Pages 31-40 - The Graduate Center, CUNY
... V of ψ(x) in Y , a neighborhood W ⊆ ψ −1 (V ) ∩ U of x, and an element s ∈ G(V ) such that s0z = sψ(z) for every z ∈ W . One verifies immediately that U 7→ G 0 (U ) satisfies the sheaf axioms. Let F be a sheaf of sets on X with morphisms u : G → ψ∗ (F), v : G 0 → ψ∗ (F). One defines u] and v [ in th ...
... V of ψ(x) in Y , a neighborhood W ⊆ ψ −1 (V ) ∩ U of x, and an element s ∈ G(V ) such that s0z = sψ(z) for every z ∈ W . One verifies immediately that U 7→ G 0 (U ) satisfies the sheaf axioms. Let F be a sheaf of sets on X with morphisms u : G → ψ∗ (F), v : G 0 → ψ∗ (F). One defines u] and v [ in th ...
Lieblich Definition 1 (Category Fibered in Groupoids). A functor F : D
... Also, using smoothness applied to p00 , ∃u00 : R → A00 with u00 (ξ) = η 00 . Set ζ = u0 ×u u00 (ξ) ∈ F (A0 ×A A00 ), this lifts (η 0 , η 00 ) and this proves (H1). For (H2), assume that A = k, A00 = k[]. We want (*) injective. Suppose that v ∈ F (A0 ×A A00 ) also restricts to η 0 and η 00 , we want ...
... Also, using smoothness applied to p00 , ∃u00 : R → A00 with u00 (ξ) = η 00 . Set ζ = u0 ×u u00 (ξ) ∈ F (A0 ×A A00 ), this lifts (η 0 , η 00 ) and this proves (H1). For (H2), assume that A = k, A00 = k[]. We want (*) injective. Suppose that v ∈ F (A0 ×A A00 ) also restricts to η 0 and η 00 , we want ...
A parametrized Borsuk-Ulam theorem for a product of - Icmc-Usp
... The definition of the Stiefel-Whitney polynomials for vector bundles with the antipodal involution was introduced by Dold in [1] and it is an useful tool in studying parametrized Borsuk-Ulam type problem. Dold used the Stiefel-Whitney polynomials to prove that if p = 2 and if m and k are the dimensi ...
... The definition of the Stiefel-Whitney polynomials for vector bundles with the antipodal involution was introduced by Dold in [1] and it is an useful tool in studying parametrized Borsuk-Ulam type problem. Dold used the Stiefel-Whitney polynomials to prove that if p = 2 and if m and k are the dimensi ...
Affine Varieties
... is a regular map, then Φ∗ (y 1 ), ..., Φ∗ (y m ) ∈ OX (X). But by Proposition 3.3, OX (X) = C[X], so Φ∗ (y 1 ) = f 1 , ..., Φ∗ (y m ) = f m for some regular functions f 1 , ..., f m ∈ C[X], and then indeed: Φ(a1 , ..., an ) = (f1 (a1 , ..., an ), ..., fm (a1 , ..., an )) for any choice of representa ...
... is a regular map, then Φ∗ (y 1 ), ..., Φ∗ (y m ) ∈ OX (X). But by Proposition 3.3, OX (X) = C[X], so Φ∗ (y 1 ) = f 1 , ..., Φ∗ (y m ) = f m for some regular functions f 1 , ..., f m ∈ C[X], and then indeed: Φ(a1 , ..., an ) = (f1 (a1 , ..., an ), ..., fm (a1 , ..., an )) for any choice of representa ...
PERIODS OF GENERIC TORSORS OF GROUPS OF
... from the fact that the Z-torsion elements of Q∗ form a G-submodule (W ∗ ) of Q∗ . Lemma 5.1. e(W ) = exponent of W . Proof. Denote the exponent of W by n. Since W n = 1, the homomorphism n // H 1 (K, W ) factors through H 1 (K, 1) = 0 for every field extension H 1 (K, W ) K/F . Hence e(W ) | n. Reca ...
