picks out the maimal ideal ( ). Finall, we let i : be the morphism induced b i # : C[]/( 4 ) C[]/( 4 ) which sends 3. One can verif that (, µ, e, i) is an algebraic group over Spec C we can compute the group (Spec C). ( ) Observe that the four points of Spec C[] correspond to the four roots of 4 ( 4 ), {,, i, i}. If a, b then µ()(a, b) : µ(a, b) ab it easil follows that (Spec C) Z 4. 8/6/5 ( ) 3. Let Spec Q[], we can still define a multiplication, identit, inverse morphism the same wa as in eample. But the group scheme structure on will be a ( 4 ) bit different ( since the ) polnomial ( 4 ) does not split over Q. Observe that Q Spec Q[] Q[] Spec Q[s,t] Q[s,t]. The ring is a reduced Artin ring, ( 4 ) ( 4 ) (s 4,t 4 ) (s 4,t 4 ) Q[s,t] hence is the product of fields. The ring has ten maimal ideals. The are (s 4,t 4 ) (s, t ), (s +, t ), (s, t + ), (s +, t + ), (s, t + ), (s +, t + ), (s +, t ), (s +, t + ), (s +, t s) (t +, t s), (s +, t + s) (t +, t + s). Note that (s +, t s) does indeed coincide with the ideal (t +, t s), the ke observation is that t + s + + (t + s)(t s). Therefore, (s 4, t 4 ) (s, t ) (s +, t ) (s, t + ) (s +, t + ) (s, t + ) (s +, t + ) (s +, t ) (s +, t + ) (s +, t s) (s +, t + s). To underst the multiplication map µ : Q we onl need to underst what each (µ # ) Q[s,t] (m) is for each of the maimal ideals m of. For eample, (s 4,t 4 ) (µ# ) ((s, t )) ( ). To verif this observe µ # ( ) st s + s(t ). Hence the maimal ideal ( ) (µ # ) ((s, t )) equalit of the ideals must follow. Similarl, µ((s, s t)) (+) since µ # (+) st+ s ++s(t s) (s +, s(t s)). Affine S-groups An affine S-group is simpl an S-group satisfing the etra condition that S is an affine morphism. Notice that to give an affine S-group is equivalent to giving an affine 3
4 morphism S morphisms of sheaves of O S -modules µ # : O O OS O e # : O O S i # : O O so that if we take the diagrams in the definition of a S-group, replace the schemes with their sheaves of regular functions, reverse the arrows we get a commutative diagrams. Caution: Not all S-groups are affine, but a Chevalle s Theorem on Algebraic roups sas that ever algebraic group over a field is constructed or built up from an affine algebraic group an Abelian variet. A classical eample of an Abelian variet is a nonsingular cubic in P whose multiplication is given b the group law. We will mostl be interested in affine algebraic groups. A ke eample of an affine algebraic group is L(n, Z) : Spec ( Z[{ ij } n i,j, D ] ) where D det( ij ). iven an arbitrar scheme S we define L(n, S) b base change, i.e., L(n, S) : L(n, Z) Z S. The multiplication map L(n, Z) L(n, Z) L(n, Z) is given b the map on rings which sends ij n k ( ik )( kj ), which mimics matri multiplication. The identit map e : Z L(n, Z) is given b sending i j δ ij, which is inspired b the identit matri. Lastl, the inverse map i : L(n, Z) L(n, Z) is given b the map on rings which sends i j ( ) j+i M ji D where M ji is the determinant of the ji-minor of the n n matri ( ij ). This map comes from the formula that gives the inverse of a nonsingular matri. Notation: We use L(n, S) L(n, Z) S, which is a scheme, we use L n (S) to denote the S-valued points of L(n, S), i.e., L n (S) : L(n, S)(S) Hom Z (S, L(n, S)). Take a look at the set L n (Z). To give an element of L n (Z) is equivalent to giving a homomorphism of rings Z[{ ij } n i,j, D ] Z. This is equivalent to assigning each of the n variables ij to an integer in such a wa that D is mapped to unit, else D has nowhere to go. Therefore to give an element of L n (Z) is equivalent to giving an integer valued n n matri A (a ij ) so that det(a) ±. Similarl, if R is a commutative