Introduction to Combinatorial Homotopy Theory
... to use standard simple Algebraic Topology and make it constructive. 3. The operadic solution where the algebraic world is enriched enough to make it equivalent to the topological world; more precisely the algebraic structure of chain complexes is sufficiently enriched, thanks to appropriate operads, ...
... to use standard simple Algebraic Topology and make it constructive. 3. The operadic solution where the algebraic world is enriched enough to make it equivalent to the topological world; more precisely the algebraic structure of chain complexes is sufficiently enriched, thanks to appropriate operads, ...
Sample pages 1 PDF
... an irreducible factor Q(x) of P(x) whose degree is greater than k. In particular, if p can be chosen such that k = n − 1, then P(x) is irreducible. Definition 2.8. Symmetric polynomials in x1 , . . . , xn are polynomials that do not change on permuting the variables x1 , . . . , xn . Elementary symm ...
... an irreducible factor Q(x) of P(x) whose degree is greater than k. In particular, if p can be chosen such that k = n − 1, then P(x) is irreducible. Definition 2.8. Symmetric polynomials in x1 , . . . , xn are polynomials that do not change on permuting the variables x1 , . . . , xn . Elementary symm ...
NOETHERIAN MODULES 1. Introduction In a finite
... In a finite-dimensional vector space, every subspace is finite-dimensional and the dimension of a subspace is at most the dimension of the whole space. Unfortunately, the naive analogue of this for modules and submodules is wrong: (1) A submodule of a finitely generated module need not be finitely g ...
... In a finite-dimensional vector space, every subspace is finite-dimensional and the dimension of a subspace is at most the dimension of the whole space. Unfortunately, the naive analogue of this for modules and submodules is wrong: (1) A submodule of a finitely generated module need not be finitely g ...
A course on finite flat group schemes and p
... 1.2. . . . and what they are good for. Special properties of p-divisible groups are used in: (1) Analysis of the local p-adic Galois action on p-torsion points of elliptic curves, see Serre’s theorem on open image for non-CM elliptic curves over number fields [Se72], and more recently in modularity ...
... 1.2. . . . and what they are good for. Special properties of p-divisible groups are used in: (1) Analysis of the local p-adic Galois action on p-torsion points of elliptic curves, see Serre’s theorem on open image for non-CM elliptic curves over number fields [Se72], and more recently in modularity ...
Algebraic Number Theory, a Computational Approach
... finitely generated free abelian group is finitely generated. Then we see that finitely generated abelian groups can be presented as quotients of finite rank free abelian groups, and such a presentation can be reinterpreted in terms of matrices over the integers. Next we describe how to use row and c ...
... finitely generated free abelian group is finitely generated. Then we see that finitely generated abelian groups can be presented as quotients of finite rank free abelian groups, and such a presentation can be reinterpreted in terms of matrices over the integers. Next we describe how to use row and c ...
SYNTHETIC PROJECTIVE GEOMETRY
... 3. This is a generalization of the previous exercise. Let S be a geometrical incidence space of dimension n ≥ 2, and let ∞S be an object not belonging to S (the axioms for set theory give us explicit choices, but the method of construction is unimportant). Define the cone on S to be S • = S ∪ {∞S }, ...
... 3. This is a generalization of the previous exercise. Let S be a geometrical incidence space of dimension n ≥ 2, and let ∞S be an object not belonging to S (the axioms for set theory give us explicit choices, but the method of construction is unimportant). Define the cone on S to be S • = S ∪ {∞S }, ...
A geometric introduction to K-theory
... 1.7. Where we are headed. Our main goal in these notes is to describe a particular subset of the mathematics surrounding Serre’s definition of multiplicity. It is possible to explore this subject purely in algebraic terms, and that is basically what Serre did in his book [S]. In contrast, our main f ...
... 1.7. Where we are headed. Our main goal in these notes is to describe a particular subset of the mathematics surrounding Serre’s definition of multiplicity. It is possible to explore this subject purely in algebraic terms, and that is basically what Serre did in his book [S]. In contrast, our main f ...
Cluster categories for topologists - University of Virginia Information
... The goal of this paper is to introduce topological triangulated orbit categories, and in particular, the motivating example of topological cluster categories. In doing so, we hope to explain the fundamental ideas of triangulated orbit categories to readers from a more homotopy-theoretic, rather than ...
... The goal of this paper is to introduce topological triangulated orbit categories, and in particular, the motivating example of topological cluster categories. In doing so, we hope to explain the fundamental ideas of triangulated orbit categories to readers from a more homotopy-theoretic, rather than ...
MONADS AND ALGEBRAIC STRUCTURES Contents 1
... = Set(X, U W ) The fact that that this bijection is obtained through canonical extensions and restrictions from set functions to linear transformations and vice-versa for all X and W ensures that this bijection is indeed natural in X and W . Hence F a U . This example is a prototype for a general si ...
