MA3056: Exercise Sheet 2 — Topological Spaces
... T = {∅} ∪ {U ⊂ X : X \ U is finite}. Show that T is indeed a topology, but that it is not Hausdorff. 17. Prove the following: (i) Any subspace of a Hausdorff space is Hausdorff. (ii) The product of two Hausdorff spaces is Hausdorff. (iii) If f : X → Y is continuous and injective, and Y Hausdorff, th ...
... T = {∅} ∪ {U ⊂ X : X \ U is finite}. Show that T is indeed a topology, but that it is not Hausdorff. 17. Prove the following: (i) Any subspace of a Hausdorff space is Hausdorff. (ii) The product of two Hausdorff spaces is Hausdorff. (iii) If f : X → Y is continuous and injective, and Y Hausdorff, th ...
Section 3.2 - Cohomology of Sheaves
... the canonical natural equivalence HA,I (X, −) = HA,I 0 (X, −). This means that the A-module structure induced on H i (X, F ) is independent of the choice of resolutions on Mod(X). Definition 5. Let (X, OX ) be a ringed space and A = Γ(X, OX ). Fix an assignment of injective resolutions J to the obje ...
... the canonical natural equivalence HA,I (X, −) = HA,I 0 (X, −). This means that the A-module structure induced on H i (X, F ) is independent of the choice of resolutions on Mod(X). Definition 5. Let (X, OX ) be a ringed space and A = Γ(X, OX ). Fix an assignment of injective resolutions J to the obje ...
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... We can consider C as a Riemann surface by simply recalling that its topological structure is that of R2 and giving it the complex structure associated to the atlas of identity maps. Taking X, Y = C in Definition 2.3.1 simply gives us that f : C → C is holomorphic as a map of Riemann surfaces exactl ...
... We can consider C as a Riemann surface by simply recalling that its topological structure is that of R2 and giving it the complex structure associated to the atlas of identity maps. Taking X, Y = C in Definition 2.3.1 simply gives us that f : C → C is holomorphic as a map of Riemann surfaces exactl ...
Complex Spaces
... Analytic algebras are factor algebras of On by proper ideals: 4.1 Definition. An analytic algebra A is a C-algebra which is different from AnAlgo the zero algebra and such there exist an n and a surjective algebra homomorphism On → A. A ring R is called a local ring if it is not the zero ring and if ...
... Analytic algebras are factor algebras of On by proper ideals: 4.1 Definition. An analytic algebra A is a C-algebra which is different from AnAlgo the zero algebra and such there exist an n and a surjective algebra homomorphism On → A. A ring R is called a local ring if it is not the zero ring and if ...
introduction to algebraic topology and algebraic geometry
... These notes assemble the contents of the introductory courses I have been giving at SISSA since 1995/96. Originally the course was intended as introduction to (complex) algebraic geometry for students with an education in theoretical physics, to help them to master the basic algebraic geometric tool ...
... These notes assemble the contents of the introductory courses I have been giving at SISSA since 1995/96. Originally the course was intended as introduction to (complex) algebraic geometry for students with an education in theoretical physics, to help them to master the basic algebraic geometric tool ...
A natural localization of Hardy spaces in several complex variables
... Here Hp denotes the pth homology space. The Tor-spaces are equipped with the quotient locally convex topology, which may not be Hausdorff. As in the algebraic case, any (C-split, topologically-) free resolution can be used to compute these spaces. d The characteristic property of the Tor-functor is ...
... Here Hp denotes the pth homology space. The Tor-spaces are equipped with the quotient locally convex topology, which may not be Hausdorff. As in the algebraic case, any (C-split, topologically-) free resolution can be used to compute these spaces. d The characteristic property of the Tor-functor is ...
Variations on the Bloch
... 2.1 Notation. In the present paper all schemes are assumed to be Noetherian and separated. By k we denote a fixed ground field. A variety over k is an integral scheme of finite type over k. To simplify the notation sometimes we will write k instead of the scheme Spec k. We will write X1 × X2 for the ...
... 2.1 Notation. In the present paper all schemes are assumed to be Noetherian and separated. By k we denote a fixed ground field. A variety over k is an integral scheme of finite type over k. To simplify the notation sometimes we will write k instead of the scheme Spec k. We will write X1 × X2 for the ...
13 Lecture 13: Uniformity and sheaf properties
... trivial valuation on Fp ) and fiber over the generic point identified with the Rieman-Zariski space Spv(Q, Z) = Spec(Z) under which the generic point corresponds to the trivial valuation and (p) corresponds to the p-adic valuation ring Z(p) in Q. It should not be a surprise that Theorem 13.3.1(i) is ...
... trivial valuation on Fp ) and fiber over the generic point identified with the Rieman-Zariski space Spv(Q, Z) = Spec(Z) under which the generic point corresponds to the trivial valuation and (p) corresponds to the p-adic valuation ring Z(p) in Q. It should not be a surprise that Theorem 13.3.1(i) is ...
