Topological modules over strictly minimal topological
... Definition 1. (R, τR ) is said to be strictly minimal if there is no separated (R, τR )-module topology on R which is strictly coarser than τR (R endowed with its canonical left R-module structure). To say that (R, τR ) is strictly minimal is equivalent to saying that, for every separated topology θ ...
... Definition 1. (R, τR ) is said to be strictly minimal if there is no separated (R, τR )-module topology on R which is strictly coarser than τR (R endowed with its canonical left R-module structure). To say that (R, τR ) is strictly minimal is equivalent to saying that, for every separated topology θ ...
1. Group actions and other topics in group theory
... Representation theory is one of the major branches of mathematics. We’ll consider representation theory of finite groups in some detail, especially over C. Example. Let F be a field, and suppose G acts on F via field automorphisms. This is precisely the situation one studies in Galois theory. Exampl ...
... Representation theory is one of the major branches of mathematics. We’ll consider representation theory of finite groups in some detail, especially over C. Example. Let F be a field, and suppose G acts on F via field automorphisms. This is precisely the situation one studies in Galois theory. Exampl ...
Help File
... 6. Exercise. We can regard π1 (X, x0 ) as the set of basepoint-preserving homotopy classes of maps (S1 , s0 ) → (X, x0 ). Let [S1 , X] be the set of homotopy classes of maps S1 → X, with no conditions on basepoints. Thus there is a natural map Φ : π1 (X, x0 ) → [S1 , X] obtained by ignoring basepoin ...
... 6. Exercise. We can regard π1 (X, x0 ) as the set of basepoint-preserving homotopy classes of maps (S1 , s0 ) → (X, x0 ). Let [S1 , X] be the set of homotopy classes of maps S1 → X, with no conditions on basepoints. Thus there is a natural map Φ : π1 (X, x0 ) → [S1 , X] obtained by ignoring basepoin ...
Tannaka Duality for Geometric Stacks
... It is natural to ask when φ is an equivalence. In the case where X and S are projective schemes, a satisfactory answer was obtained long ago. In this case, both algebraic and analytic maps may be classified by their graphs, which are closed in the product X × S. One may then deduce that any analytic ...
... It is natural to ask when φ is an equivalence. In the case where X and S are projective schemes, a satisfactory answer was obtained long ago. In this case, both algebraic and analytic maps may be classified by their graphs, which are closed in the product X × S. One may then deduce that any analytic ...
Solution 1 - D-MATH
... Solution: We first prove the lifting property as in the Hint. Let C 0 be a maximal subgroup of C such that a lift exists. Assume that C 0 6= C so that there is an x ∈ C \ C 0 . By divisibility of I also no integer multiple of x can lie in C 0 . But then we can extend the map to the group generated b ...
... Solution: We first prove the lifting property as in the Hint. Let C 0 be a maximal subgroup of C such that a lift exists. Assume that C 0 6= C so that there is an x ∈ C \ C 0 . By divisibility of I also no integer multiple of x can lie in C 0 . But then we can extend the map to the group generated b ...
Splittings of Bicommutative Hopf algebras - Mathematics
... (up to an isomorphism) whereas the Frobenius is an isomorphism. We now give some examples of short exact sequences of Hopf algebras which split both as algebras and as coalgebras but do not split as Hopf algebras. There are three reasons for doing this: (1) to show that we are actually proving somet ...
... (up to an isomorphism) whereas the Frobenius is an isomorphism. We now give some examples of short exact sequences of Hopf algebras which split both as algebras and as coalgebras but do not split as Hopf algebras. There are three reasons for doing this: (1) to show that we are actually proving somet ...
functors of artin ringso
... In many cases of interest, F is not pro-representable, but at least one may find an R and a morphism Hom(7?, ■)->■F of functors such that Hom(.R, A) -> F(A) is surjective for all A in C. If R is chosen suitably "minimal" then R is called a "hull" of F; R is then unique up to noncanonical isomorphism ...
... In many cases of interest, F is not pro-representable, but at least one may find an R and a morphism Hom(7?, ■)->■F of functors such that Hom(.R, A) -> F(A) is surjective for all A in C. If R is chosen suitably "minimal" then R is called a "hull" of F; R is then unique up to noncanonical isomorphism ...
Central Extensions in Physics
... occurs for the Galilei group: it is not a symmetry group of the (non relativistic) Schrödinger equation, but its central extension, the Bargmann group, is. Another area of physics where one encounters central extensions is the quantum theory of conserved currents of a Lagrangian. These currents spa ...
... occurs for the Galilei group: it is not a symmetry group of the (non relativistic) Schrödinger equation, but its central extension, the Bargmann group, is. Another area of physics where one encounters central extensions is the quantum theory of conserved currents of a Lagrangian. These currents spa ...
K-theory of Waldhausen categories
... Fix an exact category C, embedded in an abelian category A. The category Chb (C) of bounded chain complexes with objects in C is a Waldhausen category, where the cofibrations are the degreewise monomorphisms and weak equivalences are quasi-isomorphisms in A. Proposition 3.1 If C is closed under kern ...
... Fix an exact category C, embedded in an abelian category A. The category Chb (C) of bounded chain complexes with objects in C is a Waldhausen category, where the cofibrations are the degreewise monomorphisms and weak equivalences are quasi-isomorphisms in A. Proposition 3.1 If C is closed under kern ...
1. Introduction 1 2. Simplicial and Singular Intersection Homology 2
... Homology with Local Coefficients Some Useful Properties of Intersection Homology Sheaf-Theoretic Intersection Homology ...