... from the fact that the Z-torsion elements of Q∗ form a G-submodule (W ∗ ) of Q∗ . Lemma 5.1. e(W ) = exponent of W . Proof. Denote the exponent of W by n. Since W n = 1, the homomorphism n // H 1 (K, W ) factors through H 1 (K, 1) = 0 for every field extension H 1 (K, W ) K/F . Hence e(W ) | n. Reca ...
Class 3 - Stanford Mathematics
... have been the same function to begin with. In other words, if {Ui }i∈I is a cover of U, and f1 , f2 ∈ O(U), and resU,Ui f1 = resU,Ui f2 , then f1 = f2 . Thus I can identify functions on an open set by looking at them on a covering by small open sets. Finally, given the same U and cover Ui , take a d ...
... have been the same function to begin with. In other words, if {Ui }i∈I is a cover of U, and f1 , f2 ∈ O(U), and resU,Ui f1 = resU,Ui f2 , then f1 = f2 . Thus I can identify functions on an open set by looking at them on a covering by small open sets. Finally, given the same U and cover Ui , take a d ...
Notes on Tate's article on p-divisible groups
... This talk discusses the main results of Tate’s paper "p-Divisible Groups" [6]. From the point of view of p-adic Hodge theory, this is a foundational paper and within this setting, much of the technical work being done becomes extremely important. From our perspective, having just learned what p-divi ...
... This talk discusses the main results of Tate’s paper "p-Divisible Groups" [6]. From the point of view of p-adic Hodge theory, this is a foundational paper and within this setting, much of the technical work being done becomes extremely important. From our perspective, having just learned what p-divi ...
Sets with a Category Action Peter Webb 1. C-sets
... functors defined on the full subcategory BGroup of BCat whose objects are finite groups. The Burnside ring functor BR (C) := R ⊗Z B(C) is in fact an example of a biset functor defined on categories. Let 1 denote the category with one object and one morphism – in other words, the identity group. We s ...
... functors defined on the full subcategory BGroup of BCat whose objects are finite groups. The Burnside ring functor BR (C) := R ⊗Z B(C) is in fact an example of a biset functor defined on categories. Let 1 denote the category with one object and one morphism – in other words, the identity group. We s ...
Oct. 19, 2016 0.1. Topological groups. Let X be a topological space
... We have seen this and this is just stemming from the definition of a topological group. There is a characterization of the topology of a topological group using (a), (b), (c), (d), and this is what we haven’t discussed to the end. Proposition 3 (Prop.7.2, second part). Let G be a group, N be a nonem ...
... We have seen this and this is just stemming from the definition of a topological group. There is a characterization of the topology of a topological group using (a), (b), (c), (d), and this is what we haven’t discussed to the end. Proposition 3 (Prop.7.2, second part). Let G be a group, N be a nonem ...
the homology theory of the closed geodesic problem
... The description of the space of all closed curves on M. In [6] and [7] an algebraic description of homotopy problems via differential algebras and differential forms was given. The nature of this description is such that if a proferred formula for Λ(M) has the correct algebraic properties it must be ...
... The description of the space of all closed curves on M. In [6] and [7] an algebraic description of homotopy problems via differential algebras and differential forms was given. The nature of this description is such that if a proferred formula for Λ(M) has the correct algebraic properties it must be ...
Cohomology and K-theory of Compact Lie Groups
... The structure of the K-theory is immediate once we know that K ∗ (G) is torsion-free and apply the fact that rational cohomology ring and rational K-theory of a finite CW -complex are isomorphic through the Chern character(c.f. [AH]). In fact Theorem 1.2. If G is a compact, simply-connected Lie grou ...
... The structure of the K-theory is immediate once we know that K ∗ (G) is torsion-free and apply the fact that rational cohomology ring and rational K-theory of a finite CW -complex are isomorphic through the Chern character(c.f. [AH]). In fact Theorem 1.2. If G is a compact, simply-connected Lie grou ...