... = Set(X, U W ) The fact that that this bijection is obtained through canonical extensions and restrictions from set functions to linear transformations and vice-versa for all X and W ensures that this bijection is indeed natural in X and W . Hence F a U . This example is a prototype for a general si ...
UNIVERSAL COVERS OF TOPOLOGICAL MODULES AND A
... element 0 of additive group structure of M . We now apply the construction M (W ) of section 2 to the additive group of M and W . Proposition 3.5. Let R be a simply connected topological ring with identity 1R . Let M be a topological left R-module and W an open, path connected neighbourhood of the i ...
... element 0 of additive group structure of M . We now apply the construction M (W ) of section 2 to the additive group of M and W . Proposition 3.5. Let R be a simply connected topological ring with identity 1R . Let M be a topological left R-module and W an open, path connected neighbourhood of the i ...
Topological types of Algebraic stacks - IBS-CGP
... types, pro-objects in the homotopy category of simplicial sets, to étale topological types, proobjects in the category of simplicial sets. However, his theory still cannot not be applied to algebraic stacks (sometimes called Artin stacks). The main issue is again the use of the small étale topology. ...
... types, pro-objects in the homotopy category of simplicial sets, to étale topological types, proobjects in the category of simplicial sets. However, his theory still cannot not be applied to algebraic stacks (sometimes called Artin stacks). The main issue is again the use of the small étale topology. ...
![arXiv:0706.3441v1 [math.AG] 25 Jun 2007](http://s1.studyres.com/store/data/017784481_1-dd1160ad974ad834977ad61a663d12c3-300x300.png)
arXiv:0706.3441v1 [math.AG] 25 Jun 2007
... when U/G is paracompact then the associated separated rigid space (U/G)0 is a quotient U0 /G in the sense of rigid geometry (as defined in §2.1). We leave the details to the reader. As a special case, if k ′ /k is a finite Galois extension then descent data relative to k ′ /k on separated k ′ -analy ...
... when U/G is paracompact then the associated separated rigid space (U/G)0 is a quotient U0 /G in the sense of rigid geometry (as defined in §2.1). We leave the details to the reader. As a special case, if k ′ /k is a finite Galois extension then descent data relative to k ′ /k on separated k ′ -analy ...
Activity 3.5.4 Properties of Parallelograms
... First let’s review the definition and properties that we investigated already. 1. A parallelogram is defined as a quadrilateral with two pairs of _________________ sides. 2. Earlier in this investigation you may have discovered these properties: a. The opposite sides of a parallelogram are _________ ...
... First let’s review the definition and properties that we investigated already. 1. A parallelogram is defined as a quadrilateral with two pairs of _________________ sides. 2. Earlier in this investigation you may have discovered these properties: a. The opposite sides of a parallelogram are _________ ...
A Grothendieck site is a small category C equipped with a
... 5) The Nisnevich site Nis|S has the same underlying category as the étale site, namely all étale maps V → S and morphisms between them. A Nisnevich cover is a family of étale maps Vα → V such that every morphism Sp(K) → V lifts to some Vα where K is any field. 6) A flat cover of a scheme T is a ...
... 5) The Nisnevich site Nis|S has the same underlying category as the étale site, namely all étale maps V → S and morphisms between them. A Nisnevich cover is a family of étale maps Vα → V such that every morphism Sp(K) → V lifts to some Vα where K is any field. 6) A flat cover of a scheme T is a ...
Essential dimension and algebraic stacks
... Remark 1.2. If the functor F is limit-preserving, a condition that is satisfied in all cases that interest us, every element a ∈ F (L) has a field of definition K that is finitely generated over k, so ed a is finite. On the other hand, ed F may be infinite even in cases of interest (see for example ...
... Remark 1.2. If the functor F is limit-preserving, a condition that is satisfied in all cases that interest us, every element a ∈ F (L) has a field of definition K that is finitely generated over k, so ed a is finite. On the other hand, ed F may be infinite even in cases of interest (see for example ...
Topological realizations of absolute Galois groups
... Abstractly, it is clear that any group can be realised as the fundamental group of a topological space, by using the theory of classifying spaces. One may thus wonder what extra content Theorem 1.5 carries. We give several answers to this question. All are variants on the observation that our constr ...
... Abstractly, it is clear that any group can be realised as the fundamental group of a topological space, by using the theory of classifying spaces. One may thus wonder what extra content Theorem 1.5 carries. We give several answers to this question. All are variants on the observation that our constr ...
Abelian Categories
... • Note that in the category of monoids, category-theoretic kernels exist just as for groups, but these kernels don't carry sufficient information for algebraic purposes. Therefore, the notion of kernel studied in monoid theory is slightly different. Conversely, in the category of rings, there are no ...
