Sec 5: Affine schemes

... Recall that this is the same condition which defines sections of o⌃ over U ✓ ⌃: (x) must be given by the same rational function fg 2 K(⌃) for all x 2 U . In the case X = SpecZ, the field is K(X) = Q. So: Rn = (Xn ) ✓ K(X) = Q which we already knew. But now we should realize that it means: For each s ...

... Recall that this is the same condition which defines sections of o⌃ over U ✓ ⌃: (x) must be given by the same rational function fg 2 K(⌃) for all x 2 U . In the case X = SpecZ, the field is K(X) = Q. So: Rn = (Xn ) ✓ K(X) = Q which we already knew. But now we should realize that it means: For each s ...

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 ...

Cohomology of Categorical Self-Distributivity

... A quandle, X, is a set with a binary operation (a, b) 7→ a / b such that (I) For any a ∈ X, a / a = a. (II) For any a, b ∈ X, there is a unique c ∈ X such that a = c / b. (III) For any a, b, c ∈ X, we have (a / b) / c = (a / c) / (b / c). A rack is a set with a binary operation that satisfies (II) a ...

... A quandle, X, is a set with a binary operation (a, b) 7→ a / b such that (I) For any a ∈ X, a / a = a. (II) For any a, b ∈ X, there is a unique c ∈ X such that a = c / b. (III) For any a, b, c ∈ X, we have (a / b) / c = (a / c) / (b / c). A rack is a set with a binary operation that satisfies (II) a ...

Workshop on group schemes and p-divisible groups: Homework 1. 1

... (ii) If Gk is smooth, prove that for any subgroup Γ ⊆ G(k) the Zariski closure of Γ in G is a closed k-subgroup of G whose formation commutes with extension of the base field. (iii) If k is perfect, prove that Gred is a (closed) k-subgroup of G. Can you find a counterexample if k is not perfect? For ...

... (ii) If Gk is smooth, prove that for any subgroup Γ ⊆ G(k) the Zariski closure of Γ in G is a closed k-subgroup of G whose formation commutes with extension of the base field. (iii) If k is perfect, prove that Gred is a (closed) k-subgroup of G. Can you find a counterexample if k is not perfect? For ...

Profinite Groups - Universiteit Leiden

... where now each Gal(M/K) is a finite group. The set of such M is a partially ordered set by inclusion. We may take the composed field of two subfields, which is again finite, so this set is directed. For M ⊃ M 0 we have restriction maps Gal(M/K) → Gal(M 0 /K), so we have a projective system. Many the ...

... where now each Gal(M/K) is a finite group. The set of such M is a partially ordered set by inclusion. We may take the composed field of two subfields, which is again finite, so this set is directed. For M ⊃ M 0 we have restriction maps Gal(M/K) → Gal(M 0 /K), so we have a projective system. Many the ...

Finitely generated abelian groups, abelian categories

... Hartshorne’s “Algebraic Geometry” or ”Homological Algebra” by Hilton and Stammbach. Our first definition of abelian categories then becomes a Theorem of Peter Freyd. One requires first that M orA(C, D) is an abelian group for all objects C and D of A. If f ∈ M orA(C, D), the kernel of f can be defin ...

... Hartshorne’s “Algebraic Geometry” or ”Homological Algebra” by Hilton and Stammbach. Our first definition of abelian categories then becomes a Theorem of Peter Freyd. One requires first that M orA(C, D) is an abelian group for all objects C and D of A. If f ∈ M orA(C, D), the kernel of f can be defin ...

13 Lecture 13: Uniformity and sheaf properties

... unique possible way to extend the definition of a Huber pair associated to a rational domain to all open sets compatibly with restriction maps. Thus for any presheaf F that is adapted to a basis B on a topological space, F is a sheaf if and only if it is a sheaf when restricted to B; for the purpose ...

... unique possible way to extend the definition of a Huber pair associated to a rational domain to all open sets compatibly with restriction maps. Thus for any presheaf F that is adapted to a basis B on a topological space, F is a sheaf if and only if it is a sheaf when restricted to B; for the purpose ...

Section 3.2 - Cohomology of Sheaves

... 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 objects of Ab(X). Then for any sheaf of OX -modules F and i ≥ 0 the cohomology group H i (X, F ) ...

... 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 objects of Ab(X). Then for any sheaf of OX -modules F and i ≥ 0 the cohomology group H i (X, F ) ...

2. Basic notions of algebraic groups Now we are ready to introduce

... are disjoint, hence coincide with the connected components of the topological space G. Because of this, people usually talk just about connected components of an algebraic group (not irreducible components). The normal subgroup G0 is referred to as the identity component of G. Also note at this poin ...

... are disjoint, hence coincide with the connected components of the topological space G. Because of this, people usually talk just about connected components of an algebraic group (not irreducible components). The normal subgroup G0 is referred to as the identity component of G. Also note at this poin ...