ring, then to give an element of L n (R) : L n (Spec(R)) is equivalent to giving an n n matri with entries in R so that the determinant of that matri is a unit in R. Notation: We use m to denote the affine algebraic group L(, Z) Spec(Z[, ]). We sa that an S-group is linear if S is isomorphic to a closed subgroup of L(n, S) for some n. 8/8/5 We can define a group functor L(n, Z) : Schemes roups b sending T L n (T ). Let F be a rank n free Z-module. Let T be a scheme denote b t the morphism T t Spec(Z). Then we can define another group functor from the categor of ) schemes to the categor of groups, denoted Aut Z (F ), b sending a scheme T to Aut Z (t F where t F is the ) pullback of the free Spec(Z)-module F. If T Spec(R) for some ring R, then Aut Z (t F Aut R (F Z R) since t F is the coherent module F Z R. 4
5 Eercise: A choice of ordered basis for the free Z-module F defines an isomorphism of functors Aut Z (F ) L(n, Z). Yoneda s Lemma Representable Functors Let C be the categor of S-schemes. For an X C we can define a contravariant functor X from the categor C to sets b X(R) Hom S (T, X). So X X defines a functor from C to the categor of contravariant functors C Sets, which we denote Func C. Yoneda s Lemma sas that this functor is full faithful. So if X, X objects in C then there is a bijection between Hom S (X, X ) Hom FuncC (X, X). Consequentl, if X X are schemes we can construct an isomorphism of functors X X, there is an isomorphism of S-schemes X X. We sa that a functor F : Schemes Sets is representable if F X for some scheme X. Observe that the above eercise sas that if F is a free Z-module of rank n then Aut Z (F ) is representable b the scheme Spec(Z[{X ij } n i,j, D ]). Fi a scheme S a vector bundle E of rank n on S, i.e. a locall free sheaf of rank n on S. Now consider the group functor Aut S (E) which sends T Aut T (t E). Proposition.. The functor Aut S (E) is representable b an affine S-group Aut S (E). Sketch of proof. The result follows b the above eercise if E is trivial. Otherwise we find an open cover {U i } i I which trivializes E with trivializations ϕ i : E Ui L(n, U i ). Then Aut Ui (E Ui ) L(n, U i ) b choice of basis, i.e., b the trivializations ϕ i. Then the isomorphism ϕ j ϕ i : E Ui U j E Ui U j induce automorphisms of Aut Ui U j (E Ui U j ) which patch together to define a scheme Aut S (E). One can now show Aut S (E) is naturall represented b Aut S (E). Proposition.. If S Spec(A) is affine, E P where P is a projective A-module of rank n, then Aut S (E) is a closed subgroup of L(N, S) for some N, i.e., Aut S (E) is linear. Ke Idea. Since P is projective we can write P K F for some free A-module F. Then Aut S (E) is a subgroup of Aut S (F ) L(N, S) for some N. In fact ϕ Aut S (F ) is in Aut S (E) if onl if ϕ fies P acts identicall on K. The following eample is an eplicit calculation of Aut S (E) as a closed subgroup of L(, S) where E is a coherent line bundle over the affine scheme S Spec(C[, ]/( ( 3 ))). C[,] ( ( 3 )) Eample: Consider A look at the projective A-module (, ). Note that A is regular, in fact A is the coordinate ring of an elliptic curve in P { } A. The ideal (, ) is projective since it is locall principal, i.e., locall free of rank. We can also check that (, ) is not globall generated b single element. But I is the quotient of A via the map π which sends ( ) ( ). Since (, ) is projective, there must be a map (, ) A which splits π. To find this map, we first find K A such that A /K (, ), i.e. we find first find the kernel of π. 5
