# Algebraic Geometry

Size: px
Start display at page:

Transcription

1 MIT OpenCourseWare Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit:

2 18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009) Category theory (updated 8 Feb 09) We re going to use the language of category theory freely. Fortunately, it s easy to learn because it corresponds naturally to the way you (hopefully) already think about mathematical objects. (I could give a reference, but in fact you should be fine just looking these things up in Wikipedia.) 1 Warning: set-theoretic difficulties Category theory is a bit tricky because it tries to deal with objects like the??? of all sets, or all rings, or whatnot. Russell s paradox shows that there is in fact no set of all sets. Namely, if there were, it would have a subset consisting of those sets U for which U / U. But that would then be a set V, and if V V then V / V and vice versa. The way around this is to tamper with the axioms of set theory slightly, by introducing the notion of a class. A class is something which behaves just like a set whose members are sets, except that there is no power axiom; there is not guaranteed to be a class consisting of all subclasses of a given class. Unless, that is, your class consists just of the elements of some actual set. (You might then ask what kind of object is the??? of all classes. Never mind that for now.) We also assume there is a class of all sets, called the universe. Except for the power axiom, you may perform operations on classes like you do with sets. For instance, given a class C and a logical statement P depending on a single set, you can form the subclass of C consisting of all elements for which P is true. You can also form Cartesian products indexed by sets, although I ll hardly ever do this except for finite products. (There is also an axiom of choice at the class level.) A class is small if its elements are in bijection with some set. 2 Categories, and examples A category C consists of the following data. A class of objects, denoted Obj(C). For each ordered pair of objects (X, Y ), a set of morphisms, denoted Hom(X, Y ). (You may think of this as an element of the Cartesian product of two copies of C and one copy of the universe.) For each ordered triple of objects (X, Y, Z), a function : Hom(Y, Z) Hom(X, Y ) Hom(X, Z), called composition, which satisfies the following properties. The associative law: given an ordered quadruple of objects (X, Y, Z, W ), the two ways to compose Hom(Z, W ) Hom(Y, Z) Hom(X, Y ) to Hom(X, W ) give the same answer. 1

3 The identity law: for each object X, there must exist a morphism id X Hom(X, X) which is an identity under composition on either side. Note that id X is forced to be unique by this condition. (I have a habit of calling morphisms arrows because they are usually pictorially represented as such.) This definition is meant to capture many, if not all, basic types of structured mathematical objects. Examples: The category of sets, denoted Set, where Hom(X, Y ) is all functions from X to Y. The category of topological spaces, denoted Top, where Hom(X, Y ) is all continuous functions from X to Y. The category of (commutative, unital) rings, denoted Ring, where Hom(X, Y ) is all ring homomorphisms from X to Y. The category of topological rings, denoted TopRing, where Hom(X, Y ) is all continuous ring homomorphisms from X to Y. The category of modules over a fixed ring R, denoted Mod R, where Hom(X, Y ) is all R-module homomorphisms from X to Y. And so forth. I ll leave to your imagination the definitions of some more categories for which I might need names later: Ab (abelian groups), Grp (groups), TopGp (topological groups). However, there are other things that can be viewed as categories. An important example: given any partially ordered set S, make a category in which the objects are the elements of S, and there is exactly one morphism from X to Y if X Y and none otherwise. Important special case of the previous one: given a topological space S, we can make a category in which the objects are the open subsets of S, and the morphisms are the inclusions of one open subset into another. Another example comes from algebra. Given a group, we can make a category with only one object X, in which Hom(X, X) is the group and the composition law is the group operation. Here s a more interesting example along the lines of the previous one. Given a topological space S, make a category in which the objects are the points of S, and the morphisms from X to Y are the continuous functions f : [0, 1] S with f(0) = X and f(1) = Y. Define the composition g f for g Hom(Y, Z) and f Hom(X, Y ) to be the function h : [0, 1] S with { f(2x) x [0, 1/2] h(x) = g(2x 1) x [1/2, 1]. This is a special case of turning a groupoid (something which is like a group except that objects can only be composed if they satisfy a matching condition) into a category. This example comes from the fundamental groupoid of a topological space. 2

