PROJECTIVITY AND FLATNESS OVER THE
... conditions for the flatness and projectivity of R as an RG -module. In [12], the second author generalized the results from [9] in the following way: let k be a commutative ring, H a Hopf algebra over k, and Λ a left H-module algebra. Then we can consider the smash product Λ#H and the subring of inv ...
... conditions for the flatness and projectivity of R as an RG -module. In [12], the second author generalized the results from [9] in the following way: let k be a commutative ring, H a Hopf algebra over k, and Λ a left H-module algebra. Then we can consider the smash product Λ#H and the subring of inv ...
SUBGROUPS OF VECTOR SPACES In what follows, finite
... discrete subgroup of V then G is spanned (as a group) by a subset of V which is linearly independent (over R). In particular, G is a free abelian finitely generated group whose rank is less than or equal to the dimension of V . Proof. By replacing V with the vector subspace of V spanned by G, we can ...
... discrete subgroup of V then G is spanned (as a group) by a subset of V which is linearly independent (over R). In particular, G is a free abelian finitely generated group whose rank is less than or equal to the dimension of V . Proof. By replacing V with the vector subspace of V spanned by G, we can ...
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Completed representation ring spectra of nilpotent groups Algebraic & Geometric Topology [Logo here]
... Proposition 12 with R = A00 , we find that the map A → A0 is a multiplicative E -equivalence. Let R be a commutative S-algebra and E an R-algebra. In [3], the E -nilpotent completion (or derived completion) of an ordinary spectrum was defined; we will briefly recall this definition in the context of ...
... Proposition 12 with R = A00 , we find that the map A → A0 is a multiplicative E -equivalence. Let R be a commutative S-algebra and E an R-algebra. In [3], the E -nilpotent completion (or derived completion) of an ordinary spectrum was defined; we will briefly recall this definition in the context of ...
PERIODS OF GENERIC TORSORS OF GROUPS OF
... When Q is an algebraic torus, the period of a generic Q-torsor is studied in [7]. Based on the technique developed in [7], we generalize the results to the case where Q is a smooth group of multiplicative type (see Corollary 4.3). We also study in Section 5 the relationships among the periods of gen ...
... When Q is an algebraic torus, the period of a generic Q-torsor is studied in [7]. Based on the technique developed in [7], we generalize the results to the case where Q is a smooth group of multiplicative type (see Corollary 4.3). We also study in Section 5 the relationships among the periods of gen ...
MODULES: FINITELY GENERATED MODULES 1. Finitely
... 3.13. Corollary. Let A be a Noetherian ring. Then every finitely generated Amodule is finitely presented. Proof. Let M be a finitely generated A-module. Then we have an exact sequence A⊕n → M → 0 with n a positive integer. By Prop. 3.12 the kernel of this morphism is also finitely generated. This gi ...
... 3.13. Corollary. Let A be a Noetherian ring. Then every finitely generated Amodule is finitely presented. Proof. Let M be a finitely generated A-module. Then we have an exact sequence A⊕n → M → 0 with n a positive integer. By Prop. 3.12 the kernel of this morphism is also finitely generated. This gi ...
Direct-sum decompositions over local rings
... two minimal elements of Λ. In this way we can completely describe the decompositions of direct sums of copies of M . In §1 of this paper we review basic terminology concerning monoids and describe a natural isomorphism between +(M ) and a certain full submonoid Λ(M ) of Nn (where n is the number of ...
... two minimal elements of Λ. In this way we can completely describe the decompositions of direct sums of copies of M . In §1 of this paper we review basic terminology concerning monoids and describe a natural isomorphism between +(M ) and a certain full submonoid Λ(M ) of Nn (where n is the number of ...
T J N S
... a1 ∗ b ≤ a2 ∗ b. Since ∗ is commutative, then b ∗ a1 ≤ b ∗ a2 . Hence ∗ is isotone. (3)(b) Since a2 → b ≤ a2 → b, then (a2 → b) ∗ a2 ≤ b. So, (a2 → b) ∗ a1 ≤ b which implies that a2 → b ≤ a1 → b,i.e., → is antitone in the first variable. Since b → a1 ≤ b → a1 , then (b → a1 ) ∗ b ≤ a1 ≤ a2 . So, b → ...
... a1 ∗ b ≤ a2 ∗ b. Since ∗ is commutative, then b ∗ a1 ≤ b ∗ a2 . Hence ∗ is isotone. (3)(b) Since a2 → b ≤ a2 → b, then (a2 → b) ∗ a2 ≤ b. So, (a2 → b) ∗ a1 ≤ b which implies that a2 → b ≤ a1 → b,i.e., → is antitone in the first variable. Since b → a1 ≤ b → a1 , then (b → a1 ) ∗ b ≤ a1 ≤ a2 . So, b → ...
(pdf).
... However algebraic geometry is not the only field that influences the development of commutative algebra, as rings arise naturally from number theory, topology, combinatorics, and other fields. In fact, several fundamental theorems were inspired by the work of Hilbert in invariant theory. One of Hilb ...