6 ( ) ( ) We claim kernel of the map π : A (, ) is generated b ( ). It is clear that these two elements( lie in the kernel of π. To see these two elements generate the kernel of π we suppose that r r + r ( ( 3 )), i.e. there is a s C[, ] r ) such that r + r s + s( 3 ). This is equivalent to (r s) + (r s + s) in C[, ]. As, is a regular sequence in ( C[, ) ] there ( is ) a t ( C[, ] such ) that r s t r s r + s t. It follows that r t + s ( ), proving that ker π ) ( ) is generated b. ( ( ) ( To ) give a map ( (, ) ) A is equivalent to giving a map A A so that the two ( elements ) ( ) are mapped to. Consider f : A A which maps ( ) ( ) ( ( ) ( ) ( ) ( ) ). Then ( 3 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ( 3 )). Thus ( ) ( there is an A-linear map f : (, ) A sending ). (( We now )) check that f splits π, i.e. π f is the identit map (( on )) (, ). Well, π(f()) π ( ) + () π(f()) π ( ) + () 3, hence f( splits) π. We ( now have) that A ker π (, ) where ker π is generated b the two elements ( ) of A (, ) is generated b the two elements ( ) ( ) of A. Denote b S Spec(C[, ]/( ( 3 ))) I (, ). We can now compute Aut S (Ĩ) as a closed subgroup of AutS(Ã ) Spec (A[,,,, (D) ]) where D. B the above Proposition this is equivalent to giving an automorphism of A which fies (, ) acts identicall on the kernel of π. To give an automorphism of ( A which)( acts identicall ) ( ) on the ( kernel of π)( is represented ) b the two matri equations ( ) ( ) in A[,,,, (D) ]. To give a map (, ) (, ) is equivalent ( ) to giving a map A A such that all elements of ker π are mapped to elements in ker (( π. So to give )( an automorphism )) of (( A which)( fies (, )) ) is represented b the equations π π. Eping this information, we see that the scheme Aut S (Ĩ) is cut out b the following equations in L(n, S); 6
7 () () (3) (4) (5) (6) + ( ) ( ) + ( ) ( ( ) + ) + ( ( ) + ) ( + ) + ( + ). 7
Representations and Linear Actions
Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category
More informationwhich is a group homomorphism, such that if W V U, then
4. Sheaves Definition 4.1. Let X be a topological space. A presheaf of groups F on X is a a function which assigns to every open set U X a group F(U) and to every inclusion V U a restriction map, ρ UV
More informationAlgebraic varieties and schemes over any scheme. Non singular varieties
Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two
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 informationφ(a + b) = φ(a) + φ(b) φ(a b) = φ(a) φ(b),
16. Ring Homomorphisms and Ideals efinition 16.1. Let φ: R S be a function between two rings. We say that φ is a ring homomorphism if for every a and b R, and in addition φ(1) = 1. φ(a + b) = φ(a) + φ(b)
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset
More information3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection
3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection is called the objects of C and is denoted Obj(C). Given
More informationAPPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP
APPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP In this appendix we review some basic facts about étale cohomology, give the definition of the (cohomological) Brauer group, and discuss
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 information1. Algebraic vector bundles. Affine Varieties
0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.
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 informationDirect Limits. Mathematics 683, Fall 2013
Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry
More informationSystems of linear equations. We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K,
Systems of linear equations We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K, f 11 t 1 +... + f 1n t n = 0, f 21 t 1 +... + f 2n t n = 0,.