Orbifolds and their cohomology.

... Example 1.0.5. Let G be a finite group. Then one can form an orbifold BG := [•/G] by allowing G to act trivially on a point. In terms of groupoids, this is the category with one object and morphisms given by G. Example 1.0.6. In the definition of weighted projective space given in Example 1.0.3, all ...

... Example 1.0.5. Let G be a finite group. Then one can form an orbifold BG := [•/G] by allowing G to act trivially on a point. In terms of groupoids, this is the category with one object and morphisms given by G. Example 1.0.6. In the definition of weighted projective space given in Example 1.0.3, all ...

Manifolds and Varieties via Sheaves

... diagonal ∆ ⊂ X × X is closed. Clearly affine spaces are varieties in this sense. Projective spaces can also be seen to be varieties. Further examples can be obtained by taking open or closed subvarieties of these examples. Let (X, OX ) be an algebraic variety over k. A closed irreducible subset Y ⊂ ...

... diagonal ∆ ⊂ X × X is closed. Clearly affine spaces are varieties in this sense. Projective spaces can also be seen to be varieties. Further examples can be obtained by taking open or closed subvarieties of these examples. Let (X, OX ) be an algebraic variety over k. A closed irreducible subset Y ⊂ ...

Cobordism of pairs

... 4. Application of THOMtheory I t follows from the work of T~oM [13], (which we shall suppose known), t h a t cobordism groups for the structural group G, are given by the homotopy groups of M(Gn). Now in all the cases with which we shall be concerned, {Gn} satisfies a certain stability condition. We ...

... 4. Application of THOMtheory I t follows from the work of T~oM [13], (which we shall suppose known), t h a t cobordism groups for the structural group G, are given by the homotopy groups of M(Gn). Now in all the cases with which we shall be concerned, {Gn} satisfies a certain stability condition. We ...

Intersection homology

... Our goal in this talk is to define the most basic ingredients in this formula: the intersection (co)homology complexes ICX . These notes roughly follow the book [2] of Kirwan and Woolf. Intersection homology was invented by Goresky and MacPherson in the 1970s; the goal was to produce a homology the ...

... Our goal in this talk is to define the most basic ingredients in this formula: the intersection (co)homology complexes ICX . These notes roughly follow the book [2] of Kirwan and Woolf. Intersection homology was invented by Goresky and MacPherson in the 1970s; the goal was to produce a homology the ...

HOMOTOPICAL ENHANCEMENTS OF CYCLE CLASS MAPS 1

... S 7→ ([n] 7→ Hom(∆n , S)). This allows us to define the homotopy groups of a simplicial set, via πn (X• ) := πn (|X|). The sense in which simplicial sets are a good model for topological spaces is: if we define weak equiv−1 alences to be maps f so that πn (f ) is an isomorphism for all n, then sSet[ ...

... S 7→ ([n] 7→ Hom(∆n , S)). This allows us to define the homotopy groups of a simplicial set, via πn (X• ) := πn (|X|). The sense in which simplicial sets are a good model for topological spaces is: if we define weak equiv−1 alences to be maps f so that πn (f ) is an isomorphism for all n, then sSet[ ...

Cohomology of cyro-electron microscopy

... There has been much significant progress toward this goal in recent years [29, 35, 38, 41, 42]. Our article attempts to understand cryo-EM datasets of 2D images via Čech and singular cohomology groups. We will see that for a given molecule, the information extracted from its 2D cryo-EM images deter ...

... There has been much significant progress toward this goal in recent years [29, 35, 38, 41, 42]. Our article attempts to understand cryo-EM datasets of 2D images via Čech and singular cohomology groups. We will see that for a given molecule, the information extracted from its 2D cryo-EM images deter ...

Simplicial Objects and Singular Homology

... however, singular homology theory was introduced as a generalisation of simplicial homology theory. Simplicial homology groups are easy to understand and compute, however, they are defined only for particular triangulable spaces. Singular homology groups however are defined for arbitrary topological ...

... however, singular homology theory was introduced as a generalisation of simplicial homology theory. Simplicial homology groups are easy to understand and compute, however, they are defined only for particular triangulable spaces. Singular homology groups however are defined for arbitrary topological ...

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... Assume that G is a group scheme of finite type over k , acting on a quasi-separated k -scheme X , with a non-empty invariant open subscheme on which the action is free. Let E be a G-equivariant locally free sheaf of rank r on X . Then there exists a non-empty open G-invariant subscheme U of X , such ...

... Assume that G is a group scheme of finite type over k , acting on a quasi-separated k -scheme X , with a non-empty invariant open subscheme on which the action is free. Let E be a G-equivariant locally free sheaf of rank r on X . Then there exists a non-empty open G-invariant subscheme U of X , such ...

