Algebraic topology exam

... Answer eight questions, four from part I and four from part II. Give as much detail in your answers as you can. Part I 1. Prove the Zig-Zag lemma: let 0 C D E 0 be a short exact sequence of chain complexes with the above maps being f: C D, g : D E. Show that there is a long exact sequenc ...

... Answer eight questions, four from part I and four from part II. Give as much detail in your answers as you can. Part I 1. Prove the Zig-Zag lemma: let 0 C D E 0 be a short exact sequence of chain complexes with the above maps being f: C D, g : D E. Show that there is a long exact sequenc ...

What Is...a Topos?, Volume 51, Number 9

... functor is the stalk functor E → Ex . But new phenomena occur. Deligne has constructed examples of toposes without points (he has also given criteria for the existence of “enough points”) [SGA 4 IV 7, VI 9]. Moreover, if x and y are points of a topos T , there may exist nontrivial morphisms (of fun ...

... functor is the stalk functor E → Ex . But new phenomena occur. Deligne has constructed examples of toposes without points (he has also given criteria for the existence of “enough points”) [SGA 4 IV 7, VI 9]. Moreover, if x and y are points of a topos T , there may exist nontrivial morphisms (of fun ...

Topology Qual Winter 2000

... define functors G and H by G(X,A)=Hp(X,A), H(X,A)=Hp-1(X,A). Show that the map * is a natural transformation of G to H. Define and give an example (with proof) of a contravariant functor. 2. State and prove the Kunneth theorem for topological spaces. 3. a) Let F be the closed orientable surface of ...

... define functors G and H by G(X,A)=Hp(X,A), H(X,A)=Hp-1(X,A). Show that the map * is a natural transformation of G to H. Define and give an example (with proof) of a contravariant functor. 2. State and prove the Kunneth theorem for topological spaces. 3. a) Let F be the closed orientable surface of ...

Exercise Sheet 4

... (a) Prove that the sheaf of normal vector fields on S n−1 ⊂ Rn is isomorphic to the sheaf of functions C ∞ (−, R). (b) Give an example of a differentiable submanifold of codimension 1 where this does not hold. *2. Let X be a topological space and j : U ,→ X the embedding of an open subset. (a) Prove ...

... (a) Prove that the sheaf of normal vector fields on S n−1 ⊂ Rn is isomorphic to the sheaf of functions C ∞ (−, R). (b) Give an example of a differentiable submanifold of codimension 1 where this does not hold. *2. Let X be a topological space and j : U ,→ X the embedding of an open subset. (a) Prove ...

9. Sheaf Cohomology Definition 9.1. Let X be a topological space

... are functors H i from the category of sheaves of abelian groups on X to the category of abelian groups such that (1) H 0 (X, F) = Γ(X, F). (2) Given a short exact sequence, 0 −→ F −→ G −→ H −→ 0, there are coboundary maps H i (X, H) −→ H i+1 (X, F). which can be strung together to get a long exact s ...

... are functors H i from the category of sheaves of abelian groups on X to the category of abelian groups such that (1) H 0 (X, F) = Γ(X, F). (2) Given a short exact sequence, 0 −→ F −→ G −→ H −→ 0, there are coboundary maps H i (X, H) −→ H i+1 (X, F). which can be strung together to get a long exact s ...

0.1 A lemma of Kempf

... Theorem 1 (Serre). Let X = SpecA be an affine scheme, F a quasi-coherent sheaf. Then H i (X, F) = 0 for i ≥ 1. We shall prove this result following [?]. The idea is that X has a very nice basis: namely, the family of all sets D(f ), f ∈ A. These are themselves affine, and moreover the intersection o ...

... Theorem 1 (Serre). Let X = SpecA be an affine scheme, F a quasi-coherent sheaf. Then H i (X, F) = 0 for i ≥ 1. We shall prove this result following [?]. The idea is that X has a very nice basis: namely, the family of all sets D(f ), f ∈ A. These are themselves affine, and moreover the intersection o ...

Introduction to Sheaves

... 1. The Constant Presheaf; Let G be a abelian group and let F be the contravarient function from open sets of X to abeilan groups, such that F (U ) = G. 2. Real valued functions; Let O(U ) denote all functions f : U → R. These functions form a group under pointwise addition, and give the structure of ...

