arXiv:math.OA/9901094 v1 22 Jan 1999
... cocycle functor ZΓ : Ab(Γ) → Ab, where Ab(Γ) is the category of Γ-sheaves and Ab is the category of abelian groups, is defined as follows: Given a Γ-sheaf A, the abelian group ZΓ (A) consists of all continuous functions f : Γ → A such that f (γ) ∈ Ar(γ) (i.e. f is a continuous section of r∗ (A)) and ...
... cocycle functor ZΓ : Ab(Γ) → Ab, where Ab(Γ) is the category of Γ-sheaves and Ab is the category of abelian groups, is defined as follows: Given a Γ-sheaf A, the abelian group ZΓ (A) consists of all continuous functions f : Γ → A such that f (γ) ∈ Ar(γ) (i.e. f is a continuous section of r∗ (A)) and ...
Introduction to Teichmüller Spaces
... 3. Teichmüller Space We fix a topological surface S of genus g. Definition 3.1. A marked Riemann surface (X, f ) is a Riemann surface X together with a homemorphism f : S → X. Two marked surfaces (X, f ) ∼ (Y, g) are equivalent if gf −1 : X → Y is isotopic to an isomorphism. Definition 3.2. We defi ...
... 3. Teichmüller Space We fix a topological surface S of genus g. Definition 3.1. A marked Riemann surface (X, f ) is a Riemann surface X together with a homemorphism f : S → X. Two marked surfaces (X, f ) ∼ (Y, g) are equivalent if gf −1 : X → Y is isotopic to an isomorphism. Definition 3.2. We defi ...
2 - arXiv
... 2.3. Reduction maps. Theorem 2.10. Let X be a variety over an uncountable field, and L a nef R-divisor. There exists a rational map f : X 99K Z, called the nef reduction map, with the following properties: • f is proper over an open subset U of Z • L|F ≡ 0 on very general fibres over U • A curve C t ...
... 2.3. Reduction maps. Theorem 2.10. Let X be a variety over an uncountable field, and L a nef R-divisor. There exists a rational map f : X 99K Z, called the nef reduction map, with the following properties: • f is proper over an open subset U of Z • L|F ≡ 0 on very general fibres over U • A curve C t ...
OPERADS, FACTORIZATION ALGEBRAS, AND (TOPOLOGICAL
... shows that any associative algebra is an E1 -algebra in chain complexes. Another example of E1 -algebras is given by loop spaces: Let (X, ∗) be a pointed topological space. Then ΩX = Map(S 1 , X) is an E1 -algebra: since S 1 = [0, 1]/(0 ∼ 1), any f ∈ E1 (n) determines a map f : S 1 → S 1 q∗ . . . q∗ ...
... shows that any associative algebra is an E1 -algebra in chain complexes. Another example of E1 -algebras is given by loop spaces: Let (X, ∗) be a pointed topological space. Then ΩX = Map(S 1 , X) is an E1 -algebra: since S 1 = [0, 1]/(0 ∼ 1), any f ∈ E1 (n) determines a map f : S 1 → S 1 q∗ . . . q∗ ...
Updated October 30, 2014 CONNECTED p
... We will talk about the Serre-Tate equivalence. This is a handy tool that gives us a handle on the connected part of a p-divisible group. In particular, it does two things for us : (1) It enables us to define ‘dimension’ of a p-divisible group, which will play a role in proving the main theorem about ...
... We will talk about the Serre-Tate equivalence. This is a handy tool that gives us a handle on the connected part of a p-divisible group. In particular, it does two things for us : (1) It enables us to define ‘dimension’ of a p-divisible group, which will play a role in proving the main theorem about ...
10 Rings
... is, these sets are too large to be useful, particularly from the point of view of unique factorization. (Another hint that they might not be a good thing to work with is that there is no standard notation for√them.) For instance, how does 2√factor into the ring of all algebraic integers? It clearly ...
... is, these sets are too large to be useful, particularly from the point of view of unique factorization. (Another hint that they might not be a good thing to work with is that there is no standard notation for√them.) For instance, how does 2√factor into the ring of all algebraic integers? It clearly ...
Hochschild cohomology
... cosimplicial objects and its associated (co-)chain complex. In fact, Hochschild cohomology can be described by the cohomology of a cochain complex associated to a cosimplicial bimodule, that is a cosimplicial object in the category of bimodules. Definition. Let ∆ be the category whose objects are th ...
... cosimplicial objects and its associated (co-)chain complex. In fact, Hochschild cohomology can be described by the cohomology of a cochain complex associated to a cosimplicial bimodule, that is a cosimplicial object in the category of bimodules. Definition. Let ∆ be the category whose objects are th ...
foundations of algebraic geometry class 38
... (3) We’ll later see that this will show how cohomology groups vary in families, especially in “nice” situations. Intuitively, if we have a nice family of varieties, and a family of sheaves on them, we could hope that the cohomology varies nicely in families, and in fact in “nice” situations, this is ...
