Nilpotent groups
... D8 C D16 . The conjugation action gives a homomorphism φ : D16 → Aut(D8 ) The kernel of φ is the centralizer of D8 in D16 . This is {1, a4 } since these are the only elements of D16 which commute with both b and a2 . Thus D16 Aut(D8 ) ∼ ...
... D8 C D16 . The conjugation action gives a homomorphism φ : D16 → Aut(D8 ) The kernel of φ is the centralizer of D8 in D16 . This is {1, a4 } since these are the only elements of D16 which commute with both b and a2 . Thus D16 Aut(D8 ) ∼ ...
3. The players: rings, fields, etc.
... There are some things about the behavior of rings which we might accidentally take for granted. Let R be a ring. • Uniqueness of 0 additive identity: From the analogous discussion at the beginning of group theory, we know that there is exactly one element z = 0 with the property that r + z = r for a ...
... There are some things about the behavior of rings which we might accidentally take for granted. Let R be a ring. • Uniqueness of 0 additive identity: From the analogous discussion at the beginning of group theory, we know that there is exactly one element z = 0 with the property that r + z = r for a ...
Math 611 Homework #4 November 24, 2010
... Based on bacis algebra, it is clear that hR, +, −, 0i forms an abelian group. The addition operation,+, is commutative, associative, and there is an inverse for any element in hR, +, −, 0i. Also, hR, ·, 1i forms a monoid. The multiplication operation, ×, is associatvie and it distributes over addit ...
... Based on bacis algebra, it is clear that hR, +, −, 0i forms an abelian group. The addition operation,+, is commutative, associative, and there is an inverse for any element in hR, +, −, 0i. Also, hR, ·, 1i forms a monoid. The multiplication operation, ×, is associatvie and it distributes over addit ...
Garrett 11-04-2011 1 Recap: A better version of localization...
... Primes lying over/under [recap/cont’d] O integral over o and prime ideal p of o, there is at least one prime ideal P of O such that P ∩ o = p. P is maximal if and only if p is maximal. p · O 6= O. [Here use Nakayama, localization.] Now o is a domain, integrally closed in its field of fractions k. Fo ...
... Primes lying over/under [recap/cont’d] O integral over o and prime ideal p of o, there is at least one prime ideal P of O such that P ∩ o = p. P is maximal if and only if p is maximal. p · O 6= O. [Here use Nakayama, localization.] Now o is a domain, integrally closed in its field of fractions k. Fo ...
INTRODUCTION TO COMMUTATIVE ALGEBRA MAT6608
... Artin, Krull, van der Waerden and others. It brought about a revolution in the whole of mathematics, not just algebra. The preliminary work by Dedekind, Kronecker, Kummer, Weber, Weierstrass, Weber and others in the nineteenth century were great motivations. And of course the great mathematicians be ...
... Artin, Krull, van der Waerden and others. It brought about a revolution in the whole of mathematics, not just algebra. The preliminary work by Dedekind, Kronecker, Kummer, Weber, Weierstrass, Weber and others in the nineteenth century were great motivations. And of course the great mathematicians be ...
Chapter 3 Cartesian Tensors
... Example: suppose that two symmetric second rank tensors Rij and Sij are linearly related. Then there must be a relationship between them of the form Rij = cijkl Skl . It is clear that cijkl must be symmetric in i, j (for otherwise, Rij would not be). It is not necessarily the case that it must also ...
... Example: suppose that two symmetric second rank tensors Rij and Sij are linearly related. Then there must be a relationship between them of the form Rij = cijkl Skl . It is clear that cijkl must be symmetric in i, j (for otherwise, Rij would not be). It is not necessarily the case that it must also ...
Hochschild cohomology
... To make life a little bit easier I will give a short introduction to simplicial and cosimplicial objects and its associated (co-)chain complex. In fact, Hochschild cohomology can be described by the cohomology of a cochain complex associated to a cosimplicial bimodule, that is a cosimplicial object ...
... To make life a little bit easier I will give a short introduction to simplicial and cosimplicial objects and its associated (co-)chain complex. In fact, Hochschild cohomology can be described by the cohomology of a cochain complex associated to a cosimplicial bimodule, that is a cosimplicial object ...
An Injectivity Theorem for Casson
... would like to thank for its hospitality. 2. Preliminaries In this section we collect and prove various preliminary lemmas which we will need in the proof of our main theorem. 2.1. Strebel’s class D(R). Definition 2.1 (Strebel [Str74]). Suppose R is a commutative ring with unity. We say that a group ...
... would like to thank for its hospitality. 2. Preliminaries In this section we collect and prove various preliminary lemmas which we will need in the proof of our main theorem. 2.1. Strebel’s class D(R). Definition 2.1 (Strebel [Str74]). Suppose R is a commutative ring with unity. We say that a group ...
Chapter 2 - Cartesian Vectors and Tensors: Their Algebra Definition
... The notation vec Ω is sometimes used for ω. In summary, an antisymmetric tensor is completely characterized by the vector, vec Ω. Canonical form of a symmetric tensor We showed earlier that any second order tensor can be represented as a sum of a symmetric part and an antisymmetric part. The symmetr ...
