1 Overview 2 Sheaves on Topological Spaces
... on Y is given by the subspace topology, continuity of f : X → Y follows immediately. The other cases follow by an adaptation of the proof, using quotients of polynomials in place of quotients of homogeneous polynomials of the same degree where appropriate. Once we know that f is continuous, it follo ...
... on Y is given by the subspace topology, continuity of f : X → Y follows immediately. The other cases follow by an adaptation of the proof, using quotients of polynomials in place of quotients of homogeneous polynomials of the same degree where appropriate. Once we know that f is continuous, it follo ...
Cohomology jump loci of quasi-projective varieties Botong Wang June 27 2013
... Here the isomorphism is an isomorphism of real Lie groups, not complex Lie groups. However, we will not use this at all in this talk. Instead, we need to introduce our main player, the cohomology jump loci in MB (X ), def ...
... Here the isomorphism is an isomorphism of real Lie groups, not complex Lie groups. However, we will not use this at all in this talk. Instead, we need to introduce our main player, the cohomology jump loci in MB (X ), def ...
Garrett 02-15-2012 1 Harmonic analysis, on R, R/Z, Q , A, and A
... S 1 so that E contains no non-trivial subgroups of G. Using the b be the open compactness of G itself, let U ⊂ G b : f (G) ⊂ E} U = {f ∈ G Since E is small, f (G) = {1}. That is, f is the trivial b for compact G. homomorphism. This proves discreteness of G ...
... S 1 so that E contains no non-trivial subgroups of G. Using the b be the open compactness of G itself, let U ⊂ G b : f (G) ⊂ E} U = {f ∈ G Since E is small, f (G) = {1}. That is, f is the trivial b for compact G. homomorphism. This proves discreteness of G ...
S1-Equivariant K-Theory of CP1
... If f is a homotopy equivalence, then f ∗ is a group isomorphism, and id∗X = idKG (X ) . Thus, KG is a homotopy invariant contravariant functor from the category of compact Hausdorff G -spaces to the category of abelian groups. ...
... If f is a homotopy equivalence, then f ∗ is a group isomorphism, and id∗X = idKG (X ) . Thus, KG is a homotopy invariant contravariant functor from the category of compact Hausdorff G -spaces to the category of abelian groups. ...
MA 331 HW 15: Is the Mayflower Compact? If X is a topological
... (3) (*) Prove that a topological graph G is compact if and only if G has finitely many edges and vertices. (4) (Challenging!) Suppose that X and Y are topological spaces. Let C(X,Y ) be the set of continuous functions X → Y . We give C(X,Y ) a topology T (called the compact-open topology) as follows ...
... (3) (*) Prove that a topological graph G is compact if and only if G has finitely many edges and vertices. (4) (Challenging!) Suppose that X and Y are topological spaces. Let C(X,Y ) be the set of continuous functions X → Y . We give C(X,Y ) a topology T (called the compact-open topology) as follows ...
PDF
... X is Hausdorff if and only if the image of ∆ is closed in X × X [proof]. If we know more about the product space than we do about X, it might be easier to check if Im ∆ is closed than to verify the Hausdorff condition directly. When studying algebraic topology, the fact that we have a diagonal embed ...
... X is Hausdorff if and only if the image of ∆ is closed in X × X [proof]. If we know more about the product space than we do about X, it might be easier to check if Im ∆ is closed than to verify the Hausdorff condition directly. When studying algebraic topology, the fact that we have a diagonal embed ...
Lieblich Definition 1 (Category Fibered in Groupoids). A functor F : D
... Note: No automorphism to worry about, so we could even have a notion of equality of quotients via equality of kernels. Theorem 2. Def F ,E is a deformation functor, and satisfies (H4). If XΛ is proper and E is coherent, then Def F ,E also satisfies (H3), and so is prorepresentable. Note: For represe ...
... Note: No automorphism to worry about, so we could even have a notion of equality of quotients via equality of kernels. Theorem 2. Def F ,E is a deformation functor, and satisfies (H4). If XΛ is proper and E is coherent, then Def F ,E also satisfies (H3), and so is prorepresentable. Note: For represe ...
Version 1.0.20
... Definition 1.8. A Grothendieck topology, J , on C is an assignment to each object c of C of a collection J (c) of sieves on c such that: 1. The maximal sieve y c ∈ J (c). 2. If S ∈ J (c) and h : c 0 → c is a map, then h ∗ (c) ∈ J (c 0 ). 3. If S ∈ J (c) and if R is a sieve on c such that h ∗ (R) ∈ J ...
... Definition 1.8. A Grothendieck topology, J , on C is an assignment to each object c of C of a collection J (c) of sieves on c such that: 1. The maximal sieve y c ∈ J (c). 2. If S ∈ J (c) and h : c 0 → c is a map, then h ∗ (c) ∈ J (c 0 ). 3. If S ∈ J (c) and if R is a sieve on c such that h ∗ (R) ∈ J ...
