PRECOMPACT NONCOMPACT REFLEXIVE ABELIAN GROUPS 1
... We show in Theorem 2.5 that every pseudocompact Abelian group without infinite compact subsets is reflexive. To establish the existence of infinite pseudocompact Abelian groups without infinite compact subsets, we use the notion of h-embedded subgroup of a topological group introduced in [20]. A sub ...
... We show in Theorem 2.5 that every pseudocompact Abelian group without infinite compact subsets is reflexive. To establish the existence of infinite pseudocompact Abelian groups without infinite compact subsets, we use the notion of h-embedded subgroup of a topological group introduced in [20]. A sub ...
6. Fibre Products We start with some basic properties of schemes
... Definition 6.11. A morphism f : X −→ Y is locally of finite type if there is an open affine cover Vi = Spec Bi of Y , such that f −1 (Vi ) is a union of affine sets Uij = Spec Aij , where each Aij is a finitely generated Bi -algebra. If in addition, we can take Uij to be a finite cover of f −1 (Vi ) ...
... Definition 6.11. A morphism f : X −→ Y is locally of finite type if there is an open affine cover Vi = Spec Bi of Y , such that f −1 (Vi ) is a union of affine sets Uij = Spec Aij , where each Aij is a finitely generated Bi -algebra. If in addition, we can take Uij to be a finite cover of f −1 (Vi ) ...
Notes on étale cohomology
... only artin local Spec A over s ∈ S with residue field equal to a chosen algebraic closure of k(s). Condition (4) is the one that can be checked in abstract situations with moduli problems, and deducing the structure theorem (and hence flatness) from (4) is the hardest part of the proof that these co ...
... only artin local Spec A over s ∈ S with residue field equal to a chosen algebraic closure of k(s). Condition (4) is the one that can be checked in abstract situations with moduli problems, and deducing the structure theorem (and hence flatness) from (4) is the hardest part of the proof that these co ...
Open subgroups and Pontryagin duality
... (1.1) Lemma. The polars of compact subsets of an abelian topological group G form a basis of neighbourhoods of zero in G A. The easy proof is left for the reader. (1.2) Lemma. I f U is a neighborhood of zero in an abelian topological group G, then U ~ is a compact subset of G ^. This is a standard f ...
... (1.1) Lemma. The polars of compact subsets of an abelian topological group G form a basis of neighbourhoods of zero in G A. The easy proof is left for the reader. (1.2) Lemma. I f U is a neighborhood of zero in an abelian topological group G, then U ~ is a compact subset of G ^. This is a standard f ...
Normality on Topological Groups - Matemáticas UCM
... discrete in X. In order to see that Y is discrete, it is enough to define a continuous character in G which takes the value −1 in yα and 1 in yβ for every β ∈ A \ {α}. Consider first ϕ : Z → T, the character on Z such that ϕ(n) = eπni , ∀n ∈ Z. It gives rise to characters on ZR , just composing with ...
... discrete in X. In order to see that Y is discrete, it is enough to define a continuous character in G which takes the value −1 in yα and 1 in yβ for every β ∈ A \ {α}. Consider first ϕ : Z → T, the character on Z such that ϕ(n) = eπni , ∀n ∈ Z. It gives rise to characters on ZR , just composing with ...
Abelian Sheaves
... 4.6 Invariance by homotopy . . . . . . . . . 4.7 Cohomology of some classical manifolds . Exercises . . . . . . . . . . . . . . . . . . . . . ...
... 4.6 Invariance by homotopy . . . . . . . . . 4.7 Cohomology of some classical manifolds . Exercises . . . . . . . . . . . . . . . . . . . . . ...
Sheaves on Spaces
... C is a category and that F : C → Sets is a faithful functor. Typically F is a “forgetful” functor. For an object M ∈ Ob(C) we often call F (M ) the underlying set of the object M . If M → M 0 is a morphism in C we call F (M ) → F (M 0 ) the underlying map of sets. In fact, we will often not distingu ...
... C is a category and that F : C → Sets is a faithful functor. Typically F is a “forgetful” functor. For an object M ∈ Ob(C) we often call F (M ) the underlying set of the object M . If M → M 0 is a morphism in C we call F (M ) → F (M 0 ) the underlying map of sets. In fact, we will often not distingu ...
HIGHER CATEGORIES 1. Introduction. Categories and simplicial
... Functors Cop → D are sometimes called the contravariant functors. Definition. P (C) = Fun(Cop , Set) — the category of presheaves. Here is the origin of the name. Let X be a topological space. A sheaf on X (of, say, abelian groups) F assigns to each open set U ⊂ X an abelian group F (U ), and for V ...