... However algebraic geometry is not the only field that influences the development of commutative algebra, as rings arise naturally from number theory, topology, combinatorics, and other fields. In fact, several fundamental theorems were inspired by the work of Hilbert in invariant theory. One of Hilb ...
A basic note on group representations and Schur`s lemma
... groups. Unitary F[G]-modules (cf. Section 2.2.5), provide an special case which we discuss in the section. The results of this subsection are stated in terms orthogonal R[G]-modules, however, the analogous result holds for the case of unitary C[G]-modules. Lemma 3.4. Let V be a finite-dimensional or ...
... groups. Unitary F[G]-modules (cf. Section 2.2.5), provide an special case which we discuss in the section. The results of this subsection are stated in terms orthogonal R[G]-modules, however, the analogous result holds for the case of unitary C[G]-modules. Lemma 3.4. Let V be a finite-dimensional or ...
GROUP THEORY 1. Groups A set G is called a group if there is a
... central elements is called the center of G. Let H ⊆ G be a subgroup. Define the normalizer NH of H as {g ∈ G| gHg −1 ⊆ H}. This is the smallest subgroup of G, in which H is normal. If S ⊆ G is an arbitrary subset, then define the centralizer ZS of S as {g ∈ G | (∀x ∈ S) gx = xg}. This is the smalles ...
... central elements is called the center of G. Let H ⊆ G be a subgroup. Define the normalizer NH of H as {g ∈ G| gHg −1 ⊆ H}. This is the smallest subgroup of G, in which H is normal. If S ⊆ G is an arbitrary subset, then define the centralizer ZS of S as {g ∈ G | (∀x ∈ S) gx = xg}. This is the smalles ...
PURE–INJECTIVE AND FINITE LENGTH MODULES OVER
... isomorphic). For instance, the CB-rank of ZgR is equal to 1 and we are dealing with “finite type” case. Let F be a field of characteristic zero with a derivation ′ such that F ′ ⊂ F and whose field of constants k = {α ∈ F | α′ = 0} is algebraically closed. We also suppose that the category of finite len ...
... isomorphic). For instance, the CB-rank of ZgR is equal to 1 and we are dealing with “finite type” case. Let F be a field of characteristic zero with a derivation ′ such that F ′ ⊂ F and whose field of constants k = {α ∈ F | α′ = 0} is algebraically closed. We also suppose that the category of finite len ...
Lecture 7 - Penn Math
... 6 0, we have dim(V1 ) < dim(V ). We also have that W1 is a codimension 1 submodule of V1 . Since dim(V1 ) < dim(V ), an induction argument lets us assert V1 that V1 = W1 ⊕ Fz, for some z ∈ V1 , as g-modules. Note that Fz ∩ W = {0}, so V = W ⊕ Fz as vector spaces; the question is whether this is a g- ...
... 6 0, we have dim(V1 ) < dim(V ). We also have that W1 is a codimension 1 submodule of V1 . Since dim(V1 ) < dim(V ), an induction argument lets us assert V1 that V1 = W1 ⊕ Fz, for some z ∈ V1 , as g-modules. Note that Fz ∩ W = {0}, so V = W ⊕ Fz as vector spaces; the question is whether this is a g- ...
pdf
... to find more conceptual arguments for some of the main theorems of character sheaves. The goal of today’s lecture will be to give an introduction to character sheaves. Problem: Let G be a reductive group. The problem is to compute the character table of G(Fq ), e.g. GLn (Fq ), . . . , E8 (Fq ). Main ...
... to find more conceptual arguments for some of the main theorems of character sheaves. The goal of today’s lecture will be to give an introduction to character sheaves. Problem: Let G be a reductive group. The problem is to compute the character table of G(Fq ), e.g. GLn (Fq ), . . . , E8 (Fq ). Main ...
Homomorphisms
... There is an obvious sense in which these two groups are “the same”: You can get the second table from the first by replacing 0 with 1, 1 with a, and 2 with a2 . When are two groups the same? You might think of saying that two groups are the same if you can get one group’s table from the other by sub ...
... There is an obvious sense in which these two groups are “the same”: You can get the second table from the first by replacing 0 with 1, 1 with a, and 2 with a2 . When are two groups the same? You might think of saying that two groups are the same if you can get one group’s table from the other by sub ...
Algebra Notes
... different, but are actually related by the first isomorphism theorem of groups which states that for any homomorphism ϕ, G ∼ = Im(ϕ) ker(ϕ) This isomorphism is one of the most conceptually important facts in all of group theory; it should be part of your subconscious intuition about homomorphisms an ...
... different, but are actually related by the first isomorphism theorem of groups which states that for any homomorphism ϕ, G ∼ = Im(ϕ) ker(ϕ) This isomorphism is one of the most conceptually important facts in all of group theory; it should be part of your subconscious intuition about homomorphisms an ...
here - Halfaya
... characteristic of a function is the smallest n s.t. n · 1R = 0R . If there is no (finite) n s.t. this is true, we say that the ring has characteristic 0. This is actually something really cool. I mean, our ring elements might not even be numbers, right? But right away, we know what some ring element ...
