# INTRO TO TENSOR PRODUCTS MATH 250B

Size: px
Start display at page:

Transcription

1 INTRO TO TENSOR PRODUCTS MATH 250B ADAM TOPAZ 1. Definition of the Tensor Product Throughout this note, A will denote a commutative ring. Let M, N be two A-modules. For a third A-module Z, consider the set of A-bilinear maps M N Z. Bilin(M N, Z) Proposition 1.1. The functor Z Bilin(M N, Z) is representable by an object which we denote by M A N. This object M A N is unique upto a unique isomorphism, and it satisfies the following universal property. First, there is a representing bilinear map φ : M N M A N. And for all modules Z, the following holds. For any A-bilinear map f : M N Z, there exists a unique morphism M A N Z such that f is the composition M N φ M A N Z. Proof. The uniqueness statements all follow from general nonsense on representable functors. Thus, it suffices to determine the object M A N and the morphism φ. This is where the usual construction of the tensor product comes in: M A N = R(M N) relations where the relations are generated by (1) a(m, n) (am, n) (m, an). (2) (m + m, n) (m, n) + (m, n). (3) (m, n + n ) (m, n) + (m, n ). The map φ is given by (m, n) (m, n) =: m n M A N. Here are a few important properties of the tensor product. Proposition 1.2. The following hold: (1) is commutative and associated (up-to canonical isomorphisms). (2) is distributive (with respect to arbitrary direct sums). (3) N is a right-exact functor. (4) One has Hom(M N, Z) = Hom(M, Hom(N, Z)) canonically, i.e. N is left-adjoint to Hom(N, ). Proof. All properties can be deduced from the construction of the tensor product. But a better proof uses the fact that the tensor product represents Bilinear maps. 1

2 Example 1.3. Let N be a submodule of N. Then one has a canonical isomorphism (N/N ) M = N M/ im(n M N M). In particular, if a is an ideal of A, then one has M/a M = (A/a) M We can formulate an analogous definition for n-multi-linear maps, from M 1 M n to Z, but this will just yield the n-fold tensor product M 1 A A M n which represents the set of n-multilinear maps to Z. This n-fold tensor product makes sense since is associative. The details are left as an exercise. 2. Base Change Recall that an A-algebra is a commutative ring B endowed with a morphism A B. We can consider B as an A-module with respect to the natural action of a A on b B via the structure map and multiplication in B. For an A-module M, the base change of M to B is the B-module B A M where the action of b B on (b m) is defined as b (b m) = (bb ) m. On the other hand, if N is a B-module, then we can consider N as an A-module via the morphism A B. Proposition 2.1. The following hold: (1) One has Hom B (B A M, N) = Hom A (M, N). (2) One has N B (B A M) = N A M. Suppose that B and C are two A-algebras. Then the tensor product B A C can be given the structure of an A-algebra as follows: (1) The product in B C is given by (b c) (b c ) = (bb ) (cc ). (2) The map A B A C is given by a a 1 = 1 a. Note that one also has canonical morphisms of A-algebras B B C and C B C given by b b 1 and c 1 c. Proposition 2.2. The tensor product of algebras B A C, endowed with the morphisms B B C and C B C, is the fibered coproduct in the category of A-algebras. I.e. if E is an A-algebra which fits into the commutative diagram: then the diagram can be completed by a unique morphism.... Example 2.3. Let B be an A-algebra. Then one has B A A[t 1,..., t n ] = B[t 1,..., t n ] as B-algebras. This isn t hard to see from the construction of the tensor product. But let s try to give a different proof here. Namely, we know that A[t 1,..., t n ] represents the functor which sends an A-algebra C to C n. But then by the properties of base change, B A A[t 1,..., t n ] sends a B-algebra C to C n by the proposition above. The claim follows. 2