4 3 Interlude: is versus does The rigorous formulation of category theory exposes a dark secret of mathematics: objects in a category are rarely ever equal. For instance, we all think we agree on what the ring Z is, but if we all sat down and wrote down set-theoretic definitions, probably no two of them would exactly match. The point is that we conceive of Z, and of most mathematical objects in general, not in terms of what they literally are as sets, but by how they work, and in particular how they relate to other mathematical objects. The solution for this suggested by category theory is to characterize interesting mathematical objects using universal properties. For instance, the ring Z is characterized by the fact that it is an initial object in the category of rings: for every ring Y, there is a unique morphism from Z to Y. Any two objects with this property are uniquely isomorphic. Here are a few other arrow-theoretic properties that can be used for this purposes. I ll talk more about universal properties later. Y Obj(C) is a final object in C if for any X Obj(C), there is a unique morphism from X to Y. An object which is both initial and final is a terminal object. A morphism f Hom(X, Y ) is a monomorphism if for any g, h Hom(W, X), if f g = f h, then g = h. In the category of sets (and many other examples), f is a monomorphism if and only if f is injective. A morphism f Hom(X, Y ) is an epimorphism if for any g, h Hom(Y, Z), if g f = h f, then g = h. In the category of sets (and many other examples), f is an epimorphism if and only if f is surjective. But beware of surprises: for example, the morphism Z Q of rings is an epimorphism (and also a monomorphism). A morphism f Hom(X, Y ) is an isomorphism if it has a two-sided inverse. This im plies that it is a monomorphism and an epimorphism, but not conversely (see previous example). 4 Functors and natural transformations Functors can be thought of as functions between categories. A covariant functor from a category C 1 to a category C 2 consists of: A function F from Obj(C 1 ) to Obj(C 2 ). For each pair (X, Y ) of Obj(C 1 ), a function F X,Y : Hom(X, Y ) Hom(F (X), F (Y )), such that F commutes with composition and F carries id X to id F (X). A contravariant functor works the same way except that F X,Y carries Hom(X, Y ) to Hom(F (Y ), F (X)), that is, it reverses the sense of the morphisms. You can turn it into a covariant functor by replacing one of the two categories with its opposite category, in which all morphisms are reversed; for simplicity, let us just talk about covariant functors for the moment. 3

5 This point is actualized by the notion of a natural transformation. Given two functors F 1, F 2 from C 1 to C 2, a natural transformation of F 1 to F 2 consists of, for each X Obj(C 1 ), a morphism φ X : F 1 (X) F 2 (X) such that for every morphism f Hom(X, Y ), the diagram F 1 (f) F 1 (X) F 1 (Y ) φ X φ Y F 2 (f) F 2 (X) F 2 (Y ) is commutative (that is, if you trace around both ways you get the same answer). Natural transformations can be composed; one with an inverse (equivalently, in which the morphisms φ X are all isomorphisms) is called a natural isomorphism. For instance, the functors taking ordered triples (M 1, M 2, M 3 ) of modules over a ring R to are naturally isomorphic. (M 1 R M 2 ) R M 3 and M 1 R (M 2 R M 3 ) 5 Other properties of functors A functor is faithful if the maps F X,Y are injective. Typical examples of these are forgetful functors, in which you start with a category of objects carrying a lot of structure, and the functor strips off some structure. E.g., take groups to their underlying sets, or take rings to their additive groups, or take topological groups to their underlying topological spaces. The analogues of injectivity and surjectivity for functors are: A functor is fully faithful if the maps F X,Y are bijective. A typical example is the inclusion of a full subcategory (i.e., take some of the objects, and all of the morphisms between the chosen objects). A functor is essentially surjective if every object in C 2 is isomorphic to an object of the form F (X) for some X Obj(C 1 ). A functor is an equivalence of categories if it is fully faithful and essentially surjective. This is equivalent to the existence of a quasi-inverse functor, i.e., one for which the compositions in both directions are naturally isomorphic to the relevant identities. A typical example from last semester: take the category of affine algebraic varieties over an algebraically closed field k. The functor Ω computing regular functions is an equivalence between this category and (the opposite category of) finitely generated k-algebras which are reduced (have no nilpotent elements). One of the goals of schemes is to set up a similar equivalence between some sort of geometric objects and the category of all commutative unital rings. 4