THE BRAUER GROUP: A SURVEY Introduction Notation

... space of dimension n − 1). Of course, there are those projective bundles which come from trivial bundles, namely projectivizations of vector bundles. We wish to consider such bundles as trivial. The cohomology set H 1 (X, GLn ) is in bijective correspondence with vector bundles. There is a map H 1 ( ...

... space of dimension n − 1). Of course, there are those projective bundles which come from trivial bundles, namely projectivizations of vector bundles. We wish to consider such bundles as trivial. The cohomology set H 1 (X, GLn ) is in bijective correspondence with vector bundles. There is a map H 1 ( ...

Boundary manifolds of projective hypersurfaces Daniel C. Cohen Alexander I. Suciu

... X = CPℓ \ V , as the primary object of study. And perhaps the most thorough is to study the Milnor fibration f : Cℓ+1 \ {f (x) = 0} → C∗ . Of course, the different approaches are interrelated. For example, if the degree of f is n, then the Milnor fiber F = f −1 (1) is a cyclic n-fold cover of X. Con ...

... X = CPℓ \ V , as the primary object of study. And perhaps the most thorough is to study the Milnor fibration f : Cℓ+1 \ {f (x) = 0} → C∗ . Of course, the different approaches are interrelated. For example, if the degree of f is n, then the Milnor fiber F = f −1 (1) is a cyclic n-fold cover of X. Con ...

Very dense subsets of a topological space.

... X is very dense in X; that is, if X0 ,→ X is a quasi-homeomorphism. Proposition (10.3.2). — Let X be Jacobson, Z ⊆ X locally quasi-constructible. Then the subspace Z is Jacobson, and a point z ∈ Z is closed in Z iff it is closed in X. S Proposition (10.3.3). — Let X = α Uα be an open covering. Then ...

... X is very dense in X; that is, if X0 ,→ X is a quasi-homeomorphism. Proposition (10.3.2). — Let X be Jacobson, Z ⊆ X locally quasi-constructible. Then the subspace Z is Jacobson, and a point z ∈ Z is closed in Z iff it is closed in X. S Proposition (10.3.3). — Let X = α Uα be an open covering. Then ...

arXiv:math/0302340v2 [math.AG] 7 Sep 2003

... of mixed Hodge structure is the weight filtration. We will focus on the dual filtration in homology. We will find a relation between intersection homology defined by M. Goresky–R. MacPherson and the weight filtration. The main result of the paper says that if a variety is complete, then the pure par ...

... of mixed Hodge structure is the weight filtration. We will focus on the dual filtration in homology. We will find a relation between intersection homology defined by M. Goresky–R. MacPherson and the weight filtration. The main result of the paper says that if a variety is complete, then the pure par ...

introduction to algebraic topology and algebraic geometry

... algebraic topology, especially the singular homology of topological spaces. The future developments we have in mind are the applications to algebraic geometry, but also students interested in modern theoretical physics may find here useful material (e.g., the theory of spectral sequences). As its na ...

... algebraic topology, especially the singular homology of topological spaces. The future developments we have in mind are the applications to algebraic geometry, but also students interested in modern theoretical physics may find here useful material (e.g., the theory of spectral sequences). As its na ...

Spencer Bloch: The proof of the Mordell Conjecture

... on several c o u n t s . For o n e t h i n g , the s o l u t i o n set m i s s e s " p o i n t s at i n f i n i t y " . To avoid h a v i n g s o m e Probably most mathematicians w o u l d have agreed fiend stash all the goodies out at infinity where we with Weil (certainly I would have), until earli ...

... on several c o u n t s . For o n e t h i n g , the s o l u t i o n set m i s s e s " p o i n t s at i n f i n i t y " . To avoid h a v i n g s o m e Probably most mathematicians w o u l d have agreed fiend stash all the goodies out at infinity where we with Weil (certainly I would have), until earli ...

here

... followed by more fallow periods. The first active period was in the mid 1960’s, when Tate, himself, proved that for an abelian variety A over a finite field F of characteristic p and a prime number ` 6= p, the natural map: End(A) ⊗ Q` → EndGF (V` (A)), is bijective, where GF is the absolute Galois g ...

... followed by more fallow periods. The first active period was in the mid 1960’s, when Tate, himself, proved that for an abelian variety A over a finite field F of characteristic p and a prime number ` 6= p, the natural map: End(A) ⊗ Q` → EndGF (V` (A)), is bijective, where GF is the absolute Galois g ...

Cell-Like Maps (Lecture 5)

... 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 19. Let f : X → Y be a map of Hausdorff spaces. We will say that f is cell-like if it satisfies the fol ...

... 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 19. Let f : X → Y be a map of Hausdorff spaces. We will say that f is cell-like if it satisfies the fol ...