3 HEre is an arithmetic example: Example 2.4. Let L K be a finite Galois extension of fields with Galois group G. Then one has L K L = σ G L where G acts on the left-hand side by acting on the first component of L K L and on the right hand side by acting as permutation of the components in the direct product. This equality follows from the primitive element theorem for Galois extensions. In fact, this fact can be used to construct the Galois closure of a finite separable extension L K discussed in class. 3. Flat Modules Since M is right-exact, it makes sense to study the extent to which is fails to be exact. We say that M is a flat module provided that A M is an exact functor. Proposition 3.1. The following hold: (1) Free modules are flat. (2) The tensor product of flat modules is flat. (3) If B is an A-algebra and M is a flat A-module then the base change B A M is flat over B. (4) If B is a flat A-algebra, then every flat B-module is also flat over A. Example 3.2. Let M = Z/n as a Z-module n 2. Then M is not flat over Z. Namely, tensoring the injection Z n Z with M yields a non-injective map. Theorem 3.3. Let A be an aribtary ring and let M be an A-module. Then the following are equivalent: (1) M is flat. (2) For all ideals I of A, the canonical map I A M IM is an isomorphism. Proof. (1) implies (2) is trivial. Let us assume (2). Let N N be an injective morphism of A-modules. We must show that M preserves injectivity. First assume that N is free of finite rank, and we proceed by induiction on the rank. The case rank = 1 is the assumption. For higher rank, one has N = N 1 N 2 with both N i N. Let N 1 = N 1 N and N 2 the image of N in N 2 = N/N 1. We have a diagram: 0 N 1 N N 2 0 with exact rows and injective vertical maps. Now tensor this with M to get 0 N 1 N N 2 0 N 1 M N M N 2 M 0 0 N 1 M N M N 2 M 0 3

4 by assumption the left and right vertical morphisms are injective and the bottom row is exact since N 1 is a direct factor of N. Now the snake lemma implies that the middle morphism is injective as well. Now suppose that N is an aribtrary free module, and let N 0 be a direct factor of N of finite rank. By the above, the map N N 0 M N 0 M is injective. But every x N M comes from N N 0 M for some N 0 as above. Thus, the map N M N M is injective. Now let N be arbitrary, and take a free presentation F N of N with kernel K. Let F be the preimage of N in F. Then one has 0 K F N 0 with exact rows and injective vertical maps. Tensoring with M we get 0 K F N 0 0 K M F M N M 0 0 K M F M N M 0 whose middle map is injective. by the snake lemma, the right map is also injective. Corollary 3.4. Let A be a PID. Then an A-module M is flat if and only if M is torsion-free. Proof. exercise. 4. Tor Functors We define Tor i A(M, ) as L i (M A ) the left-derived functors of M A. Recall that to compute this, we take a projective resolution P of M, and define Here are a few immediate properties: Tor i A(M, N) = H i (P A N). Proposition 4.1. (1) A module M is flat if and only if Tor i (M, N) = 0 for all i 1 and all N. (2) One has canonical (functorial) isomorphisms Tor (M, N) = Tor (N, M). (3) Tor is additive (in both arguments). (4) A short exact sequence of modules yields a long exact sequence of Tor s. Since flat modules are acyclic with respect to Tor, we can calculate Tor s using flat resolutions. Namely, if F is a flat resolution of M, then one has Tor (M, N) = H i (F N). A particularly important case of this is using free resolustions which are, of course, flat. So in practice, one always uses a free resolution of M in order to compute Tor (M, ). Let s look at some examples: (1) Suppose that f is not a zero-divisori in A, then Tor 1 (A/(f), M) = f M the f-torsion in M. I.e. f M = {m M : fm = 0}. (2) In particular, one has Tor 1 (Z/n, M) is the n-torsion of M. 4

5 (3) One has Tor 1 (Q/Z, M) is the torsion subgroup of M. (4) One has Tor q (Q p, Z p, M) is the p-pwoer torsion group of M. (5) One has Tor (A, M) = 0 for all M and all 1 since A is free! (6) Using the above calculations, it is easy to calculate Tor (T, M) for any finitely generated abelian group T, using the fundamental theorem of finitely generated abelian groups. 5

### 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

### Notes on the definitions of group cohomology and homology.

Notes on the definitions of group cohomology and homology. Kevin Buzzard February 9, 2012 VERY sloppy notes on homology and cohomology. Needs work in several places. Last updated 3/12/07. 1 Derived functors.

