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... A regular space is a topological space in which points and closed sets can be separated by open sets; in other words, given a closed set A and a point x ∈ / A, there are disjoint open sets U and V such that x ∈ U and A ⊆ V . A T3 space is a regular T0 -space. A T3 space is necessarily also T2 , that ...
... A regular space is a topological space in which points and closed sets can be separated by open sets; in other words, given a closed set A and a point x ∈ / A, there are disjoint open sets U and V such that x ∈ U and A ⊆ V . A T3 space is a regular T0 -space. A T3 space is necessarily also T2 , that ...
0.1 A lemma of Kempf
... for all 0 < i < n and U ∈ A. Suppose α ∈ H n (X, F). Then there is a covering of X by open sets V ∈ A such that the image of α in H n (X, V F) is zero for each V . Proof. We will prove this result by induction on n. First, suppose n > 1, and that the result is valid for n − 1. The base case will be ...
... for all 0 < i < n and U ∈ A. Suppose α ∈ H n (X, F). Then there is a covering of X by open sets V ∈ A such that the image of α in H n (X, V F) is zero for each V . Proof. We will prove this result by induction on n. First, suppose n > 1, and that the result is valid for n − 1. The base case will be ...
A BRIEF INTRODUCTION TO SHEAVES References 1. Presheaves
... A ringed space (X, O) is a local ringed space if each stalk Ox (x ∈ X) is a local ring with maximal ideal mX ; a morphism (X, OX ) −→ (Y, OY ) of local ringed spaces is one where on each stalk OY,f (x) −→ f∗ OX,x is a homomorphism of local rings (i.e., a ring homomorphism h : R −→ S for which h−1 mS ...
... A ringed space (X, O) is a local ringed space if each stalk Ox (x ∈ X) is a local ring with maximal ideal mX ; a morphism (X, OX ) −→ (Y, OY ) of local ringed spaces is one where on each stalk OY,f (x) −→ f∗ OX,x is a homomorphism of local rings (i.e., a ring homomorphism h : R −→ S for which h−1 mS ...
Notes
... 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 is often used when U is considered fixed and F is allowed to vary. Example 1 ...
... 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 is often used when U is considered fixed and F is allowed to vary. Example 1 ...
