![arXiv:math/0302340v2 [math.AG] 7 Sep 2003](http://s1.studyres.com/store/data/014807023_1-69877329430c783f1dba3c51b4839685-300x300.png)
arXiv:math/0302340v2 [math.AG] 7 Sep 2003
... By [7] or [9] intersection homology is a topological invariant. It is enough to show that the decomposition into irreducible components is topologically invariant. Indeed, let Xreg be the set of points of X at which X is a topological manifold (the dimension may vary from a point to a point). The se ...
... By [7] or [9] intersection homology is a topological invariant. It is enough to show that the decomposition into irreducible components is topologically invariant. Indeed, let Xreg be the set of points of X at which X is a topological manifold (the dimension may vary from a point to a point). The se ...
Hochschild cohomology: some methods for computations
... and H 1 (A, A) = 0, see [18]. The importance of the simply connected algebras follows from the fact that usually we may reduce the study of indecomposable modules over an algebra to that for the corresponding simply connected algebras, using Galois coverings. Despite this very little is known about ...
... and H 1 (A, A) = 0, see [18]. The importance of the simply connected algebras follows from the fact that usually we may reduce the study of indecomposable modules over an algebra to that for the corresponding simply connected algebras, using Galois coverings. Despite this very little is known about ...
Tannaka Duality for Geometric Stacks
... If S is a projective scheme and X is the classifying stack of the algebraic group GLn , then Hom(S, X) classifies vector bundles on S. If S is a proper scheme, then any analytic vector bundle on S is algebraic (again by Serre’s GAGA theorem), and one may again deduce that φ is an equivalence. By com ...
... If S is a projective scheme and X is the classifying stack of the algebraic group GLn , then Hom(S, X) classifies vector bundles on S. If S is a proper scheme, then any analytic vector bundle on S is algebraic (again by Serre’s GAGA theorem), and one may again deduce that φ is an equivalence. By com ...
A Prelude to Obstruction Theory - WVU Math Department
... Let us now apply mathematical rigor to our idea of a CW-complex. Definition 1.3.1. Recall that a space holds the T2 separation axiom, or is Hausdorff, if, for any two elements of the space, there are mutually disjoint neighborhoods of these points. That is to say, for each x, y ∈ X, there exists a n ...
... Let us now apply mathematical rigor to our idea of a CW-complex. Definition 1.3.1. Recall that a space holds the T2 separation axiom, or is Hausdorff, if, for any two elements of the space, there are mutually disjoint neighborhoods of these points. That is to say, for each x, y ∈ X, there exists a n ...
Chern Character, Loop Spaces and Derived Algebraic Geometry
... elliptic cohomology and the notion of -vector bundles (parametrized version of the notion -vector spaces), even though the precise relation remains unclear at the moment. On the other hand, the fact that topological -theory is obtained as the Grothendieck group of vector bundles implies the existenc ...
... elliptic cohomology and the notion of -vector bundles (parametrized version of the notion -vector spaces), even though the precise relation remains unclear at the moment. On the other hand, the fact that topological -theory is obtained as the Grothendieck group of vector bundles implies the existenc ...
On Noether`s Normalization Lemma for projective schemes
... algebraic structures intrisecally connected to their geometric nature. It is the case of, as an example, rings of functions de ned over open subsets of a underlying topological space. In this chapter we are considering richer algebraic structures, such as modules and algebras over a ring. This will ...
... algebraic structures intrisecally connected to their geometric nature. It is the case of, as an example, rings of functions de ned over open subsets of a underlying topological space. In this chapter we are considering richer algebraic structures, such as modules and algebras over a ring. This will ...
The derived category of sheaves and the Poincare-Verdier duality
... If we set F n : F T n so that F n pX q : F pX rns q. Using the axiom TR2 we deduce that for every distinguished triangle pX, Y, Z, f, g, hq we obtain a long exact sequence in B q f rns n p1q grns n p1q hrns n 1 Ñ F npX q p1ÝÑ F pY q ÝÑ F pZ q ÝÑ F pX q Ñ (1.5) In particu ...
