HOW TO PROVE THAT A NON-REPRESENTABLE FUNCTOR IS
... which implies that g = g 0 by the universal property of idA ∈ hom(A, A). X We have shown that u = a2 has the universal property that for every square b2 in any ring B, there is a unique homomorphism A → B sending u to b2 . Now we have transferred the problem to ring theory, where we will finish it o ...
... which implies that g = g 0 by the universal property of idA ∈ hom(A, A). X We have shown that u = a2 has the universal property that for every square b2 in any ring B, there is a unique homomorphism A → B sending u to b2 . Now we have transferred the problem to ring theory, where we will finish it o ...
An introduction to stable homotopy theory “Abelian groups up to
... Thm: (Schwede-S.) If C is a Sp-model category with a (cofibrant and fibrant) small generator G then C is Quillen equivalent to (right) module spectra over ...
... Thm: (Schwede-S.) If C is a Sp-model category with a (cofibrant and fibrant) small generator G then C is Quillen equivalent to (right) module spectra over ...
estimating the states of the kauffman bracket skein module
... L1 L2 = L1 ∪L2 , for which ∅ is a unit. Since W (M ) is an ideal the multiplication descends to V (M ), which, as an algebra, is finitely generated [4, Theorem 1]. Our main result is that V (M ) maps onto the ring of SL2 (C) characters. In Section 2 we review the necessary character theory and prove ...
... L1 L2 = L1 ∪L2 , for which ∅ is a unit. Since W (M ) is an ideal the multiplication descends to V (M ), which, as an algebra, is finitely generated [4, Theorem 1]. Our main result is that V (M ) maps onto the ring of SL2 (C) characters. In Section 2 we review the necessary character theory and prove ...
Title BP operations and homological properties of
... Using (2.1) and the exactness of direct limit we obtain (2.2) every associative BPJ^BPJ-comodule is a direct limit of finitely presented associative comodules. Let G be a right BPJ*-module. We define the ίB3?£-weak dimension of G, denoted by w d i m ^ ^ G , to be less than n if Torf PJ*(G, M)=0 for ...
... Using (2.1) and the exactness of direct limit we obtain (2.2) every associative BPJ^BPJ-comodule is a direct limit of finitely presented associative comodules. Let G be a right BPJ*-module. We define the ίB3?£-weak dimension of G, denoted by w d i m ^ ^ G , to be less than n if Torf PJ*(G, M)=0 for ...
Week 3 - people.bath.ac.uk
... This gives H a = H. It now only remains to show that (b)⇔(c). But this is easy a−1 Ha = H ⇔ a · a−1 Ha = aH ⇔ Ha = aH. This finishes the proof. 2 Definition. Let G be a group with a subgroup H. The number of left cosets of H in G is called the index of H in G and is denoted [G : H]. Remark. Suppose ...
... This gives H a = H. It now only remains to show that (b)⇔(c). But this is easy a−1 Ha = H ⇔ a · a−1 Ha = aH ⇔ Ha = aH. This finishes the proof. 2 Definition. Let G be a group with a subgroup H. The number of left cosets of H in G is called the index of H in G and is denoted [G : H]. Remark. Suppose ...
Assignment 3 - UBC Physics
... objet to the i + 1’th object). Also, recall the cycle decomposition of S4 . A cycle of even length corresponds to an odd permutation and a cycle of odd length corresponds to an even permutation. For the cycle decomposition of a given permutation, the degree of the permutation is even if there is an ...
... objet to the i + 1’th object). Also, recall the cycle decomposition of S4 . A cycle of even length corresponds to an odd permutation and a cycle of odd length corresponds to an even permutation. For the cycle decomposition of a given permutation, the degree of the permutation is even if there is an ...
3. Localization.
... of A-modules there is a unique S −1 A-module homomorphism v : S −1 M → N[iSM ] such that u = viSM . The canonical map iSA : A → S −1 A has the universal property: For every homomorphism of rings ϕ : A → B where ϕ(s) is invertible in B for every element s ∈ S, there is a unique ring homomorphism χ : ...