MATH 6280 - CLASS 2 Contents 1. Categories 1 2. Functors 2 3
... (6) Homology Hn : Top → Ab which sends a space S to the n’th simplicial homology group of S. (7) Cohomology H n : Topop → Ab which sends a space S to the n’th simplicial cohomology group of S. (8) The homotopy groups functors: πn : Topop → Ab (9) If F : G → H is a group homomorphism, then it gives r ...
... (6) Homology Hn : Top → Ab which sends a space S to the n’th simplicial homology group of S. (7) Cohomology H n : Topop → Ab which sends a space S to the n’th simplicial cohomology group of S. (8) The homotopy groups functors: πn : Topop → Ab (9) If F : G → H is a group homomorphism, then it gives r ...
Generalized Broughton polynomials and characteristic varieties Nguyen Tat Thang
... The group T (f ) is determined as follows. Theorem 2.5. ([6]) Let S is a non-proper smooth curve and f : M → S be a regular function. Then the group T (f ) is computed by the following T (f ) = ⊕c∈C(h) Z/mc Z, where mc is the multiplicity of the divisor f −1 (c) and C(f ) is the set of bifurcation v ...
... The group T (f ) is determined as follows. Theorem 2.5. ([6]) Let S is a non-proper smooth curve and f : M → S be a regular function. Then the group T (f ) is computed by the following T (f ) = ⊕c∈C(h) Z/mc Z, where mc is the multiplicity of the divisor f −1 (c) and C(f ) is the set of bifurcation v ...
CW Complexes and the Projective Space
... To see this, note that it is also possible to obtain CP n as the quotient space of the (closed) disk D2n under the identifications v ∼ λv for v ∈ ∂D2n = S2n−1 . But S2n−1 modulo this relation is CP n−1 , so we are obtaining CP n by attaching a cell e2n to CP n−1 via the quotient map S2n−1 −→ CP n−1 ...
... To see this, note that it is also possible to obtain CP n as the quotient space of the (closed) disk D2n under the identifications v ∼ λv for v ∈ ∂D2n = S2n−1 . But S2n−1 modulo this relation is CP n−1 , so we are obtaining CP n by attaching a cell e2n to CP n−1 via the quotient map S2n−1 −→ CP n−1 ...
UNT UTA Algebra Symposium University of North Texas November
... commutative algebra and algebraic geometry into a noncommutative setting. In 1990, Artin, Tate and Van den Bergh made a great step forward in this endeavor, introducing the concept of a point module in their paper, Some Algebras Associated to Automorphisms of Elliptic Curves. In this talk, we will d ...
... commutative algebra and algebraic geometry into a noncommutative setting. In 1990, Artin, Tate and Van den Bergh made a great step forward in this endeavor, introducing the concept of a point module in their paper, Some Algebras Associated to Automorphisms of Elliptic Curves. In this talk, we will d ...
Math 210B. Absolute Galois groups and fundamental groups 1
... The aim of this section is to explain how the intervention of conjugation ambiguity in the functoriality of absolute Galois groups is similar to the appearance of conjugation in the functioriality of fundamental groups. It also turns out that Galois cohomology, to be discussed later in the course, e ...
... The aim of this section is to explain how the intervention of conjugation ambiguity in the functoriality of absolute Galois groups is similar to the appearance of conjugation in the functioriality of fundamental groups. It also turns out that Galois cohomology, to be discussed later in the course, e ...
Lecture 20 1 Point Set Topology
... irreducible affine variety. It follows from the fact that a finite union of constructible sets is a constructible set and every variety is a union of finite number of irreducible varieties. Thus we may assume that B is a domain. The base of induction is clear, the image of an irreducible set of dime ...
... irreducible affine variety. It follows from the fact that a finite union of constructible sets is a constructible set and every variety is a union of finite number of irreducible varieties. Thus we may assume that B is a domain. The base of induction is clear, the image of an irreducible set of dime ...