### EXT, TOR AND THE UCT

EXT, TOR AND THE UCT CHRIS KOTTKE Contents 1. Left/right exact functors 1 2. Projective resolutions 2 3. Two useful lemmas 3 4. Ext 6 5. Ext as a covariant derived functor 8 6. Universal Coefficient Theorem

### Cohomology 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)

### 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

### Notes 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

### ALGEBRA HW 3 CLAY SHONKWILER

ALGEBRA HW 3 CLAY SHONKWILER (a): Show that R[x] is a flat R-module. 1 Proof. Consider the set A = {1, x, x 2,...}. Then certainly A generates R[x] as an R-module. Suppose there is some finite linear combination

### 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

### FORMAL 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

### NOTES ON CHAIN COMPLEXES

NOTES ON CHAIN COMPLEXES ANDEW BAKE These notes are intended as a very basic introduction to (co)chain complexes and their algebra, the intention being to point the beginner at some of the main ideas which

### MODEL ANSWERS TO THE FOURTH HOMEWORK

MODEL ANSWES TO THE OUTH HOMEWOK 1. (i) We first try to define a map left to right. Since finite direct sums are the same as direct products, we just have to define a map to each factor. By the universal

### 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

### SECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS

SECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS In this section we will prove the Künneth theorem which in principle allows us to calculate the (co)homology of product spaces as soon

### 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

### 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)

### 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,

### 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.

### 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

### Math 210B:Algebra, Homework 2

Math 210B:Algebra, Homework 2 Ian Coley January 21, 2014 Problem 1. Is f = 2X 5 6X + 6 irreducible in Z[X], (S 1 Z)[X], for S = {2 n, n 0}, Q[X], R[X], C[X]? To begin, note that 2 divides all coefficients

### MATH 101A: ALGEBRA I PART C: TENSOR PRODUCT AND MULTILINEAR ALGEBRA. This is the title page for the notes on tensor products and multilinear algebra.

MATH 101A: ALGEBRA I PART C: TENSOR PRODUCT AND MULTILINEAR ALGEBRA This is the title page for the notes on tensor products and multilinear algebra. Contents 1. Bilinear forms and quadratic forms 1 1.1.

### De Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1)

II. De Rham Cohomology There is an obvious similarity between the condition d o q 1 d q = 0 for the differentials in a singular chain complex and the condition d[q] o d[q 1] = 0 which is satisfied by the

### MATH 101A: ALGEBRA I PART C: TENSOR PRODUCT AND MULTILINEAR ALGEBRA. This is the title page for the notes on tensor products and multilinear algebra.

MATH 101A: ALGEBRA I PART C: TENSOR PRODUCT AND MULTILINEAR ALGEBRA This is the title page for the notes on tensor products and multilinear algebra. Contents 1. Bilinear forms and quadratic forms 1 1.1.

### NOTES ON SPLITTING FIELDS

NOTES ON SPLITTING FIELDS CİHAN BAHRAN I will try to define the notion of a splitting field of an algebra over a field using my words, to understand it better. The sources I use are Peter Webb s and T.Y

### HOMEWORK SET 3. Local Class Field Theory - Fall For questions, remarks or mistakes write me at

HOMEWORK SET 3 Local Class Field Theory - Fall 2011 For questions, remarks or mistakes write me at sivieroa@math.leidneuniv.nl. Exercise 3.1. Suppose A is an abelian group which is torsion (every element

### Isogeny invariance of the BSD formula

Isogeny invariance of the BSD formula Bryden Cais August 1, 24 1 Introduction In these notes we prove that if f : A B is an isogeny of abelian varieties whose degree is relatively prime to the characteristic

### 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

### Math 121 Homework 5: Notes on Selected Problems

Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements

### TENSOR PRODUCTS. (5) A (distributive) multiplication on an abelian group G is a Z-balanced map G G G.

TENSOR PRODUCTS Balanced Maps. Note. One can think of a balanced map β : L M G as a multiplication taking its values in G. If instead of β(l, m) we write simply lm (a notation which is often undesirable)

### Raynaud on F -vector schemes and prolongation

Raynaud on F -vector schemes and prolongation Melanie Matchett Wood November 7, 2010 1 Introduction and Motivation Given a finite, flat commutative group scheme G killed by p over R of mixed characteristic

### Commutative Algebra. Contents. B Totaro. Michaelmas Basics Rings & homomorphisms Modules Prime & maximal ideals...

Commutative Algebra B Totaro Michaelmas 2011 Contents 1 Basics 4 1.1 Rings & homomorphisms.............................. 4 1.2 Modules........................................ 6 1.3 Prime & maximal ideals...............................

### Relative Left Derived Functors of Tensor Product Functors. Junfu Wang and Zhaoyong Huang

Relative Left Derived Functors of Tensor Product Functors Junfu Wang and Zhaoyong Huang Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, China Abstract We introduce and

### 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

### Commutative Algebra. B Totaro. Michaelmas Basics Rings & homomorphisms Modules Prime & maximal ideals...

Commutative Algebra B Totaro Michaelmas 2011 Contents 1 Basics 2 1.1 Rings & homomorphisms................... 2 1.2 Modules............................. 4 1.3 Prime & maximal ideals....................