... of A-modules there is a unique S −1 A-module homomorphism v : S −1 M → N[iSM ] such that u = viSM . The canonical map iSA : A → S −1 A has the universal property: For every homomorphism of rings ϕ : A → B where ϕ(s) is invertible in B for every element s ∈ S, there is a unique ring homomorphism χ : ...
Selected Exercises 1. Let M and N be R
... are exact. What more can be said if A is projective or B is injective ? 4. An R-module F is said to be free on the set X if there exists a set map i : X → F with the following (universal) property : Given an R-module M and a set map j : X → M , there exists a unique R-module homomorphism f : F → M s ...
... are exact. What more can be said if A is projective or B is injective ? 4. An R-module F is said to be free on the set X if there exists a set map i : X → F with the following (universal) property : Given an R-module M and a set map j : X → M , there exists a unique R-module homomorphism f : F → M s ...
∗-AUTONOMOUS CATEGORIES: ONCE MORE
... in each case, there is a ∗-autonomous category of uniform space models of the theory. In most cases, this is equivalent to the topological space models. The main tool used here is the so-called Chu construction as described in an appendix to the 1979 monograph, [Chu, 1979]. He described in detail a ...
... in each case, there is a ∗-autonomous category of uniform space models of the theory. In most cases, this is equivalent to the topological space models. The main tool used here is the so-called Chu construction as described in an appendix to the 1979 monograph, [Chu, 1979]. He described in detail a ...
HYPERELLIPTIC JACOBIANS AND SIMPLE GROUPS U3 1
... Then either End(J(Cf )) = Z or char(K) > 0 and J(Cf ) is a supersingular abelian variety. Remark 2.2. It would be interesting to find explicit examples of irreducible polynomials f (x) of degree 23m + 1 with Galois group U3 (2m ). It follows from results of Belyi [1] that such a polynomial always ex ...
... Then either End(J(Cf )) = Z or char(K) > 0 and J(Cf ) is a supersingular abelian variety. Remark 2.2. It would be interesting to find explicit examples of irreducible polynomials f (x) of degree 23m + 1 with Galois group U3 (2m ). It follows from results of Belyi [1] that such a polynomial always ex ...
F2 - Sum of Cubes
... Notice the use of the under brace to denote the groups, not parentheses. Avoid using parentheses to show the groups, for in some cases this changes the expression. Is the following expression true or false? Justify your answer using the properties of operations. (x + y)(a + b) = (a + b)(x + y) The a ...
... Notice the use of the under brace to denote the groups, not parentheses. Avoid using parentheses to show the groups, for in some cases this changes the expression. Is the following expression true or false? Justify your answer using the properties of operations. (x + y)(a + b) = (a + b)(x + y) The a ...
Category Theory Example Sheet 1
... These questions are of varying difficulty and length. Comments, corrections and clarifications can be emailed to jg352. You can find this sheet on www.dpmms.cam.ac.uk/~jg352/teaching.html. 1. (a) Show that identities in a category are unique. (b) Show that a morphism with both a right inverse and a ...
... These questions are of varying difficulty and length. Comments, corrections and clarifications can be emailed to jg352. You can find this sheet on www.dpmms.cam.ac.uk/~jg352/teaching.html. 1. (a) Show that identities in a category are unique. (b) Show that a morphism with both a right inverse and a ...
Workshop on group schemes and p-divisible groups: Homework 1. 1
... (iii) Write the ring map corresponding to the Z-group map det : GLn → Gm , and use the irreducibility of det(tij ) over any field (proof?) to deduce that the only group scheme maps from GLn to Gm over a field are detr for r ∈ Z. (iv) What is the scheme-theoretic intersection of SLn and the diagonall ...
... (iii) Write the ring map corresponding to the Z-group map det : GLn → Gm , and use the irreducibility of det(tij ) over any field (proof?) to deduce that the only group scheme maps from GLn to Gm over a field are detr for r ∈ Z. (iv) What is the scheme-theoretic intersection of SLn and the diagonall ...