... 1. The Constant Presheaf; Let G be a abelian group and let F be the contravarient function from open sets of X to abeilan groups, such that F (U ) = G. 2. Real valued functions; Let O(U ) denote all functions f : U → R. These functions form a group under pointwise addition, and give the structure of ...

1. Let G be a sheaf of abelian groups on a topological space. In this

... 1. Let G be a sheaf of abelian groups on a topological space. In this problem, we define H 1 (X, G) as the set of isomorphism classes of G-torsors on X. Let F be a G-torsor, and F 0 be a sheaf of sets with an action by G. Recall that we defined F ×G F 0 as the sheafification of the presheaf that ass ...

... 1. Let G be a sheaf of abelian groups on a topological space. In this problem, we define H 1 (X, G) as the set of isomorphism classes of G-torsors on X. Let F be a G-torsor, and F 0 be a sheaf of sets with an action by G. Recall that we defined F ×G F 0 as the sheafification of the presheaf that ass ...

LECTURE NOTES 4: CECH COHOMOLOGY 1

... for q > 0. Our work on double complexes now easily implies the desired result. 8. Proof of Proposition 8.1 The comparison of singular and Cech cohomology depended on the following result. Let U be a cover of a space X. Let A be any abelian group, and let S ∗ be the presheaf on X given by the formula ...

... for q > 0. Our work on double complexes now easily implies the desired result. 8. Proof of Proposition 8.1 The comparison of singular and Cech cohomology depended on the following result. Let U be a cover of a space X. Let A be any abelian group, and let S ∗ be the presheaf on X given by the formula ...

Homework Set 3 Solutions are due Monday, November 9th.

... corresponds to the map induced by inclusion F → F 0 . Problem 5. Deduce from the previous problem that if f : F → G is a morphism of sheaves of abelian groups on X, then Im(f ) is canonically isomorphic to the subsheaf F 0 of G, where F 0 (U ) consists of those s ∈ G(U ) such that for all x ∈ X, the ...

... corresponds to the map induced by inclusion F → F 0 . Problem 5. Deduce from the previous problem that if f : F → G is a morphism of sheaves of abelian groups on X, then Im(f ) is canonically isomorphic to the subsheaf F 0 of G, where F 0 (U ) consists of those s ∈ G(U ) such that for all x ∈ X, the ...

NOTES ON GROTHENDIECK TOPOLOGIES 1

... The canonical topology on TG turns out to be the same as the one defined earlier. So U 7→ Hom(U, Z ) are all sheaves. It turns out these are all the sheaves of sets on TG . Proposition 4.2. There is an equivalence between the category of left G-sets and the category of sheaves of sets on TG . The fu ...

... The canonical topology on TG turns out to be the same as the one defined earlier. So U 7→ Hom(U, Z ) are all sheaves. It turns out these are all the sheaves of sets on TG . Proposition 4.2. There is an equivalence between the category of left G-sets and the category of sheaves of sets on TG . The fu ...

Complex Bordism (Lecture 5)

... 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 for the fibration S(ζ) → X degenerates and gives an identification E ∗ (X) ' E ∗+n−ζ (X), given by multiplicat ...

... 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 for the fibration S(ζ) → X degenerates and gives an identification E ∗ (X) ' E ∗+n−ζ (X), given by multiplicat ...

ON TAMAGAWA NUMBERS 1. Adele geometry Let X be an

... The adele geometry is the study of the pair (XA, XQ), together with the imbedding above. Therefore, one has to define all conceivable invariants of X in terms of the pair and study relations among them or connections with other invariants of X. The Tamagawa number x (X) is an example of such invaria ...

... The adele geometry is the study of the pair (XA, XQ), together with the imbedding above. Therefore, one has to define all conceivable invariants of X in terms of the pair and study relations among them or connections with other invariants of X. The Tamagawa number x (X) is an example of such invaria ...

PDF

... Definition 2. A subset A of an irreducible variety V /K is said to be a thin algebraic set (or thin set, or “mince” set) if it is a union of a finite number of subsets of type C1 and type C2 . ...

... Definition 2. A subset A of an irreducible variety V /K is said to be a thin algebraic set (or thin set, or “mince” set) if it is a union of a finite number of subsets of type C1 and type C2 . ...