Algebraic Topology Introduction
... P can extend f to a continuous map on the underlying spaces g : |K| → |L| defined: x= ti vi ⇒ g(x) = ti f (vi ). This map is linear on the vertex set and is called a simplicial map. In the case that f is a bijection, and a set of vertices forms a simplex in K if and only if the image of the vertices ...
... P can extend f to a continuous map on the underlying spaces g : |K| → |L| defined: x= ti vi ⇒ g(x) = ti f (vi ). This map is linear on the vertex set and is called a simplicial map. In the case that f is a bijection, and a set of vertices forms a simplex in K if and only if the image of the vertices ...
Affine Varieties
... as a category with regular maps as the morphisms, then Proposition 3.5 shows that this is the same as the category of domains C[x1 , ..., xn ]/P with C-algebra homomorphisms (going in the opposite direction!) via X 7→ C[X] and Φ 7→ Φ∗ . That is, any property of affine varieties translates into an eq ...
... as a category with regular maps as the morphisms, then Proposition 3.5 shows that this is the same as the category of domains C[x1 , ..., xn ]/P with C-algebra homomorphisms (going in the opposite direction!) via X 7→ C[X] and Φ 7→ Φ∗ . That is, any property of affine varieties translates into an eq ...
THE HITCHIN FIBRATION Here X is a smooth connected projective
... regard as a homogeneous space over × with acting by right translations and by left translations. This makes an automorphism group of , when the latter is regarded as a torsor over itself. If is a variety with × -action isomorphic to , then two × -equivariant isomorphisms ...
... regard as a homogeneous space over × with acting by right translations and by left translations. This makes an automorphism group of , when the latter is regarded as a torsor over itself. If is a variety with × -action isomorphic to , then two × -equivariant isomorphisms ...
1. Lecture 4, February 21 1.1. Open immersion. Let (X,O X) be a
... Proj(S) are the open sets D+ (f ), with f ∈ S+ . On each open we define the ring S(f ) . By Lemma 1.8 we have exactness on basic opens, and therefore there is a unique sheaf of rings OX on X, such that OX (D+ (f )) = S(f ) . Now we have that X = Proj(S) is a scheme, where the structure sheaf OX (D+ ...
... Proj(S) are the open sets D+ (f ), with f ∈ S+ . On each open we define the ring S(f ) . By Lemma 1.8 we have exactness on basic opens, and therefore there is a unique sheaf of rings OX on X, such that OX (D+ (f )) = S(f ) . Now we have that X = Proj(S) is a scheme, where the structure sheaf OX (D+ ...
The support of local cohomology modules
... The aim of this paper is to describe the support of local cohomology modules in prime characteristic. Specifically, we first study the support of F finite F -modules over a regular ring R and show a computationally feasible method for computing these without the need to compute generating roots. To ...
... The aim of this paper is to describe the support of local cohomology modules in prime characteristic. Specifically, we first study the support of F finite F -modules over a regular ring R and show a computationally feasible method for computing these without the need to compute generating roots. To ...
cohomology detects failures of the axiom of choice
... choice. Our main theorem asserts that, if the first cohomology of every discrete space vanishes, for all coefficient groups, then the axiom of choice holds; in other words, cohomology (in dimension 1) is adequate for detecting failures of choice. This theorem improves a result of Diaconescu [1], who ...
... choice. Our main theorem asserts that, if the first cohomology of every discrete space vanishes, for all coefficient groups, then the axiom of choice holds; in other words, cohomology (in dimension 1) is adequate for detecting failures of choice. This theorem improves a result of Diaconescu [1], who ...
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 7 Contents
... how they glue together. The affine open U0 has coordinates (a1 , . . . , an ), and the affine open U1 has coordinates (b0 , b2 , . . . , bn ); the open subset of U0 corresponding to a1 6= 0 is glued to the open subset of U1 corresponding to b0 6= 0, and the identification is via: (1; a1 ; a2 ; . . . ...
... how they glue together. The affine open U0 has coordinates (a1 , . . . , an ), and the affine open U1 has coordinates (b0 , b2 , . . . , bn ); the open subset of U0 corresponding to a1 6= 0 is glued to the open subset of U1 corresponding to b0 6= 0, and the identification is via: (1; a1 ; a2 ; . . . ...
3. Sheaves of groups and rings.
... maps the maximal ideal in ψ ∗ (B)x = Bψ(x) to the maximal ideal in Ax for all x ∈ X. (3.10) Remark. The ringed spaces with morphism form a category, as does the locally ringed spaces with local homomorphisms. ...
... maps the maximal ideal in ψ ∗ (B)x = Bψ(x) to the maximal ideal in Ax for all x ∈ X. (3.10) Remark. The ringed spaces with morphism form a category, as does the locally ringed spaces with local homomorphisms. ...