### NOTES ON BASIC HOMOLOGICAL ALGEBRA 0 L M N 0

NOTES ON BASIC HOMOLOGICAL ALGEBRA ANDREW BAKER 1. Chain complexes and their homology Let R be a ring and Mod R the category of right R-modules; a very similar discussion can be had for the category of

### A 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

### COMMUTATIVE ALGEBRA, LECTURE NOTES

COMMUTATIVE ALGEBRA, LECTURE NOTES P. SOSNA Contents 1. Very brief introduction 2 2. Rings and Ideals 2 3. Modules 10 3.1. Tensor product of modules 15 3.2. Flatness 18 3.3. Algebras 21 4. Localisation

### Lecture 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

### Lecture 7. This set is the set of equivalence classes of the equivalence relation on M S defined by

Lecture 7 1. Modules of fractions Let S A be a multiplicative set, and A M an A-module. In what follows, we will denote by s, t, u elements from S and by m, n elements from M. Similar to the concept of

### 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

### STABLE MODULE THEORY WITH KERNELS

Math. J. Okayama Univ. 43(21), 31 41 STABLE MODULE THEORY WITH KERNELS Kiriko KATO 1. Introduction Auslander and Bridger introduced the notion of projective stabilization mod R of a category of finite

### Injective 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

### COHEN-MACAULAY RINGS SELECTED EXERCISES. 1. Problem 1.1.9

COHEN-MACAULAY RINGS SELECTED EXERCISES KELLER VANDEBOGERT 1. Problem 1.1.9 Proceed by induction, and suppose x R is a U and N-regular element for the base case. Suppose now that xm = 0 for some m M. We

### Relative FP-gr-injective and gr-flat modules

Relative FP-gr-injective and gr-flat modules Tiwei Zhao 1, Zenghui Gao 2, Zhaoyong Huang 1, 1 Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, P.R. China 2 College of Applied

### Algebraic Geometry: Limits and Colimits

Algebraic Geometry: Limits and Coits Limits Definition.. Let I be a small category, C be any category, and F : I C be a functor. If for each object i I and morphism m ij Mor I (i, j) there is an associated

### ALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS

ALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS Your Name: Conventions: all rings and algebras are assumed to be unital. Part I. True or false? If true provide a brief explanation, if false provide a counterexample

### Direct 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

### 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.

### Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime

### 4. Symmetric and Alternating Products We want to introduce some variations on the theme of tensor products.

4. Symmetric and Alternating Products We want to introduce some variations on the theme of tensor products. The idea is that one can require a bilinear map to be either symmetric or alternating. Definition

### Math 210B. Profinite group cohomology

Math 210B. Profinite group cohomology 1. Motivation Let {Γ i } be an inverse system of finite groups with surjective transition maps, and define Γ = Γ i equipped with its inverse it topology (i.e., the

### 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

### 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

### FUNCTORS AND ADJUNCTIONS. 1. Functors

FUNCTORS AND ADJUNCTIONS Abstract. Graphs, quivers, natural transformations, adjunctions, Galois connections, Galois theory. 1.1. Graph maps. 1. Functors 1.1.1. Quivers. Quivers generalize directed graphs,

### Critical Groups of Graphs with Dihedral Symmetry

Critical Groups of Graphs with Dihedral Symmetry Will Dana, David Jekel August 13, 2017 1 Introduction We will consider the critical group of a graph Γ with an action by the dihedral group D n. After defining

### Math 210B. Artin Rees and completions

Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show

### MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA 53

MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA 53 10. Completion The real numbers are the completion of the rational numbers with respect to the usual absolute value norm. This means that any Cauchy sequence

### DEFINITIONS: OPERADS, ALGEBRAS AND MODULES. Let S be a symmetric monoidal category with product and unit object κ.

DEFINITIONS: OPERADS, ALGEBRAS AND MODULES J. P. MAY Let S be a symmetric monoidal category with product and unit object κ. Definition 1. An operad C in S consists of objects C (j), j 0, a unit map η :

### Introduction to modules

Chapter 3 Introduction to modules 3.1 Modules, submodules and homomorphisms The problem of classifying all rings is much too general to ever hope for an answer. But one of the most important tools available

### Advanced Algebra II. Mar. 2, 2007 (Fri.) 1. commutative ring theory In this chapter, rings are assume to be commutative with identity.

