Free Topological Groups and the Projective Dimension of a Locally
... group topology which induces the given topology on X. PROOF. Let T be the given topology on G and i-' D the finest group topology inducing the given topology on X. By the proof of Theorem 1, A c G is closed in (G, i') if and only if each A rnXn is compact. But -r and -r' induce the same topology on ...
... group topology which induces the given topology on X. PROOF. Let T be the given topology on G and i-' D the finest group topology inducing the given topology on X. By the proof of Theorem 1, A c G is closed in (G, i') if and only if each A rnXn is compact. But -r and -r' induce the same topology on ...
On the Universal Space for Group Actions with Compact Isotropy
... for the family F is a G-CW -complex E(G, F) such that the fixed point set E(G, F)H is weakly contractible for H ∈ F and all its isotropy groups belong to F. Recall that a map f : X → Y of spaces is a weak homotopy equivalence if and only if the induced map f∗ : πn (X, x) → πn (Y, f (x)) is an isomo ...
... for the family F is a G-CW -complex E(G, F) such that the fixed point set E(G, F)H is weakly contractible for H ∈ F and all its isotropy groups belong to F. Recall that a map f : X → Y of spaces is a weak homotopy equivalence if and only if the induced map f∗ : πn (X, x) → πn (Y, f (x)) is an isomo ...
Hochschild cohomology: some methods for computations
... conditions are equivalent a) H i (A) = 0, ∀i ≥ 1; b) H 1 (A) = 0; c) Q is a tree. Example 3.3 Assume k has characteristic zero, A = kQ/I, I an homogeneous ideal (this means that I is generated by linear combinations of paths that have the same length). We want to show that if H 1 (A) = 0 then Q has ...
... conditions are equivalent a) H i (A) = 0, ∀i ≥ 1; b) H 1 (A) = 0; c) Q is a tree. Example 3.3 Assume k has characteristic zero, A = kQ/I, I an homogeneous ideal (this means that I is generated by linear combinations of paths that have the same length). We want to show that if H 1 (A) = 0 then Q has ...
Section I.7. Generating Sets and Cayley Digraphs
... Solution. For multiplication on the right by i we have: 1 · i = i, i · i = −1, −1 · i = −i, −i · i = 1, j · i = −k, −k · i = −j, −j · i = k, and k · i = j. For multiplication on the right by j we have: 1 · j = j, j · j = −1, −1 · j = −j, −j · j = 1, i · j = k, k · j = −i, −i · j = −k, and −k · j = i ...
... Solution. For multiplication on the right by i we have: 1 · i = i, i · i = −1, −1 · i = −i, −i · i = 1, j · i = −k, −k · i = −j, −j · i = k, and k · i = j. For multiplication on the right by j we have: 1 · j = j, j · j = −1, −1 · j = −j, −j · j = 1, i · j = k, k · j = −i, −i · j = −k, and −k · j = i ...
structure of abelian quasi-groups
... be any element of 9?i where a is a generating element of @. Now if \p{a) is any element of dti then(a)] is an element of both 9?i and 9t2.
Hence dti and 9?2 have a non-void crosscut and since both are minimal they
...
... be any element of 9?i where a is a generating element of @. Now if \p{a) is any element of dti then
3. Modules
... product IM and quotient N 0 : N of Definition 3.12 are exactly the product and quotient of ideals as in Construction 1.1. (b) If I is an ideal of a ring R then annR (R/I) = I. Let us recall again the linear algebra of vector spaces over a field K. At the point where we are now, i. e. after having st ...
... product IM and quotient N 0 : N of Definition 3.12 are exactly the product and quotient of ideals as in Construction 1.1. (b) If I is an ideal of a ring R then annR (R/I) = I. Let us recall again the linear algebra of vector spaces over a field K. At the point where we are now, i. e. after having st ...
Projective p-adic representations of the K-rational geometric fundamental group (with G. Frey).
... for otherwise all the pn -torsion points of the reduction AP would be K-rational, which is impossible since AP (K) is finitely generated by the Theorem of Mordell-Weil-NéronLang. Moreover, a similar conclusion holds for any subrepresentation ρV : GF → GL(V ) associated to a (non-zero) Qp [GF ]-subm ...
... for otherwise all the pn -torsion points of the reduction AP would be K-rational, which is impossible since AP (K) is finitely generated by the Theorem of Mordell-Weil-NéronLang. Moreover, a similar conclusion holds for any subrepresentation ρV : GF → GL(V ) associated to a (non-zero) Qp [GF ]-subm ...
NOTES ON THE SEPARABILITY OF C*-ALGEBRAS Chun
... The spectrum of a C*-algebra is the set Spec(A) of (spatial) equivalence classes of irreducible representations. The topology of Spec(A) is the one induced from Prim(A) through the natural map π → ker π. Again, in the abelian case we have Spec(C0 (Ω)) ∼ = Ω. However, although being locally compact, ...
... The spectrum of a C*-algebra is the set Spec(A) of (spatial) equivalence classes of irreducible representations. The topology of Spec(A) is the one induced from Prim(A) through the natural map π → ker π. Again, in the abelian case we have Spec(C0 (Ω)) ∼ = Ω. However, although being locally compact, ...
Full-Text PDF
... a sliglt generalization of L. Levy’s "Separated DivisorTheorem" for matrices over Dedekittd donains [13], we can show for Priifer domains that .I(R) is naturally isomorphic to t.,nt.sr.(n).,’vl(Rt,) it" and o.ly if R is of finite character and the valuation rings at the maximal ideals of R arc pairw ...
... a sliglt generalization of L. Levy’s "Separated DivisorTheorem" for matrices over Dedekittd donains [13], we can show for Priifer domains that .I(R) is naturally isomorphic to t.,nt.sr.(n).,’vl(Rt,) it" and o.ly if R is of finite character and the valuation rings at the maximal ideals of R arc pairw ...
THE CLASSICAL GROUPS
... and indeed the group operation, coming from composition of linear maps is continuous (even smooth). Lemma 2.14. The group SO(Rn ) (as a subspace of Hom(Rn , Rn )) is compact. Proof. Because α ∈ SO(Rn ) is an isometry, we have kα(v)k = kvk for all v ∈ Rn . But then it follows from the definition of t ...
... and indeed the group operation, coming from composition of linear maps is continuous (even smooth). Lemma 2.14. The group SO(Rn ) (as a subspace of Hom(Rn , Rn )) is compact. Proof. Because α ∈ SO(Rn ) is an isometry, we have kα(v)k = kvk for all v ∈ Rn . But then it follows from the definition of t ...
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 2, Pages 723–731
... Lemma 5. Let S be a linear map between Banach spaces X and Y , and let Tn , Rn be continuous linear maps on X and on Y , respectively, such that Rn S = STn for all n ∈ N. Let Sn be the norm closure of Rn ◦ Rn−1 ◦ · · · ◦ R1 (S), where S is the separating space of S. Then there is an integer N such t ...
... Lemma 5. Let S be a linear map between Banach spaces X and Y , and let Tn , Rn be continuous linear maps on X and on Y , respectively, such that Rn S = STn for all n ∈ N. Let Sn be the norm closure of Rn ◦ Rn−1 ◦ · · · ◦ R1 (S), where S is the separating space of S. Then there is an integer N such t ...