Solutions - NIU Math Department
... Now consider the inclusion mapping ι : Z → Q. Suppose that θ, ψ : Q → T are ring homomorphisms with θι = ψι. For any nonzero element q = ab−1 , with a, b ∈ Z, we have θ(q) = θ(ab−1 ) = θ(a)θ(b−1 ) = θ(a)(θ(b))−1 = θι(a)(θι(b))−1 = ψι(a)(ψι(b))−1 = ψ(a)(ψ(b))−1 = ψ(a)ψ(b−1 ) = ψ(ab−1 ) = ψ(q), and th ...
... Now consider the inclusion mapping ι : Z → Q. Suppose that θ, ψ : Q → T are ring homomorphisms with θι = ψι. For any nonzero element q = ab−1 , with a, b ∈ Z, we have θ(q) = θ(ab−1 ) = θ(a)θ(b−1 ) = θ(a)(θ(b))−1 = θι(a)(θι(b))−1 = ψι(a)(ψι(b))−1 = ψ(a)(ψ(b))−1 = ψ(a)ψ(b−1 ) = ψ(ab−1 ) = ψ(q), and th ...
NOTES ON DUAL SPACES In these notes we introduce the notion
... allow us to phrase many important concepts in linear algebra without the need to introduce additional structure. In particular, we will see that we can formulate many notions involving inner products in a way that does not require the use of an inner product. Let V be a vector space over a field F. ...
... allow us to phrase many important concepts in linear algebra without the need to introduce additional structure. In particular, we will see that we can formulate many notions involving inner products in a way that does not require the use of an inner product. Let V be a vector space over a field F. ...
MODULES 1. Modules Let A be a ring. A left module M over A
... Let A be a ring, M a left A-module, and N a submodule. The factor group M/N (as additive abelian group) may be made into an A-module by defining a(x + N ) = ax + N for a coset x + N ∈ M/N . The canonical epimorphism is then a module homomorphism. Let f : M → M 0 be a homomorphism of left A-modules. ...
... Let A be a ring, M a left A-module, and N a submodule. The factor group M/N (as additive abelian group) may be made into an A-module by defining a(x + N ) = ax + N for a coset x + N ∈ M/N . The canonical epimorphism is then a module homomorphism. Let f : M → M 0 be a homomorphism of left A-modules. ...
Fourier analysis on finite abelian groups
... to C× . However, an equally important use is for the trace of a group homomorphism ρ : G → GLn (k) from G to invertible n-by-n matrices with entries in a field k. In the latter sense, ...
... to C× . However, an equally important use is for the trace of a group homomorphism ρ : G → GLn (k) from G to invertible n-by-n matrices with entries in a field k. In the latter sense, ...
Honors Algebra 4, MATH 371 Winter 2010
... as the images of the generators are a generating set. Since R is a PID, it is noetherian and any submodule of a finitely generated R-module is again finitely generated (proof?). That any quotient or submodule of a p-primary module is p-primary is obvious. (b) Let ann(Tp ) := {r ∈ R : rt = 0 for all ...
... as the images of the generators are a generating set. Since R is a PID, it is noetherian and any submodule of a finitely generated R-module is again finitely generated (proof?). That any quotient or submodule of a p-primary module is p-primary is obvious. (b) Let ann(Tp ) := {r ∈ R : rt = 0 for all ...
Subfactors and Modular Tensor Categories
... associated to dihedral groups and to certain quantum groups. Based on this Evans-Gannon argued that the Haagerup subfactor should not be viewed as “exotic” at all. However there is still no general construction for Izumi-Haagerup categories, and it is not known whether Evans-Gannon’s infinite series ...
... associated to dihedral groups and to certain quantum groups. Based on this Evans-Gannon argued that the Haagerup subfactor should not be viewed as “exotic” at all. However there is still no general construction for Izumi-Haagerup categories, and it is not known whether Evans-Gannon’s infinite series ...