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves
More informationLecture 9 - Faithfully Flat Descent
Lecture 9 - Faithfully Flat Descent October 15, 2014 1 Descent of morphisms In this lecture we study the concept of faithfully flat descent, which is the notion that to obtain an object on a scheme X,
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 informationLecture 6: Etale Fundamental Group
Lecture 6: Etale Fundamental Group October 5, 2014 1 Review of the topological fundamental group and covering spaces 1.1 Topological fundamental group Suppose X is a path-connected topological space, and
More informationScheme theoretic vector bundles
Scheme theoretic vector bundles The best reference for this material is the first chapter of [Gro61]. What can be found below is a less complete treatment of the same material. 1. Introduction Let s start
More informationLecture 2 Sheaves and Functors
Lecture 2 Sheaves and Functors In this lecture we will introduce the basic concept of sheaf and we also will recall some of category theory. 1 Sheaves and locally ringed spaces The definition of sheaf
More informationCoherent sheaves on elliptic curves.
Coherent sheaves on elliptic curves. Aleksei Pakharev April 5, 2017 Abstract We describe the abelian category of coherent sheaves on an elliptic curve, and construct an action of a central extension of
More informationMath 248B. Applications of base change for coherent cohomology
Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).
More information18.312: Algebraic Combinatorics Lionel Levine. Lecture 22. Smith normal form of an integer matrix (linear algebra over Z).
18.312: Algebraic Combinatorics Lionel Levine Lecture date: May 3, 2011 Lecture 22 Notes by: Lou Odette This lecture: Smith normal form of an integer matrix (linear algebra over Z). 1 Review of Abelian
More informationFinite group schemes
Finite group schemes Johan M. Commelin October 27, 2014 Contents 1 References 1 2 Examples 2 2.1 Examples we have seen before.................... 2 2.2 Constant group schemes....................... 3 2.3
More informationFactorization of birational maps on steroids
Factorization of birational maps on steroids IAS, April 14, 2015 Dan Abramovich Brown University April 14, 2015 This is work with Michael Temkin (Jerusalem) Abramovich (Brown) Factorization of birational
More informationALGEBRAIC GROUPS JEROEN SIJSLING
ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined
More informationVector Bundles vs. Jesko Hüttenhain. Spring Abstract
Vector Bundles vs. Locally Free Sheaves Jesko Hüttenhain Spring 2013 Abstract Algebraic geometers usually switch effortlessly between the notion of a vector bundle and a locally free sheaf. I will define
More informationHARTSHORNE EXERCISES
HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing
More informationQUOT SCHEMES SIDDHARTH VENKATESH
QUOT SCHEMES SIDDHARTH VENKATESH Abstract. These are notes for a talk on Quot schemes given in the Spring 2016 MIT-NEU graduate seminar on the moduli of sheaves on K3-surfaces. In this talk, I present
More information1 Notations and Statement of the Main Results
An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main
More informationRelative Affine Schemes
Relative Affine Schemes Daniel Murfet October 5, 2006 The fundamental Spec( ) construction associates an affine scheme to any ring. In this note we study the relative version of this construction, which
More informationCohomology and Base Change
Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is half-exact if whenever 0 M M M 0 is an exact sequence of A-modules, the sequence T (M ) T (M)
More informationh M (T ). The natural isomorphism η : M h M determines an element U = η 1
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli
More informationSection Projective Morphisms
Section 2.7 - Projective Morphisms Daniel Murfet October 5, 2006 In this section we gather together several topics concerned with morphisms of a given scheme to projective space. We will show how a morphism
More informationSynopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].
Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.
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 informationALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0.
ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0. ANDREW SALCH During the last lecture, we found that it is natural (even just for doing undergraduatelevel complex analysis!)
More informationExercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti
Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of
More information14 Lecture 14: Basic generallities on adic spaces
14 Lecture 14: Basic generallities on adic spaces 14.1 Introduction The aim of this lecture and the next two is to address general adic spaces and their connection to rigid geometry. 14.2 Two open questions
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 informationSCHEMES. David Harari. Tsinghua, February-March 2005
SCHEMES David Harari Tsinghua, February-March 2005 Contents 1. Basic notions on schemes 2 1.1. First definitions and examples.................. 2 1.2. Morphisms of schemes : first properties.............