6 6 Representable functors, Yoneda s lemma, and universal properties An individual object in a category casts a sort of shadow on the entire category, via the notion of representable functors. For a fixed object X in a category C, let h X be the functor from C to Set such that h X (Y ) = Hom(X, Y ), and the image of f Hom(Y, Z) under h X carries Hom(X, Y ) to Hom(Y, Z) via postcomposition with f. It turns out that any natural transformation from h X to any other functor F : C Set is determined by specifying the image of the special element id X of Hom(X, X) = h X (X), and conversely any such choice induces a natural transformation from h X to F. This is Yoneda s lemma; proof is left as an (easy) homework problem. An arbitrary functor F : C Set is representable if it is naturally isomorphic to h X for some X. By Yoneda s lemma, if X and Y represent the same functor, then they are isomorphic in a natural way (i.e., one compatible with the action of the functor). In practice, this is usually interpreted as saying that an object of a category determined by a universal mapping property is unique up to unique isomorphism (or up to natural isomorphism). Here is an example of this which will help explain why categorical thinking is so helpful when dealing with schemes. For objects X, Y in a category C, an (absolute) product of X and Y is an object Z equipped with maps Z X and Z Y, with the following universal mapping property. Given any object W and morphisms W X and W Y, there must be a unique morphism W Z such that the diagram W Z X Y commutes. The product is unique in the sense that if Z is an other object equipped with morphisms Z X and Z Y satisfying the mapping property, there is a unique isomorphism Z Z making everything commute. In any normal category, in which objects are sets equipped with some extra structure (e.g., groups, topological groups), products exist and can be written as Cartesian products with some appropriate extra structure. But in general, products need not exists, and even if they do they might look weird. Case in point: suppose we tried to make a theory of abstract algebraic varieties over the non-algebraically closed field Q, in which the points are Galois orbits of points over Q. (This is close to what will happen with schemes, except that there will be some more points.) Then in the affine line, we have a variety consisting of the single orbit {i, i}. The product of this with itself will then consist of the two orbits {(i, i), ( i, i)} and {(i, i), ( i, i)}. 5

8 Another important example for us: let R S be a homomorphism of rings. Let F : Mod R Mod S be the functor M M R S. Let F : Mod S Mod R be the functor given by restriction of scalars from S to R. Then F and F form an adjoint pair. We can of course compose F and F both ways, and we don t in general get the identity, or even something naturally isomorphic to the identity. We do get something interesting, though. The identity map on F X corresponds to a morphism X F F X, while the identity map on F Y corresponds to a morphism F F Y Y. These morphisms are called adjunction morphisms. For example, in the previous example, for X an R-module, X F F X = X R S is the map x x 1. For Y an S-module, Y R S Y is the map y i s i y i s i. i i 7

### 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

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

### Algebraic 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Where we were 1 2. Yoneda s lemma 2 3. Limits and colimits 6 4. Adjoints 8 First, some bureaucratic details. We will move to 380-F for Monday

### COMMUTATIVE ALGEBRA LECTURE 1: SOME CATEGORY THEORY

COMMUTATIVE ALGEBRA LECTURE 1: SOME CATEGORY THEORY VIVEK SHENDE A ring is a set R with two binary operations, an addition + and a multiplication. Always there should be an identity 0 for addition, an

### Algebraic 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

### Algebraic 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

### Category Theory. Categories. Definition.

Category Theory Category theory is a general mathematical theory of structures, systems of structures and relationships between systems of structures. It provides a unifying and economic mathematical modeling

### Categories and functors

Lecture 1 Categories and functors Definition 1.1 A category A consists of a collection ob(a) (whose elements are called the objects of A) for each A, B ob(a), a collection A(A, B) (whose elements are called

### EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY

EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY 1. Categories 1.1. Generalities. I ve tried to be as consistent as possible. In particular, throughout the text below, categories will be denoted by capital

### C2.7: CATEGORY THEORY

C2.7: CATEGORY THEORY PAVEL SAFRONOV WITH MINOR UPDATES 2019 BY FRANCES KIRWAN Contents Introduction 2 Literature 3 1. Basic definitions 3 1.1. Categories 3 1.2. Set-theoretic issues 4 1.3. Functors 5

### Representable presheaves

Representable presheaves March 15, 2017 A presheaf on a category C is a contravariant functor F on C. In particular, for any object X Ob(C) we have the presheaf (of sets) represented by X, that is Hom