... The notation vec Ω is sometimes used for ω. In summary, an antisymmetric tensor is completely characterized by the vector, vec Ω. Canonical form of a symmetric tensor We showed earlier that any second order tensor can be represented as a sum of a symmetric part and an antisymmetric part. The symmetr ...
Contents - Harvard Math Department
... M = R. Then S −1 R is an R-module, and it is in fact a commutative ring in its own right. The ring structure is quite tautological: (x/s)(y/s0 ) = (xy/ss0 ). There is a map R → S −1 R sending x → x/1, which is a ring-homomorphism. Definition 1.4 For S ⊂ R a multiplicative set, the localization S −1 ...
... M = R. Then S −1 R is an R-module, and it is in fact a commutative ring in its own right. The ring structure is quite tautological: (x/s)(y/s0 ) = (xy/ss0 ). There is a map R → S −1 R sending x → x/1, which is a ring-homomorphism. Definition 1.4 For S ⊂ R a multiplicative set, the localization S −1 ...
An Introduction to Nonlinear Solid Mechanics Marino Arroyo & Anna Pandolfi
... A is said to be diagonalizable if it has n linearly independent eigenvectors. In such a case it is possible to define a right and a left eigenbasis (dual basis) with normalized eigenvectors, so that: ...
... A is said to be diagonalizable if it has n linearly independent eigenvectors. In such a case it is possible to define a right and a left eigenbasis (dual basis) with normalized eigenvectors, so that: ...
The Etingof-Kazhdan construction of Lie bialgebra deformations.
... Manin triple with Casimir element Ω, and M the Drinfeld category associated to g. Consider the functor F :M→A F (V ) = Hom(U g, V ) which is naturally isomorphic to the forgetful functor. We wish to equip F with a tensor structure: a functorial isomorphism JV W : F (V ) ⊗ F (W ) → F (V ⊗ W ) such th ...
... Manin triple with Casimir element Ω, and M the Drinfeld category associated to g. Consider the functor F :M→A F (V ) = Hom(U g, V ) which is naturally isomorphic to the forgetful functor. We wish to equip F with a tensor structure: a functorial isomorphism JV W : F (V ) ⊗ F (W ) → F (V ⊗ W ) such th ...
Grothendieck Rings for Categories of Torsion Free Modules
... extension of a known ring which is determined only by the cardinality of the set of prime ideals in W . 1.3. Locally Free Complete Decomposable Modules. Let LF be the category of locally free modules and let LFCD be the category of locally free completely decomposable modules. (Note: A rank-one modu ...
... extension of a known ring which is determined only by the cardinality of the set of prime ideals in W . 1.3. Locally Free Complete Decomposable Modules. Let LF be the category of locally free modules and let LFCD be the category of locally free completely decomposable modules. (Note: A rank-one modu ...
Solutions.
... A division ring satisfies all requirements of a field except that multiplication is not commutative. Claim 1 : When V is a cylic left R- module, then HomR (V, V ), is a division ring. ( Aside: The proof does not use the commutativity of R, so we work with a general ring R in what follows. However, t ...
... A division ring satisfies all requirements of a field except that multiplication is not commutative. Claim 1 : When V is a cylic left R- module, then HomR (V, V ), is a division ring. ( Aside: The proof does not use the commutativity of R, so we work with a general ring R in what follows. However, t ...
Some definitions that may be useful
... • A 1-morphism (A, F ) → (B, G) is a K-enriched functor E : A → B and a natural isomorphism α:F ∼ = G ◦ E, i.e. it is a triangle. • A 2-morphism is a cone on a bigon. Now again I’ll work over K = K-mod for K a ring. Pick algebras A, B and A = A-mod and B = B-mod, and take the forgetful maps as the f ...
... • A 1-morphism (A, F ) → (B, G) is a K-enriched functor E : A → B and a natural isomorphism α:F ∼ = G ◦ E, i.e. it is a triangle. • A 2-morphism is a cone on a bigon. Now again I’ll work over K = K-mod for K a ring. Pick algebras A, B and A = A-mod and B = B-mod, and take the forgetful maps as the f ...
Solution
... 2. An R-module M is called simple if it has no submodules except 0 and M. Describe, with complete proof, all simple modules for R = Z/6Z. Solution. The image of a module homomorphism is a submodule of the codomain so a homomorphism into a simple module is either zero or surjective. In particular if ...
... 2. An R-module M is called simple if it has no submodules except 0 and M. Describe, with complete proof, all simple modules for R = Z/6Z. Solution. The image of a module homomorphism is a submodule of the codomain so a homomorphism into a simple module is either zero or surjective. In particular if ...
Introduction to derived algebraic geometry
... k is just the category of cell cdgas. Let A be one. Then MB (A) should be the space of (B ⊗k A)-dg modules M such that M is projective of finite type over A, i.e. M is a direct summand of some Ap in D(A). We make MB (A) into a functor by, for a map A → A0 of cell cdgas, defining MB (A) → MB (A0 ) by ...