Endomorphisms The endomorphism ring of the abelian group Z/nZ
... (for instance the product of Z/n and Z). Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic. Group isomorphism In abstract algebra, a group isomorphism is a function between two groups that sets up a oneto-one correspondence between the elements of the groups in a way that respe ...
... (for instance the product of Z/n and Z). Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic. Group isomorphism In abstract algebra, a group isomorphism is a function between two groups that sets up a oneto-one correspondence between the elements of the groups in a way that respe ...
Groups acting on sets
... exists a g ∈ G such that g · x = y. Then ∼G is an equivalence relation, and the equivalence class containing x is the orbit G · x. Thus, two orbits of G are either disjoint or identical. Proof. Reflexive: x = 1 · x, hence x ∼G x. Symmetric: if x ∼G y, then by definition there exists a g ∈ G such tha ...
... exists a g ∈ G such that g · x = y. Then ∼G is an equivalence relation, and the equivalence class containing x is the orbit G · x. Thus, two orbits of G are either disjoint or identical. Proof. Reflexive: x = 1 · x, hence x ∼G x. Symmetric: if x ∼G y, then by definition there exists a g ∈ G such tha ...
Part II Permutations, Cosets and Direct Product
... ϕ is injective. Then the image ϕ(G) is a subgroup of G′ and ϕ induces an isomorphism between G and ϕ(G). Proof. We do not have to check associativity. Suppose x, y ∈ ϕ(G). Then x = ϕ(a), y = ϕ(b) for some a, b ∈ G. So, xy = ϕ(a) = ϕ(ab) ∈ ϕ(G). So, ϕ(G) is closed under multiplication. Let e′ ∈ G′ de ...
... ϕ is injective. Then the image ϕ(G) is a subgroup of G′ and ϕ induces an isomorphism between G and ϕ(G). Proof. We do not have to check associativity. Suppose x, y ∈ ϕ(G). Then x = ϕ(a), y = ϕ(b) for some a, b ∈ G. So, xy = ϕ(a) = ϕ(ab) ∈ ϕ(G). So, ϕ(G) is closed under multiplication. Let e′ ∈ G′ de ...
TWISTING COMMUTATIVE ALGEBRAIC GROUPS Introduction In
... twists arise naturally as “primitive” subgroup varieties of the restriction of scalars of the commutative algebraic group. We have been using and proving special cases of these results elsewhere, and believe that it would be useful to have a complete theory and complete proofs in the literature in o ...
... twists arise naturally as “primitive” subgroup varieties of the restriction of scalars of the commutative algebraic group. We have been using and proving special cases of these results elsewhere, and believe that it would be useful to have a complete theory and complete proofs in the literature in o ...
![arXiv:math/0310263v1 [math.FA] 17 Oct 2003](http://s1.studyres.com/store/data/017043321_1-3cc6558e25d00768295e612a31801f34-300x300.png)
arXiv:math/0310263v1 [math.FA] 17 Oct 2003
... On the other hand, if we have two modules for which equation (8) holds, then the preceding argument shows that they must be isomorphic. It is then possible to find, in a small simply connected neighbourhood of some fixed point w0 , a harmonic conjugate v(w) of the harmonic function u(w) := log K̃(w, ...
... On the other hand, if we have two modules for which equation (8) holds, then the preceding argument shows that they must be isomorphic. It is then possible to find, in a small simply connected neighbourhood of some fixed point w0 , a harmonic conjugate v(w) of the harmonic function u(w) := log K̃(w, ...
Equivalence of quotient Hilbert modules
... given in ([7], pp. 375–377). One of the main results in [3] states that two modules M and M̃ in Bk () are isomorphic if and only if the associated bundles are locally equivalent. While the local equivalence of bundles is completely captured in the case of line bundles by the curvature, it is more c ...
... given in ([7], pp. 375–377). One of the main results in [3] states that two modules M and M̃ in Bk () are isomorphic if and only if the associated bundles are locally equivalent. While the local equivalence of bundles is completely captured in the case of line bundles by the curvature, it is more c ...
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.