Defining Gm and Yoneda and group objects
... between the Z-points of X and Y for each scheme Z and these maps satisfy certain naturality conditions, then they arise from a map from X to Y . This correspondence also helps us make sense of a group object in a category, which is an object G and morphisms m : G × G → G, i : G → G and e : ? → G, wh ...
... between the Z-points of X and Y for each scheme Z and these maps satisfy certain naturality conditions, then they arise from a map from X to Y . This correspondence also helps us make sense of a group object in a category, which is an object G and morphisms m : G × G → G, i : G → G and e : ? → G, wh ...
AAG, LECTURE 13 If 0 → F 1 → F2 → F3 → 0 is a short exact
... but it is not necessarily the case that F2 (X) → F3 (X) is surjective. (The surjectivity of F2 → F3 implies something weaker: that for any f ∈ F3 (X) and x ∈ X there is an open x ∈ U ⊂ X and a g ∈ F2 (U ) so that g 7→ f|U . These g’s are not unique, since we can change them by sections of F1 , so on ...
... but it is not necessarily the case that F2 (X) → F3 (X) is surjective. (The surjectivity of F2 → F3 implies something weaker: that for any f ∈ F3 (X) and x ∈ X there is an open x ∈ U ⊂ X and a g ∈ F2 (U ) so that g 7→ f|U . These g’s are not unique, since we can change them by sections of F1 , so on ...
RIGID RATIONAL HOMOTOPY THEORY AND
... The first step was essentially done by Kim and Hain - in this case we can compute the rational homotopy type using log-crystalline cohomology rather than rigid cohomology. The details here are rather complicated but the basic idea is that you locally lift to characteristic zero, where exactly as in ...
... The first step was essentially done by Kim and Hain - in this case we can compute the rational homotopy type using log-crystalline cohomology rather than rigid cohomology. The details here are rather complicated but the basic idea is that you locally lift to characteristic zero, where exactly as in ...
Section 07
... where the product ranges over all (n + 1)-tuples i0 , . . . , in for which Ui0 ...in is non-empty. This is a larger complex with much redundant information (the group A(Ui0 ...in ) now occurs (n + 1)! times), hence less convenient for calculations, but it computes the same cohomology as we will now ...
... where the product ranges over all (n + 1)-tuples i0 , . . . , in for which Ui0 ...in is non-empty. This is a larger complex with much redundant information (the group A(Ui0 ...in ) now occurs (n + 1)! times), hence less convenient for calculations, but it computes the same cohomology as we will now ...
Notes
... iii) (Isomorphisms) If V → U is an isomorphism, then the one-element family {V → U } is a covering family. A category together with a choice of Grothendieck topology is called a site. Strictly this is the notion of a Grothendieck pretopology; in this talk we can do without the more general notion of ...
... iii) (Isomorphisms) If V → U is an isomorphism, then the one-element family {V → U } is a covering family. A category together with a choice of Grothendieck topology is called a site. Strictly this is the notion of a Grothendieck pretopology; in this talk we can do without the more general notion of ...
What Is...a Topos?, Volume 51, Number 9
... compare this with the fact that a group can be defined by generators and relations in many different ways. The site is some kind of system of generators and relations for the topos. And in the same way in which a group G can be defined by a set of generators that is G itself, a topos T can be define ...
... compare this with the fact that a group can be defined by generators and relations in many different ways. The site is some kind of system of generators and relations for the topos. And in the same way in which a group G can be defined by a set of generators that is G itself, a topos T can be define ...
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 ...
The equivariant spectral sequence and cohomology with local coefficients Alexander I. Suciu
... monodromy action and the Aomoto complex, established in Theorem 2. Theorem 3. Let N be the kernel of an epimorphism ν : π Z. Suppose π is 1-formal, and b1 (N, C) < ∞. Then the eigenvalue 1 of the monodromy action of Z on H1 (N, C) has only 1 × 1 Jordan blocks. Given a reduced polynomial function f ...
... monodromy action and the Aomoto complex, established in Theorem 2. Theorem 3. Let N be the kernel of an epimorphism ν : π Z. Suppose π is 1-formal, and b1 (N, C) < ∞. Then the eigenvalue 1 of the monodromy action of Z on H1 (N, C) has only 1 × 1 Jordan blocks. Given a reduced polynomial function f ...
Pages 31-40 - The Graduate Center, CUNY
... (3.7.1) We will show that when one takes K to be the category of sets, the inverse image by ψ of every K-valued presheaf G must exist (the notation and hypotheses for X, Y , ψ being those of (3.5.3)). Indeed, for every open U ⊆ X, G 0 (U ) is defined as follows: an element s0 of G 0 (U ) is a family ...