... • Note that in the category of monoids, category-theoretic kernels exist just as for groups, but these kernels don't carry sufficient information for algebraic purposes. Therefore, the notion of kernel studied in monoid theory is slightly different. Conversely, in the category of rings, there are no ...
´Etale cohomology of schemes and analytic spaces
... Moreover, all these notions behave in the usual way. • The category of modules over a given ring (not necessarily commutative) is abelian. • The category of sheaves of abelian groups on a site C is abelian. This means in particular that one can define the notion of the image of a morphism, and that ...
... Moreover, all these notions behave in the usual way. • The category of modules over a given ring (not necessarily commutative) is abelian. • The category of sheaves of abelian groups on a site C is abelian. This means in particular that one can define the notion of the image of a morphism, and that ...
Introduction The following thesis plays a central role in deformation
... In order to adequately treat cases (b) through (d), it is important to note that for 0 ≤ n ≤ ∞, an En -ring is an essentially homotopy-theoretic object, and should therefore be treated using the formalism of higher category theory. In §2 we will give an overview of this formalism; in particular, we ...
... In order to adequately treat cases (b) through (d), it is important to note that for 0 ≤ n ≤ ∞, an En -ring is an essentially homotopy-theoretic object, and should therefore be treated using the formalism of higher category theory. In §2 we will give an overview of this formalism; in particular, we ...
When are induction and conduction functors isomorphic
... arises : “if the functors Ind and Coind are isomorphic, does it follow that the ring R is strongly graded ?” A simple example (see Remark 3.3) shows that the answer to this question is negative. So we may ask this other question : “if R is a graded ring and the functors Ind and Coind are isomorphic, ...
... arises : “if the functors Ind and Coind are isomorphic, does it follow that the ring R is strongly graded ?” A simple example (see Remark 3.3) shows that the answer to this question is negative. So we may ask this other question : “if R is a graded ring and the functors Ind and Coind are isomorphic, ...
Topology and robot motion planning
... For example, a continuous function f : X → Y induces a homomorphism of groups f∗ : π2 (X) → π2 (Y ). If f : X → Y and g : Y → Z are continuous, their composition g ◦ f : X → Z induces the composition g∗ ◦ f∗ : π2 (X) → π2 (Z). The identity function IdX : X → X induces the identity homomorphism Idπ2 ...
... For example, a continuous function f : X → Y induces a homomorphism of groups f∗ : π2 (X) → π2 (Y ). If f : X → Y and g : Y → Z are continuous, their composition g ◦ f : X → Z induces the composition g∗ ◦ f∗ : π2 (X) → π2 (Z). The identity function IdX : X → X induces the identity homomorphism Idπ2 ...
7.1. Sheaves and sheafification. The first thing we have to do to
... on every affine open subset U = Spec R of X. Almost all sheaves that we will consider are of this form. This reduces local computations regarding these sheaves to computations in commutative algebra. A quasi-coherent sheaf on X is called locally free of rank r if it is locally isomorphic to OX⊕r . L ...
... on every affine open subset U = Spec R of X. Almost all sheaves that we will consider are of this form. This reduces local computations regarding these sheaves to computations in commutative algebra. A quasi-coherent sheaf on X is called locally free of rank r if it is locally isomorphic to OX⊕r . L ...
Some structure theorems for algebraic groups
... algebraic groups of a specific geometric nature, such as smooth, connected, affine, proper... A first result in this direction asserts that every algebraic group G has a largest connected normal subgroup scheme G0 , the quotient G/G0 is finite and étale, and the formation of G0 commutes with field ...
... algebraic groups of a specific geometric nature, such as smooth, connected, affine, proper... A first result in this direction asserts that every algebraic group G has a largest connected normal subgroup scheme G0 , the quotient G/G0 is finite and étale, and the formation of G0 commutes with field ...
Étale Cohomology
... Grothendieck was the first to suggest étale cohomology (1960) as an attempt to solve the Weil Conjectures. By that time, it was already known by Serre that if one had a suitable cohomology theory for abstract varieties defined over the complex numbers, then one could deduce the conjectures using th ...
... Grothendieck was the first to suggest étale cohomology (1960) as an attempt to solve the Weil Conjectures. By that time, it was already known by Serre that if one had a suitable cohomology theory for abstract varieties defined over the complex numbers, then one could deduce the conjectures using th ...
Rings and modules
... Examples. Every abelian group is a Z -module, so the class of abelian groups coincide with the class of Z -modules. Every vector space over a field F is an F -module. 2.2. A map f : M → N is called a homomorphism of A -modules if f (x + y) = f (x) + f (y) for every x, y ∈ M and f (ax) = af (x) for e ...
... Examples. Every abelian group is a Z -module, so the class of abelian groups coincide with the class of Z -modules. Every vector space over a field F is an F -module. 2.2. A map f : M → N is called a homomorphism of A -modules if f (x + y) = f (x) + f (y) for every x, y ∈ M and f (ax) = af (x) for e ...