... (3) We’ll later see that this will show how cohomology groups vary in families, especially in “nice” situations. Intuitively, if we have a nice family of varieties, and a family of sheaves on them, we could hope that the cohomology varies nicely in families, and in fact in “nice” situations, this is ...
7. A1 -homotopy theory 7.1. Closed model categories. We begin with
... which is the identity on objects and whose set of morphisms from X to Y equals the set of homotopy classes of morphisms from some fibrant/cofibrant replacement of X to some fibrant/cofibrant replacement of Y : HomHo(C) (X, Y ) = π(RQX, RQY ). If F : C → D if a functor with the property that F sends ...
... which is the identity on objects and whose set of morphisms from X to Y equals the set of homotopy classes of morphisms from some fibrant/cofibrant replacement of X to some fibrant/cofibrant replacement of Y : HomHo(C) (X, Y ) = π(RQX, RQY ). If F : C → D if a functor with the property that F sends ...
s principle
... aims to assist the development of those subj ects by revealing characteristic ways in which their categories differ from others . ( Such considerations will be important in order to carry out Grothendieck ’ s 1 973 program [ 2 ] for s implifying the foundations of algebraic geometry . ) An axiomatic ...
... aims to assist the development of those subj ects by revealing characteristic ways in which their categories differ from others . ( Such considerations will be important in order to carry out Grothendieck ’ s 1 973 program [ 2 ] for s implifying the foundations of algebraic geometry . ) An axiomatic ...
Notes 1
... (1) Let f ∈ k[x1 , . . . , xn ] be a non-constant polynomial. Then V (f ) ⊂ An is called the hypersurface defined by f . If the degree of f is one, then the corresponding hypersurface is called a hyperplane. If the degree of f is two, then the corresponding hypersurface is called a quadric hypersurf ...
... (1) Let f ∈ k[x1 , . . . , xn ] be a non-constant polynomial. Then V (f ) ⊂ An is called the hypersurface defined by f . If the degree of f is one, then the corresponding hypersurface is called a hyperplane. If the degree of f is two, then the corresponding hypersurface is called a quadric hypersurf ...
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 ...
Lesson 34 – Coordinate Ring of an Affine Variety
... , the restriction of to defines a map from to ; that is, . Under the usual pointwise operations of addition and multiplication, these functions form a -algebra which we call the coordinate ring of ...
... , the restriction of to defines a map from to ; that is, . Under the usual pointwise operations of addition and multiplication, these functions form a -algebra which we call the coordinate ring of ...
CORE VARIETIES, EXTENSIVITY, AND RIG GEOMETRY 1
... aims to assist the development of those subjects by revealing characteristic ways in which their categories differ from others. (Such considerations will be important in order to carry out Grothendieck’s 1973 program [2] for simplifying the foundations of algebraic geometry.) An axiomatic theory oft ...
... aims to assist the development of those subjects by revealing characteristic ways in which their categories differ from others. (Such considerations will be important in order to carry out Grothendieck’s 1973 program [2] for simplifying the foundations of algebraic geometry.) An axiomatic theory oft ...
PDF
... treatment of simplicial complexes is much simpler in the finite case and so for now we will assume that V is a finite set of cardinality k. We introduce the vector space RV of formal R–linear combinations of elements of V ; i.e., RV := {a1 V1 + a2 V2 + · · · + ak Vk | ai ∈ R, Vi ∈ V }, and the vecto ...
... treatment of simplicial complexes is much simpler in the finite case and so for now we will assume that V is a finite set of cardinality k. We introduce the vector space RV of formal R–linear combinations of elements of V ; i.e., RV := {a1 V1 + a2 V2 + · · · + ak Vk | ai ∈ R, Vi ∈ V }, and the vecto ...
Topological Field Theories
... Historically, it all began with cobordism, before topological field theories were even defined. Pontrjagin invented cobordism theory in the 1930’s in order to compute homotopy groups of spheres. Back then they knew that πi S n = 0 for i < n (proof by cellular approximation) and that πn S n ∼ = Z, cl ...
... Historically, it all began with cobordism, before topological field theories were even defined. Pontrjagin invented cobordism theory in the 1930’s in order to compute homotopy groups of spheres. Back then they knew that πi S n = 0 for i < n (proof by cellular approximation) and that πn S n ∼ = Z, cl ...
Fibre Bundles and Homotopy Exact Sequence
... Let I = [0, 1] and I n = I × · · · × I be the unit n-cube, its boundary is the set ∂I = {(x1 , . . . , xn ) ∈ I n | ∃i : xi = 0, 1} A pointed space (X, x0 ) is a space X with a base point x0 , a pair (X, A) of spaces is a pointed space X and its pointed subspace A: x0 ∈ A ∈ X. A continuous map of pa ...
... Let I = [0, 1] and I n = I × · · · × I be the unit n-cube, its boundary is the set ∂I = {(x1 , . . . , xn ) ∈ I n | ∃i : xi = 0, 1} A pointed space (X, x0 ) is a space X with a base point x0 , a pair (X, A) of spaces is a pointed space X and its pointed subspace A: x0 ∈ A ∈ X. A continuous map of pa ...