### Assume the left square is a pushout. Then the right square is a pushout if and only if the big rectangle is.

COMMUTATIVE ALGERA LECTURE 2: MORE CATEGORY THEORY VIVEK SHENDE Last time we learned about Yoneda s lemma, and various universal constructions initial and final objects, products and coproducts (which

### ALGEBRAIC 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

### CATEGORY THEORY. Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths.

CATEGORY THEORY PROFESSOR PETER JOHNSTONE Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths. Definition 1.1. A category C consists

### Category Theory (UMV/TK/07)

P. J. Šafárik University, Faculty of Science, Košice Project 2005/NP1-051 11230100466 Basic information Extent: 2 hrs lecture/1 hrs seminar per week. Assessment: Written tests during the semester, written

Adjunctions! Everywhere! Carnegie Mellon University Thursday 19 th September 2013 Clive Newstead Abstract What do free groups, existential quantifiers and Stone-Čech compactifications all have in common?

### 48 CHAPTER 2. COMPUTATIONAL METHODS

48 CHAPTER 2. COMPUTATIONAL METHODS You get a much simpler result: Away from 2, even projective spaces look like points, and odd projective spaces look like spheres! I d like to generalize this process

### 3. The Sheaf of Regular Functions

24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

### Lectures on Homological Algebra. Weizhe Zheng

Lectures on Homological Algebra Weizhe Zheng Morningside Center of Mathematics Academy of Mathematics and Systems Science, Chinese Academy of Sciences Beijing 100190, China University of the Chinese Academy

### VERY ROUGH NOTES ON SPECTRAL DEFORMATION THEORY

VERY ROUGH NOTES ON SPECTRAL DEFORMATION THEORY 1. Classical Deformation Theory I want to begin with some classical deformation theory, before moving on to the spectral generalizations that constitute

### MATH 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

### Homology and Cohomology of Stacks (Lecture 7)

Homology and Cohomology of Stacks (Lecture 7) February 19, 2014 In this course, we will need to discuss the l-adic homology and cohomology of algebro-geometric objects of a more general nature than algebraic

### Formal 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

### Algebraic 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

### Math 762 Spring h Y (Z 1 ) (1) h X (Z 2 ) h X (Z 1 ) Φ Z 1. h Y (Z 2 )

Math 762 Spring 2016 Homework 3 Drew Armstrong Problem 1. Yoneda s Lemma. We have seen that the bifunctor Hom C (, ) : C C Set is analogous to a bilinear form on a K-vector space, : V V K. Recall that

### FOUNDATIONS 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

### sset(x, Y ) n = sset(x [n], Y ).

1. Symmetric monoidal categories and enriched categories In practice, categories come in nature with more structure than just sets of morphisms. This extra structure is central to all of category theory,

### Symbol Index Group GermAnal Ring AbMonoid

Symbol Index 409 Symbol Index Symbols of standard and uncontroversial usage are generally not included here. As in the word index, boldface page-numbers indicate pages where definitions are given. If a

### Adjoints, naturality, exactness, small Yoneda lemma. 1. Hom(X, ) is left exact

(April 8, 2010) Adjoints, naturality, exactness, small Yoneda lemma Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ The best way to understand or remember left-exactness or right-exactness

### 1 Categorical Background

1 Categorical Background 1.1 Categories and Functors Definition 1.1.1 A category C is given by a class of objects, often denoted by ob C, and for any two objects A, B of C a proper set of morphisms C(A,

### ALGEBRAIC 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!)

### SECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES

SECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there

### Universal Properties

A categorical look at undergraduate algebra and topology Julia Goedecke Newnham College 24 February 2017, Archimedeans Julia Goedecke (Newnham) 24/02/2017 1 / 30 1 Maths is Abstraction : more abstraction

### MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23

MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23 6.4. Homotopy uniqueness of projective resolutions. Here I proved that the projective resolution of any R-module (or any object of an abelian category

Monomorphism - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/monomorphism 1 of 3 24/11/2012 02:01 Monomorphism From Wikipedia, the free encyclopedia In the context of abstract algebra or

### PART I. Abstract algebraic categories

PART I Abstract algebraic categories It should be observed first that the whole concept of category is essentially an auxiliary one; our basic concepts are those of a functor and a natural transformation.

### ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY

ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ARDA H. DEMIRHAN Abstract. We examine the conditions for uniqueness of differentials in the abstract setting of differential geometry. Then we ll come up

### Exercises on chapter 0

Exercises on chapter 0 1. A partially ordered set (poset) is a set X together with a relation such that (a) x x for all x X; (b) x y and y x implies that x = y for all x, y X; (c) x y and y z implies that

### Lecture 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

### 1 Introduction. 2 Categories. Mitchell Faulk June 22, 2014 Equivalence of Categories for Affine Varieties

Mitchell Faulk June 22, 2014 Equivalence of Categories for Affine Varieties 1 Introduction Recall from last time that every affine algebraic variety V A n determines a unique finitely generated, reduced

### Categories and Modules

Categories and odules Takahiro Kato arch 2, 205 BSTRCT odules (also known as profunctors or distributors) and morphisms among them subsume categories and functors and provide more general and abstract

### Operads. Spencer Liang. March 10, 2015

Operads Spencer Liang March 10, 2015 1 Introduction The notion of an operad was created in order to have a well-defined mathematical object which encodes the idea of an abstract family of composable n-ary

### A TALE OF TWO FUNCTORS. Marc Culler. 1. Hom and Tensor

A TALE OF TWO FUNCTORS Marc Culler 1. Hom and Tensor It was the best of times, it was the worst of times, it was the age of covariance, it was the age of contravariance, it was the epoch of homology, it

### 1. Introduction. Let C be a Waldhausen category (the precise definition

K-THEORY OF WLDHUSEN CTEGORY S SYMMETRIC SPECTRUM MITY BOYRCHENKO bstract. If C is a Waldhausen category (i.e., a category with cofibrations and weak equivalences ), it is known that one can define its

### 1 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

### Morita Equivalence. Eamon Quinlan

Morita Equivalence Eamon Quinlan Given a (not necessarily commutative) ring, you can form its category of right modules. Take this category and replace the names of all the modules with dots. The resulting

### Elementary (ha-ha) Aspects of Topos Theory

Elementary (ha-ha) Aspects of Topos Theory Matt Booth June 3, 2016 Contents 1 Sheaves on topological spaces 1 1.1 Presheaves on spaces......................... 1 1.2 Digression on pointless topology..................

### Homework 3: Relative homology and excision

Homework 3: Relative homology and excision 0. Pre-requisites. The main theorem you ll have to assume is the excision theorem, but only for Problem 6. Recall what this says: Let A V X, where the interior

### DESCENT 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,

### Lecture 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,

### S n 1 i D n l S n 1 is the identity map. Associated to this sequence of maps is the sequence of group homomorphisms

ALGEBRAIC TOPOLOGY Contents 1. Informal introduction 1 1.1. What is algebraic topology? 1 1.2. Brower fixed point theorem 2 2. Review of background material 3 2.1. Algebra 3 2.2. Topological spaces 5 2.3.

### Basic categorial constructions

November 9, 2010) Basic categorial constructions Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ 1. Categories and functors 2. Standard boring) examples 3. Initial and final objects

### OVERVIEW OF SPECTRA. Contents

OVERVIEW OF SPECTRA Contents 1. Motivation 1 2. Some recollections about Top 3 3. Spanier Whitehead category 4 4. Properties of the Stable Homotopy Category HoSpectra 5 5. Topics 7 1. Motivation There

### An introduction to locally finitely presentable categories

An introduction to locally finitely presentable categories MARU SARAZOLA A document born out of my attempt to understand the notion of locally finitely presentable category, and my annoyance at constantly

### 58 CHAPTER 2. COMPUTATIONAL METHODS

58 CHAPTER 2. COMPUTATIONAL METHODS 23 Hom and Lim We will now develop more properties of the tensor product: its relationship to homomorphisms and to direct limits. The tensor product arose in our study

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Some constructions using universal properties, old and new 1 2. Adjoint functors 3 3. Sheaves 6 Last day: What is algebraic geometry? Crash

### Etale cohomology of fields by Johan M. Commelin, December 5, 2013

Etale cohomology of fields by Johan M. Commelin, December 5, 2013 Etale cohomology The canonical topology on a Grothendieck topos Let E be a Grothendieck topos. The canonical topology T on E is given in