More informationThe Nori Fundamental Group Scheme
The Nori Fundamental Group Scheme Angelo Vistoli Scuola Normale Superiore, Pisa Alfréd Rényi Institute of Mathematics, Budapest, August 2014 1/64 Grothendieck s theory of the fundamental group Let X be
More informationALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES
ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to
More informationIntroduction and preliminaries Wouter Zomervrucht, Februari 26, 2014
Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally
More informationConstruction of M B, M Dol, M DR
Construction of M B, M Dol, M DR Hendrik Orem Talbot Workshop, Spring 2011 Contents 1 Some Moduli Space Theory 1 1.1 Moduli of Sheaves: Semistability and Boundedness.............. 1 1.2 Geometric Invariant
More informationLectures on algebraic stacks
Rend. Mat. Appl. (7). Volume 38, (2017), 1 169 RENDICONTI DI MATEMATICA E DELLE SUE APPLICAZIONI Lectures on algebraic stacks Alberto Canonaco Abstract. These lectures give a detailed and almost self-contained
More informationFORMAL GLUEING OF MODULE CATEGORIES
FORMAL GLUEING OF MODULE CATEGORIES BHARGAV BHATT Fix a noetherian scheme X, and a closed subscheme Z with complement U. Our goal is to explain a result of Artin that describes how coherent sheaves on
More informationInjective Modules and Matlis Duality
Appendix A Injective Modules and Matlis Duality Notes on 24 Hours of Local Cohomology William D. Taylor We take R to be a commutative ring, and will discuss the theory of injective R-modules. The following
More informationMath 797W Homework 4
Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition
More informationWhat are stacks and why should you care?
What are stacks and why should you care? Milan Lopuhaä October 12, 2017 Todays goal is twofold: I want to tell you why you would want to study stacks in the first place, and I want to define what a stack
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 informationAzumaya Algebras. Dennis Presotto. November 4, Introduction: Central Simple Algebras
Azumaya Algebras Dennis Presotto November 4, 2015 1 Introduction: Central Simple Algebras Azumaya algebras are introduced as generalized or global versions of central simple algebras. So the first part
More informationNote that the first map is in fact the zero map, as can be checked locally. It follows that we get an isomorphism
11. The Serre construction Suppose we are given a globally generated rank two vector bundle E on P n. Then the general global section σ of E vanishes in codimension two on a smooth subvariety Y. If E is
More informationAlgebraic Geometry I Lectures 14 and 15
Algebraic Geometry I Lectures 14 and 15 October 22, 2008 Recall from the last lecture the following correspondences {points on an affine variety Y } {maximal ideals of A(Y )} SpecA A P Z(a) maximal ideal
More informationAlgebra Qualifying Exam Solutions January 18, 2008 Nick Gurski 0 A B C 0
1. Show that if B, C are flat and Algebra Qualifying Exam Solutions January 18, 2008 Nick Gurski 0 A B C 0 is exact, then A is flat as well. Show that the same holds for projectivity, but not for injectivity.
More informationLECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES
LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EAMPLES VIVEK SHENDE Historically, sheaves come from topology and analysis; subsequently they have played a fundamental role in algebraic geometry and certain
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 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 informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
More informationWe can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle
More informationHere is another way to understand what a scheme is 1.GivenaschemeX, and a commutative ring R, the set of R-valued points
Chapter 7 Schemes III 7.1 Functor of points Here is another way to understand what a scheme is 1.GivenaschemeX, and a commutative ring R, the set of R-valued points X(R) =Hom Schemes (Spec R, X) This is
More information0.1 Spec of a monoid
These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.