... If we set F n : F T n so that F n pX q : F pX rns q. Using the axiom TR2 we deduce that for every distinguished triangle pX, Y, Z, f, g, hq we obtain a long exact sequence in B q f rns n p1q grns n p1q hrns n 1 Ñ F npX q p1ÝÑ F pY q ÝÑ F pZ q ÝÑ F pX q Ñ (1.5) In particu ...
Homology and Cohomology
... (2) Inductively, form n-skeleton X n from X n−1 by attaching n-cells Dαn via maps φα :∂Dn → X n−1 . This means that X n is the quotient space of the disjoint union of X n−1 with a`collection of n−disks Dαn under the identifications x ∼ φα (x) for x ∈ ∂Dαn , i.e. X n−1 α Dαn /x ∼ φα (x) for all x ∈ ∂ ...
... (2) Inductively, form n-skeleton X n from X n−1 by attaching n-cells Dαn via maps φα :∂Dn → X n−1 . This means that X n is the quotient space of the disjoint union of X n−1 with a`collection of n−disks Dαn under the identifications x ∼ φα (x) for x ∈ ∂Dαn , i.e. X n−1 α Dαn /x ∼ φα (x) for all x ∈ ∂ ...
CATEGORIES AND COHOMOLOGY THEORIES
... inverse. (“ Homotopy-Cartesian” means that the induced map from A, to the homotopytheoretic fibre-product of A, and IPA I over I A I is a homotopy-equivalence.) PA is the usual “ simplicial path-space” of A, i.e. it is the composite A 0 P, where P:A.-,Atakes[n]to[n+1]and~:[m]-t[n]to~:[m+l]-t[n+1],wh ...
... inverse. (“ Homotopy-Cartesian” means that the induced map from A, to the homotopytheoretic fibre-product of A, and IPA I over I A I is a homotopy-equivalence.) PA is the usual “ simplicial path-space” of A, i.e. it is the composite A 0 P, where P:A.-,Atakes[n]to[n+1]and~:[m]-t[n]to~:[m+l]-t[n+1],wh ...
Seminar in Topology and Actions of Groups. Topological Groups
... said to be compatible if they satisfy (i) and (ii). Example 1. 1.The discrete topology on a group G is compatible with the group structure. A topological group whose topology is discrete is called a discrete group. 2. The trivial topology on G is compatible with the group structure of G. 3. Every no ...
... said to be compatible if they satisfy (i) and (ii). Example 1. 1.The discrete topology on a group G is compatible with the group structure. A topological group whose topology is discrete is called a discrete group. 2. The trivial topology on G is compatible with the group structure of G. 3. Every no ...
Algebraic Topology Lecture Notes Jarah Evslin and
... generated by any two relatively prime numbers, so the number of generators depends on the choice of generators. ZN can be generated by only N elements of the lattice, for example the N elements whose coordinates are all 0 except for a single 1. On the other hand the groups Q and R require an infinit ...
... generated by any two relatively prime numbers, so the number of generators depends on the choice of generators. ZN can be generated by only N elements of the lattice, for example the N elements whose coordinates are all 0 except for a single 1. On the other hand the groups Q and R require an infinit ...
MA3056 — Exercise Sheet 2: Topological Spaces
... 12. Let X and Y be topological spaces and suppose that E ⊂ X and F ⊂ Y are closed. Show that E × F is closed in the topological product X × Y . 13. Let X = [−1, 1] equipped with the usual topology. (a)∗ Let f : X → [0, 1] be the function f (x) = |x|. Show that quotient topology induced on [0, 1] by ...
... 12. Let X and Y be topological spaces and suppose that E ⊂ X and F ⊂ Y are closed. Show that E × F is closed in the topological product X × Y . 13. Let X = [−1, 1] equipped with the usual topology. (a)∗ Let f : X → [0, 1] be the function f (x) = |x|. Show that quotient topology induced on [0, 1] by ...
Sheaves of Groups and Rings
... morphism i Gi −→ F P (coproduct in Ab(X)). We can describe the subsheaf i Gi explicitly as follows. P Lemma 11. Let {Gi }i∈I be a nonempty set of subsheaves of a sheaf F . Then the subsheaf i Gi P is defined by the following condition: given an open set U and s ∈ F (U ) we have s ∈ ( i Gi )(U ) if a ...