18. Fibre products of schemes The main result of this section is
... every point, except the origin. Note that this is like the classical construction of a topological space, which is locally a manifold, but which is not Hausdorff. Of course no scheme is ever Hausdorff (apart from the most trivial examples) and it turns out that there is an appropriate condition for ...
... every point, except the origin. Note that this is like the classical construction of a topological space, which is locally a manifold, but which is not Hausdorff. Of course no scheme is ever Hausdorff (apart from the most trivial examples) and it turns out that there is an appropriate condition for ...
2. Direct and inverse images.
... U0 = idZ , and (ρG )Ui the projection on the i’the factor, for i = 1, 2. It is clear that the map u : F → G given by uX : Z → Z ⊕ Z with uX (n) = (n, n), by u∅ = id{0} , and where uUi = idZ for i = 0, 1, 2, is a homomorphism of sheaves. We have that H(∅) = {0}, H(X) is isomorphic to Z, H(U0 ) = {0}, ...
... U0 = idZ , and (ρG )Ui the projection on the i’the factor, for i = 1, 2. It is clear that the map u : F → G given by uX : Z → Z ⊕ Z with uX (n) = (n, n), by u∅ = id{0} , and where uUi = idZ for i = 0, 1, 2, is a homomorphism of sheaves. We have that H(∅) = {0}, H(X) is isomorphic to Z, H(U0 ) = {0}, ...
A quasi-coherent sheaf of notes
... G(f −1 (V )) to build the morphism of sheaves. There exists an adjoint map f −1 . Proposition 1.3. Let C be the category of abelian groups or rings. Let F ∈ Sh(Y, C). Then the functor G → HomSh(X,C) (G, f∗ F) is corepresentable. In particular f∗ admits an adjoint, and f −1 F exists. Proof. The proof ...
... G(f −1 (V )) to build the morphism of sheaves. There exists an adjoint map f −1 . Proposition 1.3. Let C be the category of abelian groups or rings. Let F ∈ Sh(Y, C). Then the functor G → HomSh(X,C) (G, f∗ F) is corepresentable. In particular f∗ admits an adjoint, and f −1 F exists. Proof. The proof ...
1. Scheme A ringed space is a pair (X,OX), where X is a topological
... A ringed space is a pair (X, OX ), where X is a topological space X and OX is a sheaf of rings on X. We call X and OX the underlying topological space and the structure sheaf of (X, OX ) respectively. Example 1.1. Let X be any topological space and Z be the ring of integers with the discrete topolog ...
... A ringed space is a pair (X, OX ), where X is a topological space X and OX is a sheaf of rings on X. We call X and OX the underlying topological space and the structure sheaf of (X, OX ) respectively. Example 1.1. Let X be any topological space and Z be the ring of integers with the discrete topolog ...
Dualities in Mathematics: Locally compact abelian groups
... Let G be a locally compact group. Then G admits a left Haar measure. This measure is unique up to an overall factor. • G has a left Haar measure iff it has a right Haar measure. • A non-zero Haar measure is positive on all open sets. • µ(G) is finite iff G is compact. ...
... Let G be a locally compact group. Then G admits a left Haar measure. This measure is unique up to an overall factor. • G has a left Haar measure iff it has a right Haar measure. • A non-zero Haar measure is positive on all open sets. • µ(G) is finite iff G is compact. ...
The Weil-étale topology for number rings
... Proof. We show first that the cohomological dimension of a profinite space X is zero. To do this it suffices to show (by using alternating cochains) that every open cover has a refinement by a disjoint cover. It is immediate that X has a base for its topology consisting of sets Ui that are both open ...
... Proof. We show first that the cohomological dimension of a profinite space X is zero. To do this it suffices to show (by using alternating cochains) that every open cover has a refinement by a disjoint cover. It is immediate that X has a base for its topology consisting of sets Ui that are both open ...
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 ...
Mini-course on K3 surfaces Antonio Laface Universidad de
... two generators [F1 ] and [F2 ] have intersections Fi · Fj = δij . Hence the Picard lattice of X is represented by the Gram matrix ...
... two generators [F1 ] and [F2 ] have intersections Fi · Fj = δij . Hence the Picard lattice of X is represented by the Gram matrix ...
Cell-Like Maps (Lecture 5)
... Remark 18. Definition 15 is perhaps only appropriate when the topological space X is paracompact; for many purposes, it is the conclusions of Proposition 17 which are important. For us, this will be irrelevant: we will only be interested in the case where X is a compact Hausdorff space. Definition ...
... Remark 18. Definition 15 is perhaps only appropriate when the topological space X is paracompact; for many purposes, it is the conclusions of Proposition 17 which are important. For us, this will be irrelevant: we will only be interested in the case where X is a compact Hausdorff space. Definition ...