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 ...
Generalities About Sheaves - Lehrstuhl B für Mathematik
... Let {Vi } be an open covering of U ⊆ X (open). A presheaf F on X is a sheaf if for all i: s ∈ F(U ) and s|Vi = 0 then s = 0, given si ∈ F(Vi ) that match on the overlaps: si |Vi ∩Vj = sj |Vi ∩Vj there is a unique s ∈ F(U ) with s|Vi = si . Sheaves are defined by local data. ...
... Let {Vi } be an open covering of U ⊆ X (open). A presheaf F on X is a sheaf if for all i: s ∈ F(U ) and s|Vi = 0 then s = 0, given si ∈ F(Vi ) that match on the overlaps: si |Vi ∩Vj = sj |Vi ∩Vj there is a unique s ∈ F(U ) with s|Vi = si . Sheaves are defined by local data. ...
Exercise Sheet 4 - D-MATH
... Let TX1 be the topology given by TX1 “ tU Ď X1 | f1 pU q is open in Y1 u Observe that a) f1 is a topological sheaf and the maps φ˘ :“ f1 |R0 Yt0˘ u define a smooth atlas on X1 , but the induced topology is not Hausdorff. b) More generally, any topological sheaf f : X Ñ Rn automatically acquires a s ...
... Let TX1 be the topology given by TX1 “ tU Ď X1 | f1 pU q is open in Y1 u Observe that a) f1 is a topological sheaf and the maps φ˘ :“ f1 |R0 Yt0˘ u define a smooth atlas on X1 , but the induced topology is not Hausdorff. b) More generally, any topological sheaf f : X Ñ Rn automatically acquires a s ...
LECTURE NOTES 4: CECH COHOMOLOGY 1
... F : Open(X)op → C. More generally, if E is a category, then a presheaf on E with values in C is just a contravariant functor F : E op → C. Example 1.2. Suppose that X and T are topological spaces. If U is an open set, let T (U ) = map(U, T ) be the set of continuous functions from U to T . For f ∈ T ...
... F : Open(X)op → C. More generally, if E is a category, then a presheaf on E with values in C is just a contravariant functor F : E op → C. Example 1.2. Suppose that X and T are topological spaces. If U is an open set, let T (U ) = map(U, T ) be the set of continuous functions from U to T . For f ∈ T ...
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 ...
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... construct a stable homotopy theory, so that the homotopy category of spectra is canonically defined in the classical manner. Therefore, for any given construction of an Ω–spectrum one is able to canonically define an associated cohomology theory; thus, one defines the cohomology groups of a CW-compl ...
... construct a stable homotopy theory, so that the homotopy category of spectra is canonically defined in the classical manner. Therefore, for any given construction of an Ω–spectrum one is able to canonically define an associated cohomology theory; thus, one defines the cohomology groups of a CW-compl ...
OX(D) (or O(D)) for a Cartier divisor D on a scheme X (1) on
... The construction of O(1) actually makes sense starting from any graded ring (so one denes O(1) not just for projective space), and what's essential is the notion of degree. But in full generality, I think that there is no map OX → O(1). See the dierences with OX (D) above. OPnk (H) ' OPnk (1) on p ...
... The construction of O(1) actually makes sense starting from any graded ring (so one denes O(1) not just for projective space), and what's essential is the notion of degree. But in full generality, I think that there is no map OX → O(1). See the dierences with OX (D) above. OPnk (H) ' OPnk (1) on p ...
0.1 A lemma of Kempf
... have assumed inductively that the result is valid for n − 1. So α maps to some β ∈ H n−1 (X, H); this means there is an open cover of X by various V ∈ A such that β maps to zero in H n−1 (X, V H). This means that α maps to zero in these H n (X, V F) by naturality. This completes the proof of the ind ...
... have assumed inductively that the result is valid for n − 1. So α maps to some β ∈ H n−1 (X, H); this means there is an open cover of X by various V ∈ A such that β maps to zero in H n−1 (X, V H). This means that α maps to zero in these H n (X, V F) by naturality. This completes the proof of the ind ...
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 ...
LECTURE 21 - SHEAF THEORY II 1. Stalks
... are neighborhoods of x and ρV U is the restriction map, then τ U = τ V ◦ ρV U . (3) To simplify notation, we will usually use the symbol sx to denote [(s, U )]∼ . Examples 1.5. (1) Let A be an abelian group and let F be a constant presheaf with value A. Then Fx ∼ = A for every x. Indeed, if U is any ...
... are neighborhoods of x and ρV U is the restriction map, then τ U = τ V ◦ ρV U . (3) To simplify notation, we will usually use the symbol sx to denote [(s, U )]∼ . Examples 1.5. (1) Let A be an abelian group and let F be a constant presheaf with value A. Then Fx ∼ = A for every x. Indeed, if U is any ...
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 ...
Exercise Sheet 4
... 1. Let X ⊂ Rn be a differentiable submanifold. Let F be the sheaf of normal vector fields on X, i.e., of C ∞ -functions X → Rn whose values at each x ∈ X are orthogonal to the tangent space TX,x . (a) Prove that the sheaf of normal vector fields on S n−1 ⊂ Rn is isomorphic to the sheaf of functions ...
... 1. Let X ⊂ Rn be a differentiable submanifold. Let F be the sheaf of normal vector fields on X, i.e., of C ∞ -functions X → Rn whose values at each x ∈ X are orthogonal to the tangent space TX,x . (a) Prove that the sheaf of normal vector fields on S n−1 ⊂ Rn is isomorphic to the sheaf of functions ...
Homework Set 3 Solutions are due Monday, November 9th.
... Problem 4. Let X be a topological space. i) Suppose that F is a presheaf on X that satisfies the first presheaf axiom, that is, for every open cover U = ∪i Ui of an open subset U of X, and for every s, t ∈ F(U ) such that s|Ui = t|Ui for all i, we have s = t. Show that if θ : F → F + is the canonica ...
... Problem 4. Let X be a topological space. i) Suppose that F is a presheaf on X that satisfies the first presheaf axiom, that is, for every open cover U = ∪i Ui of an open subset U of X, and for every s, t ∈ F(U ) such that s|Ui = t|Ui for all i, we have s = t. Show that if θ : F → F + is the canonica ...
Universal spaces in birational geometry
... Universal spaces in birational geometry — Fedor Bogomolov, October 8, 2010 I want to discuss our joint results with Yuri Tschinkel. The Bloch-Kato conjecture implies that cohomology elements with finite constant coefficients of an algebraic variety can be induced from abelian quotient of the fundame ...
... Universal spaces in birational geometry — Fedor Bogomolov, October 8, 2010 I want to discuss our joint results with Yuri Tschinkel. The Bloch-Kato conjecture implies that cohomology elements with finite constant coefficients of an algebraic variety can be induced from abelian quotient of the fundame ...