... morphism i Gi −→ F P (coproduct in Ab(X)). We can describe the subsheaf i Gi explicitly as follows. P Lemma 11. Let {Gi }i∈I be a nonempty set of subsheaves of a sheaf F . Then the subsheaf i Gi P is defined by the following condition: given an open set U and s ∈ F (U ) we have s ∈ ( i Gi )(U ) if a ...
ABELIAN VARIETIES A canonical reference for the subject is
... Proof. Choose a nonempty open affine U ⊂ X and let D = X \ U with the reduced structure. We claim that D is a Cartier divisor (that is, its coherent ideal sheaf is invertible). Since X is regular, it is equivalent to say that all generic points of D have codimension 1. Mumford omits this explanation ...
... Proof. Choose a nonempty open affine U ⊂ X and let D = X \ U with the reduced structure. We claim that D is a Cartier divisor (that is, its coherent ideal sheaf is invertible). Since X is regular, it is equivalent to say that all generic points of D have codimension 1. Mumford omits this explanation ...
STABLE COHOMOLOGY OF FINITE AND PROFINITE GROUPS 1
... finite groups are continious on Gal(K) with respect to the above topology, and we are going to consider only continious maps and continious cochains on the Gal(K). Since V L → V L /G corresponds to a finite Galois extension k(V L ) : k(V L /G) with G as a Galois group, we have a natural surjection ...
... finite groups are continious on Gal(K) with respect to the above topology, and we are going to consider only continious maps and continious cochains on the Gal(K). Since V L → V L /G corresponds to a finite Galois extension k(V L ) : k(V L /G) with G as a Galois group, we have a natural surjection ...
Very dense subsets of a topological space.
... is an isomorphism. Since f −1 is exact, it follows that the canonical maps in cohomology H i (Y, F) → H i (X, f −1 (F)) are isomorphisms. (ii) If f is a quasi-homeomorphism then f −1 gives an equivalence between the categories of sheaves of rings on X and on Y . If f = (ψ, θ) : (X, OX ) → (Y, OY ) i ...
... is an isomorphism. Since f −1 is exact, it follows that the canonical maps in cohomology H i (Y, F) → H i (X, f −1 (F)) are isomorphisms. (ii) If f is a quasi-homeomorphism then f −1 gives an equivalence between the categories of sheaves of rings on X and on Y . If f = (ψ, θ) : (X, OX ) → (Y, OY ) i ...
Connes–Karoubi long exact sequence for Fréchet sheaves
... Lemma 2.3. Let (X, OX ) denote the formal scheme obtained by completing an integral noetherian scheme X ′ of finite type over C along a closed primary integral subscheme X, as in example (a) above. Then, for each open subset U of X, the ring OX (U ) is an ultrametric Banach algebra, i.e., (X, OX ) d ...
... Lemma 2.3. Let (X, OX ) denote the formal scheme obtained by completing an integral noetherian scheme X ′ of finite type over C along a closed primary integral subscheme X, as in example (a) above. Then, for each open subset U of X, the ring OX (U ) is an ultrametric Banach algebra, i.e., (X, OX ) d ...
NOTES ON FORMAL SCHEMES, SHEAVES ON R
... Exercise 5. Check that this is equivalent to the definition in [Lur10, lecture 11]. Alternatively, show that they are different and correct these notes. Let G be a formal group over R. The abelian group structure induces a unit map e : SpecR R → G. The Lie algebra g of G is e∗ TG — a line bundle on ...
... Exercise 5. Check that this is equivalent to the definition in [Lur10, lecture 11]. Alternatively, show that they are different and correct these notes. Let G be a formal group over R. The abelian group structure induces a unit map e : SpecR R → G. The Lie algebra g of G is e∗ TG — a line bundle on ...
Group cohomology - of Alexey Beshenov
... If one takes f(g, h) = 1 for all g, h ∈ G, then (L/K, f) ' Mn (K). The cross product algebras up to isomorphism correspond to H 2 (Gal(L/K), L× ). Later on we will see how to calculate such cohomology groups. In the example with quaternions actually H 2 (Gal(C/R), C× ) ' Z/2, so there are two algebr ...