... Functors Cop → D are sometimes called the contravariant functors. Definition. P (C) = Fun(Cop , Set) — the category of presheaves. Here is the origin of the name. Let X be a topological space. A sheaf on X (of, say, abelian groups) F assigns to each open set U ⊂ X an abelian group F (U ), and for V ...
free topological groups with no small subgroups
... (i) (J C V and (ii) U2 , C U , tot n £ N. By Theorem 8.2 of [6], these sets ...
... (i) (J C V and (ii) U2 , C U , tot n £ N. By Theorem 8.2 of [6], these sets ...
Properties of topological groups and Haar measure
... (iv)If H is a subgroup of G, then G is the disjoint union of cosets of H, one of which can be taken to be H itself. One therefore gets G = (∪g∈H / gH) ∪ H. By translation invariance, each coset gH is open, and therefore the union of such sets is open as well. Then the previous equation shows that th ...
... (iv)If H is a subgroup of G, then G is the disjoint union of cosets of H, one of which can be taken to be H itself. One therefore gets G = (∪g∈H / gH) ∪ H. By translation invariance, each coset gH is open, and therefore the union of such sets is open as well. Then the previous equation shows that th ...
Sheaves on Spaces
... C is a category and that F : C → Sets is a faithful functor. Typically F is a “forgetful” functor. For an object M ∈ Ob(C) we often call F (M ) the underlying set of the object M . If M → M 0 is a morphism in C we call F (M ) → F (M 0 ) the underlying map of sets. In fact, we will often not distingu ...
... C is a category and that F : C → Sets is a faithful functor. Typically F is a “forgetful” functor. For an object M ∈ Ob(C) we often call F (M ) the underlying set of the object M . If M → M 0 is a morphism in C we call F (M ) → F (M 0 ) the underlying map of sets. In fact, we will often not distingu ...
... This thesis would have remained a dream had the Almighty God not been with me with His blessings, protection, and guidance throughout this period. I could never have accomplished this without the faith I have in Him. It is with immense gratitude that I acknowledge His help. I owe my deepest gratitud ...
A Coherence Criterion for Fréchet Modules
... In the literature, one finds essentially two general criteria to get the finiteness of the cohomology groups of complexes of locally convex topological vector spaces. They are (a) If u· : G· −→ F · is a compact morphism of complexes of Fréchet spaces then dim H k (F · ) < +∞ for any k ∈ ZZ such tha ...
... In the literature, one finds essentially two general criteria to get the finiteness of the cohomology groups of complexes of locally convex topological vector spaces. They are (a) If u· : G· −→ F · is a compact morphism of complexes of Fréchet spaces then dim H k (F · ) < +∞ for any k ∈ ZZ such tha ...
Non-archimedean analytic spaces
... Here X/Ker(f ) is provided with the quotient norm, and Im(f ) is provided with the norm induced from Y . (In particular, if f is admissible, Im(f ) is closed in Y .) Recall that, by the Banach openness theorem, if the valuation on k is nontrivial, every surjective bounded k-linear map between Banach ...
... Here X/Ker(f ) is provided with the quotient norm, and Im(f ) is provided with the norm induced from Y . (In particular, if f is admissible, Im(f ) is closed in Y .) Recall that, by the Banach openness theorem, if the valuation on k is nontrivial, every surjective bounded k-linear map between Banach ...
BP as a multiplicative Thom spectrum
... theory a spectrum that represents it? Satisfactorily, the answer is ’yes’ though this way is not that easy. In the case of cohomology, it follows from Brown’s representability theorem which we will present in the following. In the case of homology however, things are again a little bit more complica ...
... theory a spectrum that represents it? Satisfactorily, the answer is ’yes’ though this way is not that easy. In the case of cohomology, it follows from Brown’s representability theorem which we will present in the following. In the case of homology however, things are again a little bit more complica ...
Math 396. Paracompactness and local compactness 1. Motivation
... For example, the covering of R by open intervals (n − 1, n + 1) for n ∈ Z is locally finite, whereas the covering of (−1, 1) by intervals (−1/n, 1/n) (for n ≥ 1) barely fails to be locally finite: there is a problem at the origin (but nowhere else). Definition 2.5. A topological space X is paracompa ...
... For example, the covering of R by open intervals (n − 1, n + 1) for n ∈ Z is locally finite, whereas the covering of (−1, 1) by intervals (−1/n, 1/n) (for n ≥ 1) barely fails to be locally finite: there is a problem at the origin (but nowhere else). Definition 2.5. A topological space X is paracompa ...