### Algebraic 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

### MAS435 Algebraic Topology Part A: Semester 1 Exercises

MAS435 Algebraic Topology 2011-12 Part A: Semester 1 Exercises Dr E. L. G. Cheng Office: J24 E-mail: e.cheng@sheffield.ac.uk http://cheng.staff.shef.ac.uk/mas435/ You should hand in your solutions to Exercises

### LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT

LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral D-module, unless explicitly stated

### Algebraic 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

### Algebra and Topology

Algebra and Topology Course at Paris VI University, 2007/2008 1 Pierre Schapira http://www.math.jussieu.fr/ schapira/lectnotes schapira@math.jussieu.fr 1/9/2011, v2 1 To the students: the material covered

### MATH 101: ALGEBRA I WORKSHEET, DAY #1. We review the prerequisites for the course in set theory and beginning a first pass on group. 1.

MATH 101: ALGEBRA I WORKSHEET, DAY #1 We review the prerequisites for the course in set theory and beginning a first pass on group theory. Fill in the blanks as we go along. 1. Sets A set is a collection

### ALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!

ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.

### Review of category theory

Review of category theory Proseminar on stable homotopy theory, University of Pittsburgh Friday 17 th January 2014 Friday 24 th January 2014 Clive Newstead Abstract This talk will be a review of the fundamentals

### and this makes M into an R-module by (1.2). 2

1. Modules Definition 1.1. Let R be a commutative ring. A module over R is set M together with a binary operation, denoted +, which makes M into an abelian group, with 0 as the identity element, together

### GALOIS CATEGORIES MELISSA LYNN

GALOIS CATEGORIES MELISSA LYNN Abstract. In abstract algebra, we considered finite Galois extensions of fields with their Galois groups. Here, we noticed a correspondence between the intermediate fields

### CATEGORICAL 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.

### Some remarks on Frobenius and Lefschetz in étale cohomology

Some remarks on obenius and Lefschetz in étale cohomology Gabriel Chênevert January 5, 2004 In this lecture I will discuss some more or less related issues revolving around the main idea relating (étale)

### 14 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

### Derived Algebraic Geometry IX: Closed Immersions

Derived Algebraic Geometry I: Closed Immersions November 5, 2011 Contents 1 Unramified Pregeometries and Closed Immersions 4 2 Resolutions of T-Structures 7 3 The Proof of Proposition 1.0.10 14 4 Closed

### arxiv:math/ v1 [math.at] 6 Oct 2004

arxiv:math/0410162v1 [math.at] 6 Oct 2004 EQUIVARIANT UNIVERSAL COEFFICIENT AND KÜNNETH SPECTRAL SEQUENCES L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL Abstract. We construct hyper-homology spectral sequences

### The Adjoint Functor Theorem.

The Adjoint Functor Theorem. Kevin Buzzard February 7, 2012 Last modified 17/06/2002. 1 Introduction. The existence of free groups is immediate from the Adjoint Functor Theorem. Whilst I ve long believed

### FILTERED 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

### MTH 428/528. Introduction to Topology II. Elements of Algebraic Topology. Bernard Badzioch

MTH 428/528 Introduction to Topology II Elements of Algebraic Topology Bernard Badzioch 2016.12.12 Contents 1. Some Motivation.......................................................... 3 2. Categories

### Finite 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

### Commutative Algebra Lecture 3: Lattices and Categories (Sept. 13, 2013)

Commutative Algebra Lecture 3: Lattices and Categories (Sept. 13, 2013) Navid Alaei September 17, 2013 1 Lattice Basics There are, in general, two equivalent approaches to defining a lattice; one is rather

### UMASS AMHERST MATH 300 SP 05, F. HAJIR HOMEWORK 8: (EQUIVALENCE) RELATIONS AND PARTITIONS

UMASS AMHERST MATH 300 SP 05, F. HAJIR HOMEWORK 8: (EQUIVALENCE) RELATIONS AND PARTITIONS 1. Relations Recall the concept of a function f from a source set X to a target set Y. It is a rule for mapping

### Recall: a mapping f : A B C (where A, B, C are R-modules) is called R-bilinear if f is R-linear in each coordinate, i.e.,

23 Hom and We will do homological algebra over a fixed commutative ring R. There are several good reasons to take a commutative ring: Left R-modules are the same as right R-modules. [In general a right