... If one takes f(g, h) = 1 for all g, h ∈ G, then (L/K, f) ' Mn (K). The cross product algebras up to isomorphism correspond to H 2 (Gal(L/K), L× ). Later on we will see how to calculate such cohomology groups. In the example with quaternions actually H 2 (Gal(C/R), C× ) ' Z/2, so there are two algebr ...
LECTURE 2 1. Finitely Generated Abelian Groups We discuss the
... Theorem 1.5. If A is a finitely generated torsion-free abelian group that has a minimal set of generators with q elements, then A is isomorphic to the free abelian group of rank q. Proof. By induction on the minimal number of generators of A. If A is cyclic (that is, generated by one non-zero elemen ...
... Theorem 1.5. If A is a finitely generated torsion-free abelian group that has a minimal set of generators with q elements, then A is isomorphic to the free abelian group of rank q. Proof. By induction on the minimal number of generators of A. If A is cyclic (that is, generated by one non-zero elemen ...
Sheaf theory - Department of Mathematics
... Observe that a presheaf F is separated if and only if, for any object U of Top(X) and any s, t ∈ F (U ), we have s = t if and only if sx = tx for all x ∈ U . A morphism of presheaves f : F → G induces a morphism fx : Fx → Gx on stalks for each x ∈ X by taking the direct limit of the morphisms f (U ) ...
... Observe that a presheaf F is separated if and only if, for any object U of Top(X) and any s, t ∈ F (U ), we have s = t if and only if sx = tx for all x ∈ U . A morphism of presheaves f : F → G induces a morphism fx : Fx → Gx on stalks for each x ∈ X by taking the direct limit of the morphisms f (U ) ...
Abelian topological groups and (A/k)C ≈ k 1. Compact
... Given non-trivial ψ ∈ (A/k)b, the k-vectorspace k · ψ inside (A/k)b injects to a copy of k · ψ inside A Assuming for a moment that the image in A is essentially the same as the diagonal copy of k, the quotient (A/k)b/k injects to the compact A/k. The topology of (A/k)b is discrete, and the quotient ...
... Given non-trivial ψ ∈ (A/k)b, the k-vectorspace k · ψ inside (A/k)b injects to a copy of k · ψ inside A Assuming for a moment that the image in A is essentially the same as the diagonal copy of k, the quotient (A/k)b/k injects to the compact A/k. The topology of (A/k)b is discrete, and the quotient ...
1 The affine superscheme
... of the morphisms ψU , there is an induced map on the stalks ψx : OY,ϕ(x) → OX,x , which we require to be a local morphism. That is, if mϕ(x) is the maximal ideal of OY,ϕ(x) and mx is the maximal ideal of OX,x then ψx (mϕ(x) ) ⊂ mx , or equivalently, ψx−1 (mx ) = mϕ(x) . Definition 2.1. Let X be a to ...
... of the morphisms ψU , there is an induced map on the stalks ψx : OY,ϕ(x) → OX,x , which we require to be a local morphism. That is, if mϕ(x) is the maximal ideal of OY,ϕ(x) and mx is the maximal ideal of OX,x then ψx (mϕ(x) ) ⊂ mx , or equivalently, ψx−1 (mx ) = mϕ(x) . Definition 2.1. Let X be a to ...
Intersection homology
... with coefficients in a local system L on Xm To define the restriction map, we begin with a singular simplex P σ on U . We will use σ to define a set J(σ) of simplices on V and set r(σ) = σ0 ∈J(σ) σ 0 . The procedure is inductive: 1. If the image of σ lies entirely in U \ V then J(σ) = ∅. 2. If the i ...
... with coefficients in a local system L on Xm To define the restriction map, we begin with a singular simplex P σ on U . We will use σ to define a set J(σ) of simplices on V and set r(σ) = σ0 ∈J(σ) σ 0 . The procedure is inductive: 1. If the image of σ lies entirely in U \ V then J(σ) = ∅. 2. If the i ...