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 informationAppendix by Brian Conrad: The Shimura construction in weight 2
CHAPTER 5 Appendix by Brian Conrad: The Shimura construction in weight 2 The purpose of this appendix is to explain the ideas of Eichler-Shimura for constructing the two-dimensional -adic representations
More informationTheta divisors and the Frobenius morphism
Theta divisors and the Frobenius morphism David A. Madore Abstract We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following
More informationQUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS
QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS SAM RASKIN 1. Differential operators on stacks 1.1. We will define a D-module of differential operators on a smooth stack and construct a symbol map when
More informationA GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander
A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander During the first three days of September, 1997, I had the privilege of giving a series of five lectures at the beginning of the School on Algebraic
More informationAN ABSTRACT CHARACTERIZATION OF NONCOMMUTATIVE PROJECTIVE LINES
AN ABSTRACT CHARACTERIZATION OF NONCOMMUTATIVE PROJECTIVE LINES A. NYMAN Abstract. Let k be a field. We describe necessary and sufficient conditions for a k-linear abelian category to be a noncommutative
More informationLectures on Galois Theory. Some steps of generalizations
= Introduction Lectures on Galois Theory. Some steps of generalizations Journée Galois UNICAMP 2011bis, ter Ubatuba?=== Content: Introduction I want to present you Galois theory in the more general frame
More informationCOURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA
COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties
More informationFormal power series rings, inverse limits, and I-adic completions of rings
Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More informationA Primer on Homological Algebra
A Primer on Homological Algebra Henry Y Chan July 12, 213 1 Modules For people who have taken the algebra sequence, you can pretty much skip the first section Before telling you what a module is, you probably
More informationRepresentations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III
Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group
More informationSheaves. S. Encinas. January 22, 2005 U V. F(U) F(V ) s s V. = s j Ui Uj there exists a unique section s F(U) such that s Ui = s i.
Sheaves. S. Encinas January 22, 2005 Definition 1. Let X be a topological space. A presheaf over X is a functor F : Op(X) op Sets, such that F( ) = { }. Where Sets is the category of sets, { } denotes
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 informationAn Atlas For Bun r (X)
An Atlas For Bun r (X) As told by Dennis Gaitsgory to Nir Avni October 28, 2009 1 Bun r (X) Is Not Of Finite Type The goal of this lecture is to find a smooth atlas locally of finite type for the stack
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 informationTHE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3
THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3 KAZUNORI NAKAMOTO AND TAKESHI TORII Abstract. There exist 26 equivalence classes of k-subalgebras of M 3 (k) for any algebraically closed field
More informationCOMPACTIFICATIONS OF MODULI OF ABELIAN VARIETIES: AN INTRODUCTION. 1. Introduction
COMPACTIFICATIONS OF MODULI OF ABELIAN VARIETIES: AN INTRODUCTION. MARTIN OLSSON Abstract. In this expository paper, we survey the various approaches to compactifying moduli stacks of polarized abelian
More informationTOPICS IN ALGEBRA COURSE NOTES AUTUMN Contents. Preface Notations and Conventions
TOPICS IN ALGEBRA COURSE NOTES AUTUMN 2003 ROBERT E. KOTTWITZ WRITTEN UP BY BRIAN D. SMITHLING Preface Notations and Conventions Contents ii ii 1. Grothendieck Topologies and Sheaves 1 1.1. A Motivating
More informationSUMMER COURSE IN MOTIVIC HOMOTOPY THEORY
SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY MARC LEVINE Contents 0. Introduction 1 1. The category of schemes 2 1.1. The spectrum of a commutative ring 2 1.2. Ringed spaces 5 1.3. Schemes 10 1.4. Schemes
More informationFourier Mukai transforms II Orlov s criterion
Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and
More informationPreliminary Exam Topics Sarah Mayes
Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition
More informationLecture 3: Flat Morphisms
Lecture 3: Flat Morphisms September 29, 2014 1 A crash course on Properties of Schemes For more details on these properties, see [Hartshorne, II, 1-5]. 1.1 Open and Closed Subschemes If (X, O X ) is a
More informationCoherent Sheaves; Invertible Sheaves
CHAPTER 13 Coherent Sheaves; Invertible Sheaves In this chapter, k is an arbitrary field. a Coherent sheaves Let V D SpmA be an affine variety over k, and let M be a finitely generated A-module. There
More informationLocality for qc-sheaves associated with tilting
Locality for qc-sheaves associated with tilting - ICART 2018, Faculty of Sciences, Mohammed V University in Rabat Jan Trlifaj Univerzita Karlova, Praha Jan Trlifaj (Univerzita Karlova, Praha) Locality
More informationSynopsis of material from EGA Chapter II, 3
Synopsis of material from EGA Chapter II, 3 3. Homogeneous spectrum of a sheaf of graded algebras 3.1. Homogeneous spectrum of a graded quasi-coherent O Y algebra. (3.1.1). Let Y be a prescheme. A sheaf
More informationThe Picard Scheme and the Dual Abelian Variety
The Picard Scheme and the Dual Abelian Variety Gabriel Dorfsman-Hopkins May 3, 2015 Contents 1 Introduction 2 1.1 Representable Functors and their Applications to Moduli Problems............... 2 1.2 Conditions
More informationBRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,
CONNECTIONS, CURVATURE, AND p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves
More informationCurtis Heberle MTH 189 Final Paper 12/14/2010. Algebraic Groups
Algebraic Groups Curtis Heberle MTH 189 Final Paper 12/14/2010 The primary objects of study in algebraic geometry are varieties. Having become acquainted with these objects, it is interesting to consider
More informationSection Higher Direct Images of Sheaves
Section 3.8 - Higher Direct Images of Sheaves Daniel Murfet October 5, 2006 In this note we study the higher direct image functors R i f ( ) and the higher coinverse image functors R i f! ( ) which will
More informationMathematics of Cryptography Part I
CHAPTER 2 Mathematics of Crptograph Part I (Solution to Practice Set) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinit to positive infinit. The set of
More informationPERVERSE SHEAVES. Contents
PERVERSE SHEAVES SIDDHARTH VENKATESH Abstract. These are notes for a talk given in the MIT Graduate Seminar on D-modules and Perverse Sheaves in Fall 2015. In this talk, I define perverse sheaves on a
More informationLECTURE 3: RELATIVE SINGULAR HOMOLOGY
LECTURE 3: RELATIVE SINGULAR HOMOLOGY In this lecture we want to cover some basic concepts from homological algebra. These prove to be very helpful in our discussion of singular homology. The following
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5 RAVI VAKIL CONTENTS 1. The inverse image sheaf 1 2. Recovering sheaves from a sheaf on a base 3 3. Toward schemes 5 4. The underlying set of affine schemes 6 Last
More information1 Flat, Smooth, Unramified, and Étale Morphisms
1 Flat, Smooth, Unramified, and Étale Morphisms 1.1 Flat morphisms Definition 1.1. An A-module M is flat if the (right-exact) functor A M is exact. It is faithfully flat if a complex of A-modules P N Q
More informationCATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS. Contents. 1. The ring K(R) and the group Pic(R)
CATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS J. P. MAY Contents 1. The ring K(R) and the group Pic(R) 1 2. Symmetric monoidal categories, K(C), and Pic(C) 2 3. The unit endomorphism ring R(C ) 5 4.
More informationLevel Structures of Drinfeld Modules Closing a Small Gap
Level Structures of Drinfeld Modules Closing a Small Gap Stefan Wiedmann Göttingen 2009 Contents 1 Drinfeld Modules 2 1.1 Basic Definitions............................ 2 1.2 Division Points and Level Structures................
More informationTCC Homological Algebra: Assignment #3 (Solutions)
TCC Homological Algebra: Assignment #3 (Solutions) David Loeffler, d.a.loeffler@warwick.ac.uk 30th November 2016 This is the third of 4 problem sheets. Solutions should be submitted to me (via any appropriate
More informationCHAPTER 1. Étale cohomology
CHAPTER 1 Étale cohomology This chapter summarizes the theory of the étale topology on schemes, culminating in the results on l-adic cohomology that are needed in the construction of Galois representations
More information