... (3.7.1) We will show that when one takes K to be the category of sets, the inverse image by ψ of every K-valued presheaf G must exist (the notation and hypotheses for X, Y , ψ being those of (3.5.3)). Indeed, for every open U ⊆ X, G 0 (U ) is defined as follows: an element s0 of G 0 (U ) is a family ...
Complex Bordism (Lecture 5)
... fiber ζx . This dependence is not very strong, since the orthogonal group O(n) has only two components: the resulting elements of E 0 ({x}) are off by a sign if we choose trivializations with different orientations. Remark 4. A consequence of Definition 2 is that the Leray-Serre spectral sequence fo ...
... fiber ζx . This dependence is not very strong, since the orthogonal group O(n) has only two components: the resulting elements of E 0 ({x}) are off by a sign if we choose trivializations with different orientations. Remark 4. A consequence of Definition 2 is that the Leray-Serre spectral sequence fo ...
Problem Set 5 - Stony Brook Mathematics
... space is a homology n-manifold, then (a) X consists entirely of n-simplices and their faces, (b) Every (n − 1)-simplex is a face of precisely two n-simplices. Problem 2. Suppose that X is a compact triangulable homology n-manifold. (a) Show that if X is orientable, then Hn−1 (X) is torsion-free, and ...
... space is a homology n-manifold, then (a) X consists entirely of n-simplices and their faces, (b) Every (n − 1)-simplex is a face of precisely two n-simplices. Problem 2. Suppose that X is a compact triangulable homology n-manifold. (a) Show that if X is orientable, then Hn−1 (X) is torsion-free, and ...
NOTES ON GROTHENDIECK TOPOLOGIES 1
... Definition 3.5. The sheaf F # in Theorem 3.4 is called the sheafification or the associated sheaf. We describe the construction. First, given a presheaf F , we define a presheaf F + as follows. Let U ∈ T and let {Ui → U } be any covering of U. Consider a tuple of sections f i ∈ F (Ui ) such that the ...
... Definition 3.5. The sheaf F # in Theorem 3.4 is called the sheafification or the associated sheaf. We describe the construction. First, given a presheaf F , we define a presheaf F + as follows. Let U ∈ T and let {Ui → U } be any covering of U. Consider a tuple of sections f i ∈ F (Ui ) such that the ...
A BRIEF INTRODUCTION TO SHEAVES References 1. Presheaves
... 0 −→ F 0 −→ F 1 −→ F 1 −→ 0. This construction can now be continued to form a resolution 0 −→ F −→ F 0 −→ F 1 −→ · · · −→ F s −→ · · · by flabby sheaves with exact sequences 0 −→ F s−1 −→ F s −→ F s −→ 0. H ∗ (X, F) can be calculated using this resolution. Another type of cohomology is associated to ...
... 0 −→ F 0 −→ F 1 −→ F 1 −→ 0. This construction can now be continued to form a resolution 0 −→ F −→ F 0 −→ F 1 −→ · · · −→ F s −→ · · · by flabby sheaves with exact sequences 0 −→ F s−1 −→ F s −→ F s −→ 0. H ∗ (X, F) can be calculated using this resolution. Another type of cohomology is associated to ...
PDF
... presheaf on T is a contravariant functor F : T → C. If U → V is a morphism in T , we call F (i) : F (V ) → F (U ) the restriction map obtained from i. A morphism of presheaves is a natural transformation. To understand this definition, and what it has to do with the more familiar definition of a pre ...
... presheaf on T is a contravariant functor F : T → C. If U → V is a morphism in T , we call F (i) : F (V ) → F (U ) the restriction map obtained from i. A morphism of presheaves is a natural transformation. To understand this definition, and what it has to do with the more familiar definition of a pre ...
Lecture 1: Lie algebra cohomology
... particular the de Rham cohomology H• (M) has a well-defined multiplication induced from the wedge product. If M is riemannian, compact and orientable one has the celebrated Hodge decomposition theorem stating that in every de Rham cohomology class there is a unique smooth harmonic form. The second e ...
... particular the de Rham cohomology H• (M) has a well-defined multiplication induced from the wedge product. If M is riemannian, compact and orientable one has the celebrated Hodge decomposition theorem stating that in every de Rham cohomology class there is a unique smooth harmonic form. The second e ...
Note - Math
... In other words, if X is a topological space and F, G, H are three sheaves over X, then the sequence of morphisms 0 −→ F −→ G −→ H −→ 0 is exact if and only if for every x ∈ X, the following sequence, induced by these morphisms, is exact: 0 −→ Fx −→ Gx −→ Hx −→ 0 Example 1. Let X, Y be two topologica ...
... In other words, if X is a topological space and F, G, H are three sheaves over X, then the sequence of morphisms 0 −→ F −→ G −→ H −→ 0 is exact if and only if for every x ∈ X, the following sequence, induced by these morphisms, is exact: 0 −→ Fx −→ Gx −→ Hx −→ 0 Example 1. Let X, Y be two topologica ...