### Algebra 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.

### WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES

WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels,

### LECTURE 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

### Adjunctions, the Stone-Čech compactification, the compact-open topology, the theorems of Ascoli and Arzela

Adjunctions, the Stone-Čech compactification, the compact-open topology, the theorems of Ascoli and Arzela John Terilla Fall 2014 Contents 1 Adjoint functors 2 2 Example: Product-Hom adjunction in Set

### RUDIMENTARY CATEGORY THEORY NOTES

RUDIMENTARY CATEGORY THEORY NOTES GREG STEVENSON Contents 1. Categories 1 2. First examples 3 3. Properties of morphisms 5 4. Functors 6 5. Natural transformations 9 6. Equivalences 12 7. Adjunctions 14

### Category Theory. Travis Dirle. December 12, 2017

Category Theory 2 Category Theory Travis Dirle December 12, 2017 2 Contents 1 Categories 1 2 Construction on Categories 7 3 Universals and Limits 11 4 Adjoints 23 5 Limits 31 6 Generators and Projectives

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 3

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 3 RAVI VAKIL CONTENTS 1. Kernels, cokernels, and exact sequences: A brief introduction to abelian categories 1 2. Sheaves 7 3. Motivating example: The sheaf of differentiable

### NOTES ON ATIYAH S TQFT S

NOTES ON ATIYAH S TQFT S J.P. MAY As an example of categorification, I presented Atiyah s axioms [1] for a topological quantum field theory (TQFT) to undergraduates in the University of Chicago s summer

### Lecture 9: Sheaves. February 11, 2018

Lecture 9: Sheaves February 11, 2018 Recall that a category X is a topos if there exists an equivalence X Shv(C), where C is a small category (which can be assumed to admit finite limits) equipped with

### Sample algebra qualifying exam

Sample algebra qualifying exam University of Hawai i at Mānoa Spring 2016 2 Part I 1. Group theory In this section, D n and C n denote, respectively, the symmetry group of the regular n-gon (of order 2n)

### Supercategories. Urs July 5, Odd flows and supercategories 4. 4 Braided monoidal supercategories 7

Supercategories Urs July 5, 2007 ontents 1 Introduction 1 2 Flows on ategories 2 3 Odd flows and supercategories 4 4 Braided monoidal supercategories 7 1 Introduction Motivated by the desire to better

### A Grothendieck site is a small category C equipped with a Grothendieck topology T. A Grothendieck topology T consists of a collection of subfunctors

Contents 5 Grothendieck topologies 1 6 Exactness properties 10 7 Geometric morphisms 17 8 Points and Boolean localization 22 5 Grothendieck topologies A Grothendieck site is a small category C equipped

### 1 Replete topoi. X = Shv proét (X) X is locally weakly contractible (next lecture) X is replete. D(X ) is left complete. K D(X ) we have R lim

Reference: [BS] Bhatt, Scholze, The pro-étale topology for schemes In this lecture we consider replete topoi This is a nice class of topoi that include the pro-étale topos, and whose derived categories

### Math 901 Notes 3 January of 33

Math 901 Notes 3 January 2012 1 of 33 Math 901 Fall 2011 Course Notes Information: These are class notes for the second year graduate level algebra course at UNL (Math 901) as taken in class and later

### Basic Modern Algebraic Geometry

Version of December 13, 1999. Final version for M 321. Audun Holme Basic Modern Algebraic Geometry Introduction to Grothendieck s Theory of Schemes Basic Modern Algebraic Geometry Introduction to Grothendieck

### Derived Differential Geometry

Derived Differential Geometry Lecture 1 of 3: Dominic Joyce, Oxford University Derived Algebraic Geometry and Interactions, Toulouse, June 2017 For references, see http://people.maths.ox.ac.uk/ joyce/dmanifolds.html,

### 2. Prime and Maximal Ideals

18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the so-called prime and maximal ideals. Let

### COURSE NOTES: HOMOLOGICAL ALGEBRA

COURSE NOTES: HOMOLOGICAL ALGEBRA AMNON YEKUTIELI Contents 1. Introduction 1 2. Categories 2 3. Free Modules 13 4. Functors 18 5. Natural Transformations 23 6. Equivalence of Categories 29 7. Opposite