Notes
... We often refer to the elements s ∈ F(U ) as the sections of F over U . This language is supposed to evoke the idea that the set F(U ) lives ‘above’ U in some sense. Sections of F over X are the global sections. An alternate notation, common in algebraic geometry, is to write Γ(U, F) for F(U ). This ...
... We often refer to the elements s ∈ F(U ) as the sections of F over U . This language is supposed to evoke the idea that the set F(U ) lives ‘above’ U in some sense. Sections of F over X are the global sections. An alternate notation, common in algebraic geometry, is to write Γ(U, F) for F(U ). This ...
Homology and cohomology theories on manifolds
... In this paper, we study generalized homology and cohomology theories on the categories of smooth, PL, and topological manifolds. These are, by definition, absolute theories satisfying a Mayer-Vietoris property (see Section 4). Recently, the first author constructed such a theory (of ordinary type) o ...
... In this paper, we study generalized homology and cohomology theories on the categories of smooth, PL, and topological manifolds. These are, by definition, absolute theories satisfying a Mayer-Vietoris property (see Section 4). Recently, the first author constructed such a theory (of ordinary type) o ...
Homology and cohomology theories on manifolds
... In this paper, we study generalized homology and cohomology theories on the categories of smooth, PL, and topological manifolds. These are, by definition, absolute theories satisfying a Mayer-Vietoris property (see Section 4). Recently, the first author constructed such a theory (of ordinary type) o ...
... In this paper, we study generalized homology and cohomology theories on the categories of smooth, PL, and topological manifolds. These are, by definition, absolute theories satisfying a Mayer-Vietoris property (see Section 4). Recently, the first author constructed such a theory (of ordinary type) o ...
The Weil-étale topology for number rings
... zeta-function conjecture is, and what relation this bears to Bloch-Kato. 1. Cohomology of topological groups Let G be a topological group. We define a Grothendieck topology T .G/ as follows: Let Cat.T .G// be the category of G-spaces and G-morphisms. A collection of maps fi W Xi ! X g will be calle ...
... zeta-function conjecture is, and what relation this bears to Bloch-Kato. 1. Cohomology of topological groups Let G be a topological group. We define a Grothendieck topology T .G/ as follows: Let Cat.T .G// be the category of G-spaces and G-morphisms. A collection of maps fi W Xi ! X g will be calle ...
MANIFOLDS, COHOMOLOGY, AND SHEAVES
... the partial derivatives ∂ k f /∂r j1 · · · ∂r jk exist on U for all integers k ≥ 1 and all j1 , . . . , jk . A vector-valued function f = (f 1 , . . . , f m ) : U → Rm is smooth if each component f i is smooth on U . In these lectures we use the words “smooth” and “C ∞ ” interchangeably. A topologic ...
... the partial derivatives ∂ k f /∂r j1 · · · ∂r jk exist on U for all integers k ≥ 1 and all j1 , . . . , jk . A vector-valued function f = (f 1 , . . . , f m ) : U → Rm is smooth if each component f i is smooth on U . In these lectures we use the words “smooth” and “C ∞ ” interchangeably. A topologic ...
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... α − β in this group, and define this morphism to be diff. Note that exactness of the sequence (??) is an element free condition, and therefore makes sense for any abelian category A, even if A is not concrete. Accordingly, for any abelian category A, we define a sheaf to be a presheaf F for which th ...
... α − β in this group, and define this morphism to be diff. Note that exactness of the sequence (??) is an element free condition, and therefore makes sense for any abelian category A, even if A is not concrete. Accordingly, for any abelian category A, we define a sheaf to be a presheaf F for which th ...
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... In this section we define the homology and cohomology groups associated to a simplicial complex K. We do so not because the homology of a simplicial complex is so intrinsically interesting in and of itself, but because the resulting homology theory is identical to the singular homology of the associ ...
... In this section we define the homology and cohomology groups associated to a simplicial complex K. We do so not because the homology of a simplicial complex is so intrinsically interesting in and of itself, but because the resulting homology theory is identical to the singular homology of the associ ...