... k is just the category of cell cdgas. Let A be one. Then MB (A) should be the space of (B ⊗k A)-dg modules M such that M is projective of finite type over A, i.e. M is a direct summand of some Ap in D(A). We make MB (A) into a functor by, for a map A → A0 of cell cdgas, defining MB (A) → MB (A0 ) by ...
arXiv:math.OA/9901094 v1 22 Jan 1999
... When Γ = Γ(X, σ), we have Γ0 = X (under the identification (x, 0, x) 7→ x) and we shall show that there is an isomorphism ZΓn (A) ' H n (X, A) for any Γ-sheaf A. Note that H 0 (X, A) = S(A), the group of continuous sections of A, and that H n (X, ·) is the nth right derived functor of S (when it is ...
... When Γ = Γ(X, σ), we have Γ0 = X (under the identification (x, 0, x) 7→ x) and we shall show that there is an isomorphism ZΓn (A) ' H n (X, A) for any Γ-sheaf A. Note that H 0 (X, A) = S(A), the group of continuous sections of A, and that H n (X, ·) is the nth right derived functor of S (when it is ...
SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR
... idF 0 : F 0 → F 0 would be such extensions. By the uniqueness up to chain homotopy for such chain maps, again by the previous lemma, it follows that h and h0 are inverse chain homotopy equivalences as intended. ...
... idF 0 : F 0 → F 0 would be such extensions. By the uniqueness up to chain homotopy for such chain maps, again by the previous lemma, it follows that h and h0 are inverse chain homotopy equivalences as intended. ...
Theory of Modules UW-Madison Modules Basic Definitions We now
... We now move on to the study of modules. Modules are generalizations of the notion of a vector space over a field. Instead, modules are defined over an arbitrary ring. Through this notes R will denote a ring with unity. Definition. A right R-module M is an abelian group (written additively), together ...
... We now move on to the study of modules. Modules are generalizations of the notion of a vector space over a field. Instead, modules are defined over an arbitrary ring. Through this notes R will denote a ring with unity. Definition. A right R-module M is an abelian group (written additively), together ...
Modules - University of Oregon
... in the commutative case the standard way to view any left R-module as a right R-module (or vice versa): if M is a left R-module, define a right action of R on M by mr := rm. In view of this, when working with commutative rings, I allow myself to be especially careless and usually just talk about R-m ...
... in the commutative case the standard way to view any left R-module as a right R-module (or vice versa): if M is a left R-module, define a right action of R on M by mr := rm. In view of this, when working with commutative rings, I allow myself to be especially careless and usually just talk about R-m ...
18.1 Multiplying Polynomial Expressions by Monomials
... Remember that the Distributive Property states that multiplying a term by a sum is the same thing as multiplying the term by each part of the sum then adding the results. ...
... Remember that the Distributive Property states that multiplying a term by a sum is the same thing as multiplying the term by each part of the sum then adding the results. ...
LECTURE NOTES 1. Basic definitions Let K be a field. Definition 1.1
... of the module defined by X together with those of the module defined by Z 0 . Since X is similar to X 0 the former pair are isomorphic k[G]modules. To see that the latter pair have the same composition factors one may use an induction hypothesis on the dimension of M over K (the theorem is clearly t ...
... of the module defined by X together with those of the module defined by Z 0 . Since X is similar to X 0 the former pair are isomorphic k[G]modules. To see that the latter pair have the same composition factors one may use an induction hypothesis on the dimension of M over K (the theorem is clearly t ...
Lecture 1 Linear Superalgebra
... (This of course implies that θi2 = 0 for all i.) In other words we can view A as the ordinary tensor product k[t1 . . . tp ] ⊗ ∧(θ1 . . . θq ), where ∧(θ1 . . . θq ) is the exterior algebra generated by θ1 . . . θq . We claim that A is a supercommutative algebra. In fact, ...
... (This of course implies that θi2 = 0 for all i.) In other words we can view A as the ordinary tensor product k[t1 . . . tp ] ⊗ ∧(θ1 . . . θq ), where ∧(θ1 . . . θq ) is the exterior algebra generated by θ1 . . . θq . We claim that A is a supercommutative algebra. In fact, ...
Sets with a Category Action Peter Webb 1. C-sets
... The Burnside ring functor BR (C) := R ⊗Z B(C) is in fact an example of a biset functor defined on categories. Let 1 denote the category with one object and one morphism – in other words, the identity group. We see that if C is any category, C-sets may be identified as the same thing as (C, 1)-bisets ...
... The Burnside ring functor BR (C) := R ⊗Z B(C) is in fact an example of a biset functor defined on categories. Let 1 denote the category with one object and one morphism – in other words, the identity group. We see that if C is any category, C-sets may be identified as the same thing as (C, 1)-bisets ...
Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps (module homomorphisms). The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology and algebraic geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.