Advanced Algebra II Mar. 2, 2007 (Fri.) 1. commutative ring theory In this chapter, rings are assume to be commutative with identity. 1.1. basic definitions. We recall some basic definitions in the section.

### CONTINUITY. 1. Continuity 1.1. Preserving limits and colimits. Suppose that F : J C and R: C D are functors. Consider the limit diagrams.

CONTINUITY Abstract. Continuity, tensor products, complete lattices, the Tarski Fixed Point Theorem, existence of adjoints, Freyd s Adjoint Functor Theorem 1. Continuity 1.1. Preserving limits and colimits.

### COARSENINGS, INJECTIVES AND HOM FUNCTORS

COARSENINGS, INJECTIVES AND HOM FUNCTORS FRED ROHRER It is characterized when coarsening functors between categories of graded modules preserve injectivity of objects, and when they commute with graded

### 38 Irreducibility criteria in rings of polynomials

38 Irreducibility criteria in rings of polynomials 38.1 Theorem. Let p(x), q(x) R[x] be polynomials such that p(x) = a 0 + a 1 x +... + a n x n, q(x) = b 0 + b 1 x +... + b m x m and a n, b m 0. If b m

### Schemes via Noncommutative Localisation

Schemes via Noncommutative Localisation Daniel Murfet September 18, 2005 In this note we give an exposition of the well-known results of Gabriel, which show how to define affine schemes in terms of the

### THE TATE MODULE. Seminar: Elliptic curves and the Weil conjecture. Yassin Mousa. Z p

THE TATE MODULE Seminar: Elliptic curves and the Weil conjecture Yassin Mousa Abstract This paper refers to the 10th talk in the seminar Elliptic curves and the Weil conjecture supervised by Prof. Dr.

### Derivations and differentials

Derivations and differentials Johan Commelin April 24, 2012 In the following text all rings are commutative with 1, unless otherwise specified. 1 Modules of derivations Let A be a ring, α : A B an A algebra,

### MA 252 notes: Commutative algebra

MA 252 notes: Commutative algebra (Distilled from [Atiyah-MacDonald]) Dan Abramovich Brown University February 4, 2017 Abramovich MA 252 notes: Commutative algebra 1 / 13 Rings of fractions Fractions Theorem

### 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

### arxiv: v4 [math.rt] 14 Jun 2016

TWO HOMOLOGICAL PROOFS OF THE NOETHERIANITY OF FI G LIPING LI arxiv:163.4552v4 [math.rt] 14 Jun 216 Abstract. We give two homological proofs of the Noetherianity of the category F I, a fundamental result

### THE GENERALIZED HOMOLOGY OF PRODUCTS

THE GENERALIZED HOMOLOGY OF PRODUCTS MARK HOVEY Abstract. We construct a spectral sequence that computes the generalized homology E ( Q X ) of a product of spectra. The E 2 -term of this spectral sequence

### Ring Theory Problems. A σ

Ring Theory Problems 1. Given the commutative diagram α A σ B β A σ B show that α: ker σ ker σ and that β : coker σ coker σ. Here coker σ = B/σ(A). 2. Let K be a field, let V be an infinite dimensional

### Level raising. Kevin Buzzard April 26, v1 written 29/3/04; minor tinkering and clarifications written

Level raising Kevin Buzzard April 26, 2012 [History: 3/6/08] v1 written 29/3/04; minor tinkering and clarifications written 1 Introduction What s at the heart of level raising, when working with 1-dimensional

### NOTES ON LINEAR ALGEBRA OVER INTEGRAL DOMAINS. Contents. 1. Introduction 1 2. Rank and basis 1 3. The set of linear maps 4. 1.

NOTES ON LINEAR ALGEBRA OVER INTEGRAL DOMAINS Contents 1. Introduction 1 2. Rank and basis 1 3. The set of linear maps 4 1. Introduction These notes establish some basic results about linear algebra over

### LECTURE 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

### Modules over a Ringed Space

Modules over a Ringed Space Daniel Murfet October 5, 2006 In these notes we collect some useful facts about sheaves of modules on a ringed space that are either left as exercises in [Har77] or omitted

### LECTURES ON DEFORMATIONS OF GALOIS REPRESENTATIONS. Mark Kisin

LECTURES ON DEFORMATIONS OF GALOIS REPRESENTATIONS Mark Kisin Lecture 5: Flat deformations (5.1) Flat deformations: Let K/Q p be a finite extension with residue field k. Let W = W (k) and K 0 = FrW. We