Picard groups and class groups of algebraic varieties
... (b) The zero set Z(xy(x − 1), xy(y − 1)) ⊂ C2 consists of the two coordinate axes and the point (1, 1), so the ideal (xy(x − 1), xy(y − 1)) is not prime. Loosely speaking, a complex algebraic variety is obtained by gluing together affine varieties along Zariski open sets with regular functions, much ...
... (b) The zero set Z(xy(x − 1), xy(y − 1)) ⊂ C2 consists of the two coordinate axes and the point (1, 1), so the ideal (xy(x − 1), xy(y − 1)) is not prime. Loosely speaking, a complex algebraic variety is obtained by gluing together affine varieties along Zariski open sets with regular functions, much ...
Cell-Like Maps (Lecture 5)
... Remark 12. Any piecewise linear homeomorphism of polyhedra f : X → Y is a simple homotopy equivalence (with respect to the simple homotopy structures of Example 9). Remark 13. The collection of all polyhedra can be organized into a category, where the morphisms are given by piecewise linear maps. Th ...
... Remark 12. Any piecewise linear homeomorphism of polyhedra f : X → Y is a simple homotopy equivalence (with respect to the simple homotopy structures of Example 9). Remark 13. The collection of all polyhedra can be organized into a category, where the morphisms are given by piecewise linear maps. Th ...
Garrett 10-03-2011 1 We will later elaborate the ideas mentioned earlier: relations
... algebraic. Specifically, do not try to explicitly find a polynomial P with rational coefficients and P (α + β) = 0, in terms of the minimal polynomials of α, β. The methodological point in the latter is first that it is not required to explicitly determine the minimal polynomial of α + β. Second, ab ...
... algebraic. Specifically, do not try to explicitly find a polynomial P with rational coefficients and P (α + β) = 0, in terms of the minimal polynomials of α, β. The methodological point in the latter is first that it is not required to explicitly determine the minimal polynomial of α + β. Second, ab ...
Chap 0
... If F : X ⇥ Z ! Y is continuous then we need to show that (F ) is also continuous. The key step in the proof is given by: Lemma 0.6.4 (Hot Dog Lemma). Let W be an open subset of X ⇥ Y . If W contains K ⇥ y0 where K is a compact subset of X and y0 2 Y then there are open subsets U ⇢ X and V ⇢ Y so tha ...
... If F : X ⇥ Z ! Y is continuous then we need to show that (F ) is also continuous. The key step in the proof is given by: Lemma 0.6.4 (Hot Dog Lemma). Let W be an open subset of X ⇥ Y . If W contains K ⇥ y0 where K is a compact subset of X and y0 2 Y then there are open subsets U ⇢ X and V ⇢ Y so tha ...
Homology and cohomology theories on manifolds
... In this paper, we study generalized homology and cohomology theories on the categories of smooth, PL, and topological manifolds. These are, by definition, absolute theories satisfying a Mayer-Vietoris property (see Section 4). Recently, the first author constructed such a theory (of ordinary type) o ...
... In this paper, we study generalized homology and cohomology theories on the categories of smooth, PL, and topological manifolds. These are, by definition, absolute theories satisfying a Mayer-Vietoris property (see Section 4). Recently, the first author constructed such a theory (of ordinary type) o ...
Homology and cohomology theories on manifolds
... In this paper, we study generalized homology and cohomology theories on the categories of smooth, PL, and topological manifolds. These are, by definition, absolute theories satisfying a Mayer-Vietoris property (see Section 4). Recently, the first author constructed such a theory (of ordinary type) o ...
... In this paper, we study generalized homology and cohomology theories on the categories of smooth, PL, and topological manifolds. These are, by definition, absolute theories satisfying a Mayer-Vietoris property (see Section 4). Recently, the first author constructed such a theory (of ordinary type) o ...
Introduction to abstract algebra: definitions, examples, and exercises
... In the case of sets, an invertible map of sets f : X → Y is the same as a bijective map, i.e., one which is one-to-one and onto. Definition 28. A permutation of a set X is an invertible (equivalently bijective) map p : X → X. Let Perm(X) denote the set of permutations of X. Exercise 29. Show that P ...
... In the case of sets, an invertible map of sets f : X → Y is the same as a bijective map, i.e., one which is one-to-one and onto. Definition 28. A permutation of a set X is an invertible (equivalently bijective) map p : X → X. Let Perm(X) denote the set of permutations of X. Exercise 29. Show that P ...
ARITHMETIC OF CURVES OVER TWO DIMENSIONAL LOCAL
... filtration of degenerating abelian varieties on local fields. In this work, we use this approach to investigate the group π1c.s (X) . As mentioned by Yoshida in [12, section 2] Grothendieck’s theory of monodromy-weight filtration on Tate module of abelian varieties are valid where the residue field ...
... filtration of degenerating abelian varieties on local fields. In this work, we use this approach to investigate the group π1c.s (X) . As mentioned by Yoshida in [12, section 2] Grothendieck’s theory of monodromy-weight filtration on Tate module of abelian varieties are valid where the residue field ...