E∞-Comodules and Topological Manifolds A Dissertation presented
... The first story begins with a question of Steenrod. He asked if the product in the cohomology of a triangulated space, which is associative and graded commutative, can be induced from a cochain level product satisfying the same two properties. He answered it in the negative after identifying homolog ...
... The first story begins with a question of Steenrod. He asked if the product in the cohomology of a triangulated space, which is associative and graded commutative, can be induced from a cochain level product satisfying the same two properties. He answered it in the negative after identifying homolog ...
Global Calculus:Basic Motivations
... any subset U of M (open or not), we can consider the set S U of functions from U to S. Define, now, a presheaf A as follows: • For every open set U , associate the set of functions from U to S. That is A(U ) = S U . • If V ⊆ U define the restriction map from S U to S V simply as function restriction ...
... any subset U of M (open or not), we can consider the set S U of functions from U to S. Define, now, a presheaf A as follows: • For every open set U , associate the set of functions from U to S. That is A(U ) = S U . • If V ⊆ U define the restriction map from S U to S V simply as function restriction ...
algebraic geometry and the generalisation of bezout`s theorem
... 3. Every ideal I of R is finitely generated, that is, there exist r1 , r2 , ... , rn ∈ R such that I = (r1 , r2 , ... , rn ). A ring R satisfying the above conditions is called a Noetherian ring Proof. 1 → 2. Suppose not, say S does not have a maximal element. Then given any J1 ∈ S there exists a J2 ...
... 3. Every ideal I of R is finitely generated, that is, there exist r1 , r2 , ... , rn ∈ R such that I = (r1 , r2 , ... , rn ). A ring R satisfying the above conditions is called a Noetherian ring Proof. 1 → 2. Suppose not, say S does not have a maximal element. Then given any J1 ∈ S there exists a J2 ...
topological group
... While every group can be made into a topological group, the same cannot be said of every topological space. In this section we mention some of the properties that the underlying topological space must have. Every topological group is bihomogeneous and completely regular. Note that our earlier claim ...
... While every group can be made into a topological group, the same cannot be said of every topological space. In this section we mention some of the properties that the underlying topological space must have. Every topological group is bihomogeneous and completely regular. Note that our earlier claim ...
4. Morphisms
... as a ringed space with the structure sheaf being the restriction OX |U as in Definition 3.18. With this idea that the regular functions make up the structure of an affine variety the obvious idea to define a morphism f : X → Y between affine varieties (or more generally ringed spaces) is now that th ...
... as a ringed space with the structure sheaf being the restriction OX |U as in Definition 3.18. With this idea that the regular functions make up the structure of an affine variety the obvious idea to define a morphism f : X → Y between affine varieties (or more generally ringed spaces) is now that th ...
DG AFFINITY OF DQ-MODULES 1. Introduction Many classical
... This paper is organised as follows. In the first part, we recall some classical material concerning generators in triangulated category. We review, following [8], the notion of cohomological completeness and its link with the functor of ~-graduation. We finally state some results specific to deforma ...
... This paper is organised as follows. In the first part, we recall some classical material concerning generators in triangulated category. We review, following [8], the notion of cohomological completeness and its link with the functor of ~-graduation. We finally state some results specific to deforma ...
THE GEOMETRY OF TORIC VARIETIES
... is that there are many algebraic varieties which it is most reasonable to embed not in projective space P" but in a suitable toric variety; it becomes more natural in such a case to compare the properties of the variety and the ambient space. This also applies to the choice of compactification of a ...
... is that there are many algebraic varieties which it is most reasonable to embed not in projective space P" but in a suitable toric variety; it becomes more natural in such a case to compare the properties of the variety and the ambient space. This also applies to the choice of compactification of a ...
"One-parameter subgroups of topological abelian groups". Topology
... other the relation between CHom(R, G) and the group G given by the evaluation mapping CHom(R, G) −→ G, ϕ −→ ϕ(1) The vector space CHom(R, G) endowed with the compact open topology is called the Lie algebra of the topological group G and denoted by L(G) in analogy with the classical theory of Lie gr ...
... other the relation between CHom(R, G) and the group G given by the evaluation mapping CHom(R, G) −→ G, ϕ −→ ϕ(1) The vector space CHom(R, G) endowed with the compact open topology is called the Lie algebra of the topological group G and denoted by L(G) in analogy with the classical theory of Lie gr ...