RIGID RATIONAL HOMOTOPY THEORY AND

... Suppose that k is a finite field, and X{k is a geometrically connected variety (“ separated scheme of finite type). Question. Is there a way to ’do algebraic topology on X’ ? More specifically, we can ask if there are cohomology functors H i pXq which is some way behave like the singular cohomology ...

... Suppose that k is a finite field, and X{k is a geometrically connected variety (“ separated scheme of finite type). Question. Is there a way to ’do algebraic topology on X’ ? More specifically, we can ask if there are cohomology functors H i pXq which is some way behave like the singular cohomology ...

Generalities About Sheaves - Lehrstuhl B für Mathematik

... ker(ϕ) is a subsheaf of F. ϕ is injective ⇐⇒ ker(ϕ) = 0. The sheaf im(ϕ) is im(ϕ) := impresheaf (ϕ)+ ϕ is surjective ⇐⇒ im(ϕ) = G. ϕ is surjective ⇐⇒ ∀x ∈ X ...

... ker(ϕ) is a subsheaf of F. ϕ is injective ⇐⇒ ker(ϕ) = 0. The sheaf im(ϕ) is im(ϕ) := impresheaf (ϕ)+ ϕ is surjective ⇐⇒ im(ϕ) = G. ϕ is surjective ⇐⇒ ∀x ∈ X ...

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 ...

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 ...

Sheaf Cohomology 1. Computing by acyclic resolutions

... For the particular functor ‘take global sections’, other conditions on a sheaf still guarantee acyclicity and at the same time are ‘intrinsic’ in that they do not refer to any ‘ambient category’. Threfore, in principle these other conditions are more readily verifiable. For the moment, we merely cat ...

... For the particular functor ‘take global sections’, other conditions on a sheaf still guarantee acyclicity and at the same time are ‘intrinsic’ in that they do not refer to any ‘ambient category’. Threfore, in principle these other conditions are more readily verifiable. For the moment, we merely cat ...

THE GEOMETRY OF FORMAL VARIETIES IN ALGEBRAIC

... Shifting gears slightly, one of the classical presentations of infinitesimal elements are elements that “square to zero.” This turns out to be a fairly difficult thing to make both precise and analytically useful, which is why most analysts spend their time thinking about other things — but it’s a v ...

... Shifting gears slightly, one of the classical presentations of infinitesimal elements are elements that “square to zero.” This turns out to be a fairly difficult thing to make both precise and analytically useful, which is why most analysts spend their time thinking about other things — but it’s a v ...

Notes

... Proof. We construct a quasi-inverse. Let S be a continuous G-set. Let s 2 S be a point. Write H for the stabiliser of s. It is a open subgroup. By elementary group theory, we have an isomorphism of G-sets G=H ! Gs. This shows that S is the disjoint union of nite orbits. We map Gs to Spec(kH ), and ...

... Proof. We construct a quasi-inverse. Let S be a continuous G-set. Let s 2 S be a point. Write H for the stabiliser of s. It is a open subgroup. By elementary group theory, we have an isomorphism of G-sets G=H ! Gs. This shows that S is the disjoint union of nite orbits. We map Gs to Spec(kH ), and ...

PH Kropholler Olympia Talelli

... of the coinduced module CoindyH and contains G-fixed points in the form of the constant functions. The action of G is given by @(g’) = 4( gg’). We prove that Z is ZF-free for each finite F using a result of Nobeling that [s2, Z] is free abelian for any set 0; see [ 1, Corollary 97.41. Let F be a fin ...

... of the coinduced module CoindyH and contains G-fixed points in the form of the constant functions. The action of G is given by @(g’) = 4( gg’). We prove that Z is ZF-free for each finite F using a result of Nobeling that [s2, Z] is free abelian for any set 0; see [ 1, Corollary 97.41. Let F be a fin ...

Algebraic Geometry

... geometric object to every commutative ring. Unfortunately, A 7! spm A is not functorial in this generality: if 'W A ! B is a homomorphism of rings, then ' 1 .m/ for m maximal need not be maximal — consider for example the inclusion Z ,! Q. Thus he was forced to replace spm.A/ with spec.A/, the set o ...

... geometric object to every commutative ring. Unfortunately, A 7! spm A is not functorial in this generality: if 'W A ! B is a homomorphism of rings, then ' 1 .m/ for m maximal need not be maximal — consider for example the inclusion Z ,! Q. Thus he was forced to replace spm.A/ with spec.A/, the set o ...