### COURSE 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

### where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset

Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

### 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

### Lecture 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

### MATH 233B, FLATNESS AND SMOOTHNESS.

MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)

### Matsumura: Commutative Algebra Part 2

Matsumura: Commutative Algebra Part 2 Daniel Murfet October 5, 2006 This note closely follows Matsumura s book [Mat80] on commutative algebra. Proofs are the ones given there, sometimes with slightly more

### 12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.

12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End

### Q N id β. 2. Let I and J be ideals in a commutative ring A. Give a simple description of

Additional Problems 1. Let A be a commutative ring and let 0 M α N β P 0 be a short exact sequence of A-modules. Let Q be an A-module. i) Show that the naturally induced sequence is exact, but that 0 Hom(P,

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

### THE SMOOTH BASE CHANGE THEOREM

THE SMOOTH BASE CHANGE THEOREM AARON LANDESMAN CONTENTS 1. Introduction 2 1.1. Statement of the smooth base change theorem 2 1.2. Topological smooth base change 4 1.3. A useful case of smooth base change

### AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES

AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES MATTHEW H. BAKER AND JÁNOS A. CSIRIK This paper was written in conjunction with R. Hartshorne s Spring 1996 Algebraic Geometry course at

### Topological K-theory, Lecture 3

Topological K-theory, Lecture 3 Matan Prasma March 2, 2015 1 Applications of the classification theorem continued Let us see how the classification theorem can further be used. Example 1. The bundle γ

### INJECTIVE MODULES AND THE INJECTIVE HULL OF A MODULE, November 27, 2009

INJECTIVE ODULES AND THE INJECTIVE HULL OF A ODULE, November 27, 2009 ICHIEL KOSTERS Abstract. In the first section we will define injective modules and we will prove some theorems. In the second section,

### APPENDIX 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

### arxiv: v1 [math.rt] 12 Jul 2018

ON THE LIFTING OF THE DADE GROUP CAROLINE LASSUEUR AND JACQUES THÉVENAZ arxiv:1807.04487v1 [math.rt] 12 Jul 2018 Abstract. For the group of endo-permutation modules of a finite p-group, there is a surjective

### Math 210B: Algebra, Homework 4

Math 210B: Algebra, Homework 4 Ian Coley February 5, 2014 Problem 1. Let S be a multiplicative subset in a commutative ring R. Show that the localisation functor R-Mod S 1 R-Mod, M S 1 M, is exact. First,

### In the special case where Y = BP is the classifying space of a finite p-group, we say that f is a p-subgroup inclusion.

Definition 1 A map f : Y X between connected spaces is called a homotopy monomorphism at p if its homotopy fibre F is BZ/p-local for every choice of basepoint. In the special case where Y = BP is the classifying

### GORENSTEIN DIMENSIONS OF UNBOUNDED COMPLEXES AND CHANGE OF BASE (WITH AN APPENDIX BY DRISS BENNIS)

GORENSTEIN DIMENSIONS OF UNBOUNDED COMPLEXES AND CHANGE OF BASE (WITH AN APPENDIX BY DRISS BENNIS) LARS WINTHER CHRISTENSEN, FATIH KÖKSAL, AND LI LIANG Abstract. For a commutative ring R and a faithfully

### An extension of Dwyer s and Palmieri s proof of Ohkawa s theorem on Bousfield classes

An extension of Dwyer s and Palmieri s proof of Ohkawa s theorem on Bousfield classes Greg Stevenson Abstract We give a proof that in any compactly generated triangulated category with a biexact coproduct

### Homological Methods in Commutative Algebra

Homological Methods in Commutative Algebra Olivier Haution Ludwig-Maximilians-Universität München Sommersemester 2017 1 Contents Chapter 1. Associated primes 3 1. Support of a module 3 2. Associated primes

### Topics in Module Theory

Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and constructions concerning modules over (primarily) noncommutative rings that will be needed to study

### Homological Dimension

Homological Dimension David E V Rose April 17, 29 1 Introduction In this note, we explore the notion of homological dimension After introducing the basic concepts, our two main goals are to give a proof

### A short proof of Klyachko s theorem about rational algebraic tori

A short proof of Klyachko s theorem about rational algebraic tori Mathieu Florence Abstract In this paper, we give another proof of a theorem by Klyachko ([?]), which asserts that Zariski s conjecture