... Homology with Local Coefficients Some Useful Properties of Intersection Homology Sheaf-Theoretic Intersection Homology ...
Manifolds and Varieties via Sheaves
... 1. An n-dimensional topological manifold is defined as above but with (Rn , C ∞ ) replaced by (Rn , ContRn ,R ). 2. An n-dimensional complex manifold can be defined by replacing (R n , C ∞ ) by (Cn , O). One dimensional complex manifolds are usually called Riemann surfaces. Definition 1.2.6. A C ∞ m ...
... 1. An n-dimensional topological manifold is defined as above but with (Rn , C ∞ ) replaced by (Rn , ContRn ,R ). 2. An n-dimensional complex manifold can be defined by replacing (R n , C ∞ ) by (Cn , O). One dimensional complex manifolds are usually called Riemann surfaces. Definition 1.2.6. A C ∞ m ...
Finite flat group schemes course
... The example to bear in mind if you’re worrying about set-theoretic issues is the category of sets: there is no “set of all sets” because its subset, the set of all sets that don’t contain themselves as elements, gives an easy contradiction. However there is, as far as I am concerned, a category of a ...
... The example to bear in mind if you’re worrying about set-theoretic issues is the category of sets: there is no “set of all sets” because its subset, the set of all sets that don’t contain themselves as elements, gives an easy contradiction. However there is, as far as I am concerned, a category of a ...
Lie algebra cohomology and Macdonald`s conjectures
... is called trivial.) The only element of V that is fixed by all ρ(X) is 0, for ρ(0) = 0. Yet there is a notion of g-invariant vectors. Namely, an element v of V is g-invariant if ∀X ∈ g : ρ(X)v = 0. Later on it will become clear why this is a reasonable definition. These invariants also form a g-subm ...
... is called trivial.) The only element of V that is fixed by all ρ(X) is 0, for ρ(0) = 0. Yet there is a notion of g-invariant vectors. Namely, an element v of V is g-invariant if ∀X ∈ g : ρ(X)v = 0. Later on it will become clear why this is a reasonable definition. These invariants also form a g-subm ...
Notes - Mathematics and Statistics
... law, (x, y ) 7→ xy , with a two-sided inverse. Define a category C , with a single object ∗ and with Mor(∗, ∗) = S, where composition is given by multiplication. (So morphisms need not be functions.) (11) The category of linear representations of a finite group G over a field k, Repk (G). Let G be a ...
... law, (x, y ) 7→ xy , with a two-sided inverse. Define a category C , with a single object ∗ and with Mor(∗, ∗) = S, where composition is given by multiplication. (So morphisms need not be functions.) (11) The category of linear representations of a finite group G over a field k, Repk (G). Let G be a ...
Homological algebra
... Theorem 3.7. The category of R-modules has sufficiently many injectives. I.e., every R-module embeds in an injective R-module. As in the case of projective modules this theorem will tell us that every R-module M has an injective co-resolution which is an exact sequence: 0 → M → Q0 → Q1 → Q2 → · · · ...
... Theorem 3.7. The category of R-modules has sufficiently many injectives. I.e., every R-module embeds in an injective R-module. As in the case of projective modules this theorem will tell us that every R-module M has an injective co-resolution which is an exact sequence: 0 → M → Q0 → Q1 → Q2 → · · · ...
Derived splinters in positive characteristic
... Derived splinters in positive characteristic Returning to affine D-splinters, we note that in positive characteristic p, by Theorem 1.4 this class of singularities is closely related to other classes of singularities, the so-called F-singularities, defined using the Frobenius action. For example, l ...
... Derived splinters in positive characteristic Returning to affine D-splinters, we note that in positive characteristic p, by Theorem 1.4 this class of singularities is closely related to other classes of singularities, the so-called F-singularities, defined using the Frobenius action. For example, l ...
MATH 436 Notes: Finitely generated Abelian groups.
... A dual property holds for direct sums as long as we restrict ourselves to Abelian groups. For the rest of this section, we will be dealing with Abelian groups so it is useful to review some conventions: If A is an Abelian group then it is common to denote the group operation ⋆ by +. It is then also ...
... A dual property holds for direct sums as long as we restrict ourselves to Abelian groups. For the rest of this section, we will be dealing with Abelian groups so it is useful to review some conventions: If A is an Abelian group then it is common to denote the group operation ⋆ by +. It is then also ...
EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY 1
... text below, categories will be denoted by capital script letters such as A , B, C (possibly with subscripts); objects of categories will be denoted by capital Roman letters such as X, Y, Z; morphisms in categories will be denoted by lowercase Roman letters such as f, g, h; functors between categorie ...
... text below, categories will be denoted by capital script letters such as A , B, C (possibly with subscripts); objects of categories will be denoted by capital Roman letters such as X, Y, Z; morphisms in categories will be denoted by lowercase Roman letters such as f, g, h; functors between categorie ...
The Hurewicz Theorem
... (f + g)(s1 , s2 , ..., sn ) = g(2s1 − 1, s2 , ..., sn ) s1 ∈ [ 12 , 1] If we have a homotopy ft between f and some other map f 0 , then we can define a homotopy (f + g)t from f + g to f 0 + g by replacing (f + g) by (f + g)t on the left hand side of the above definition and f by ft on the right hand ...
... (f + g)(s1 , s2 , ..., sn ) = g(2s1 − 1, s2 , ..., sn ) s1 ∈ [ 12 , 1] If we have a homotopy ft between f and some other map f 0 , then we can define a homotopy (f + g)t from f + g to f 0 + g by replacing (f + g) by (f + g)t on the left hand side of the above definition and f by ft on the right hand ...