... characteristic of a function is the smallest n s.t. n · 1R = 0R . If there is no (finite) n s.t. this is true, we say that the ring has characteristic 0. This is actually something really cool. I mean, our ring elements might not even be numbers, right? But right away, we know what some ring element ...
Square Deal: Lower Bounds and Improved Relaxations for Tensor
... structures simultaneously: it is low-rank along each of the K modes. In practical applications, many other such simultaneously structured objects could also be of interest. For sparse phase retrieval problems in signal processing (Oymak et al., 2012), the task can be rephrased to infer a block spars ...
... structures simultaneously: it is low-rank along each of the K modes. In practical applications, many other such simultaneously structured objects could also be of interest. For sparse phase retrieval problems in signal processing (Oymak et al., 2012), the task can be rephrased to infer a block spars ...
A counterexample to the local-global principle of linear dependence
... answer to this problem is always affirmative, but this is wrong. In this note we present a counterexample. A counterexample to the analogous statement for tori was given by Schinzel in [Sch75]. We have recently been informed that Banaszak and Krasoń found different counterexamples, which will appea ...
... answer to this problem is always affirmative, but this is wrong. In this note we present a counterexample. A counterexample to the analogous statement for tori was given by Schinzel in [Sch75]. We have recently been informed that Banaszak and Krasoń found different counterexamples, which will appea ...
*These are notes + solutions to herstein problems(second edition
... 14)Suppose a finite set G is closed under associative product and both cancellation laws hold. PT G is a group Since G is finite let G={x1,x2..xn} Look at S(x1)= {x1.x1, x1.x2, x1.x3,…..x1.xn} All these are distinct because of left cancellation law So S (x1) in some order is G Let xi be the element ...
... 14)Suppose a finite set G is closed under associative product and both cancellation laws hold. PT G is a group Since G is finite let G={x1,x2..xn} Look at S(x1)= {x1.x1, x1.x2, x1.x3,…..x1.xn} All these are distinct because of left cancellation law So S (x1) in some order is G Let xi be the element ...
Math 230 – 2003-04 – Assignment 2 Due
... show that if we take two different representatives x and y for some coset, we don’t get conflicting answers for θ. If x and y are both representatives for some coset, then x − y ∈ 60Z, i.e. x = y + 60k for some k ∈ Z. If y = 6q + r where q, r ∈ Z and 0 ≤ r < 6, we have x = y + 60k = (6q + r) + 60 = ...
... show that if we take two different representatives x and y for some coset, we don’t get conflicting answers for θ. If x and y are both representatives for some coset, then x − y ∈ 60Z, i.e. x = y + 60k for some k ∈ Z. If y = 6q + r where q, r ∈ Z and 0 ≤ r < 6, we have x = y + 60k = (6q + r) + 60 = ...
Completion of rings and modules Let (Λ, ≤) be a directed set. This is
... and for every i ≥ 0 a map Xi+1 → Xi . The other maps needed can be obtained from these by composition. In the cases of the categories mentioned above, to give an element of the inverse limit is the same a giving a sequence of elements x0 , x1 , x2 , . . . such that for all i, xi ∈ Xi , and xi+1 maps ...
... and for every i ≥ 0 a map Xi+1 → Xi . The other maps needed can be obtained from these by composition. In the cases of the categories mentioned above, to give an element of the inverse limit is the same a giving a sequence of elements x0 , x1 , x2 , . . . such that for all i, xi ∈ Xi , and xi+1 maps ...
Natural associativity and commutativity
... while associativity is an isomorphism a natural in its arguments A,B, and C. The general associative law again shows that any two iterated products F and F' of the n arguments A,, ...,A, are naturally isomorphic, under a natural isomorphism F z F' given by "iteration" of a. We then ask: what conditi ...
... while associativity is an isomorphism a natural in its arguments A,B, and C. The general associative law again shows that any two iterated products F and F' of the n arguments A,, ...,A, are naturally isomorphic, under a natural isomorphism F z F' given by "iteration" of a. We then ask: what conditi ...
Categories and functors
... This sections is a review of basic concepts in category theory. It will follow as an immediate consequence of basic definitions that all left adjoint functors are right exact and similarly for right adjoint functors. Definition 22.1. A category C is a collection Ob(C) of objects A and morphism sets ...
... This sections is a review of basic concepts in category theory. It will follow as an immediate consequence of basic definitions that all left adjoint functors are right exact and similarly for right adjoint functors. Definition 22.1. A category C is a collection Ob(C) of objects A and morphism sets ...
the predual theorem to the jacobson-bourbaki theorem
... Following Cartan-Eilenberg, we use AM to indicate that M is a left Amodule and MA indicates that M is a right .¿-module. If B is another ring, we use AMB to indicate that AM, MB and the following condition is satisfied: ...
... Following Cartan-Eilenberg, we use AM to indicate that M is a left Amodule and MA indicates that M is a right .¿-module. If B is another ring, we use AMB to indicate that AM, MB and the following condition is satisfied: ...
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.