universal covering spaces and fundamental groups in algebraic
... More concretely, consider the following procedure: (1) define trivial covering space. (2) Define covering space. (3) Find a large class of schemes which admit a simply connected covering space, where a simply connected scheme is a scheme whose covering spaces are all trivial. (4) Use (3) and the adj ...
... More concretely, consider the following procedure: (1) define trivial covering space. (2) Define covering space. (3) Find a large class of schemes which admit a simply connected covering space, where a simply connected scheme is a scheme whose covering spaces are all trivial. (4) Use (3) and the adj ...
Toroidal deformations and the homotopy type of Berkovich spaces
... There is a strong deformation retraction of Xan onto S(X0 , X) ∪ Dan . By induction on the dimension, this result implies that the analytification of any algebraic variety over k has a strong deformation retraction onto a closed polyhedral subspace. Similar arguments apply more generally for any dis ...
... There is a strong deformation retraction of Xan onto S(X0 , X) ∪ Dan . By induction on the dimension, this result implies that the analytification of any algebraic variety over k has a strong deformation retraction onto a closed polyhedral subspace. Similar arguments apply more generally for any dis ...
Homological Conjectures and lim Cohen
... If f, g : N1 → (0, ∞) where N1 ⊆ N contains all n 0, f (n) = O g(n) if f (n)/g(n) is bounded and f (n) = o g(n) if f (n)/g(n) → 0 as n → ∞. A big Cohen-Macaulay module over a local ring (R, m, K) is a (not necessarily finitely generated) module M such that mM 6= M and every system of parameters f ...
... If f, g : N1 → (0, ∞) where N1 ⊆ N contains all n 0, f (n) = O g(n) if f (n)/g(n) is bounded and f (n) = o g(n) if f (n)/g(n) → 0 as n → ∞. A big Cohen-Macaulay module over a local ring (R, m, K) is a (not necessarily finitely generated) module M such that mM 6= M and every system of parameters f ...
MATH 436 Notes: Finitely generated Abelian groups.
... If |X| < ∞ then | ⊕x∈X Z/2Z| = 2|X| . Thus | ⊕y∈Y Z/2Z| is also finite and so |Y | < ∞ and 2|Y | = 2|X| which yields |X| = |Y | as desired. So it only remains to consider the case |X| = |Y | = ∞. Let Pf inite (X) = {S ⊂ X||S| < ∞} be the set of finite subsets of X. Then there is a bijection Θ : Pf i ...
... If |X| < ∞ then | ⊕x∈X Z/2Z| = 2|X| . Thus | ⊕y∈Y Z/2Z| is also finite and so |Y | < ∞ and 2|Y | = 2|X| which yields |X| = |Y | as desired. So it only remains to consider the case |X| = |Y | = ∞. Let Pf inite (X) = {S ⊂ X||S| < ∞} be the set of finite subsets of X. Then there is a bijection Θ : Pf i ...
Algebraic Groups
... The proof shows that R∗ is a special open set of R. In particular, R∗ is irreducible of dimension dim R∗ = dim R. 1.2. Isomorphisms and products. It follows from our definition that an algebraic group G is an affine variety with a group structure. These two structures are related in the usual way. N ...
... The proof shows that R∗ is a special open set of R. In particular, R∗ is irreducible of dimension dim R∗ = dim R. 1.2. Isomorphisms and products. It follows from our definition that an algebraic group G is an affine variety with a group structure. These two structures are related in the usual way. N ...
Study Guide for the Midterm Exam
... 1. From our textbook: consequences of negating Euclid’s fifth postulate on page 76, elementary facts about congruence (Theorems 3.2.1 through 3.2.4) Pasch Axiom (Theorem 3.2.5), Crossbar Theorem (Theorem 3.2.6), equivalent forms of Euclid’s fifth postulate (theorems in section 3.4), if there is one ...
... 1. From our textbook: consequences of negating Euclid’s fifth postulate on page 76, elementary facts about congruence (Theorems 3.2.1 through 3.2.4) Pasch Axiom (Theorem 3.2.5), Crossbar Theorem (Theorem 3.2.6), equivalent forms of Euclid’s fifth postulate (theorems in section 3.4), if there is one ...
LOCALIZATION OF ALGEBRAS OVER COLOURED OPERADS
... A common feature of these examples is that they can be described in terms of algebras over operads or, in some cases, algebras over coloured operads. Coloured operads first appeared in the book of Boardman and Vogt [BV73] on homotopy invariant algebraic structures on topological spaces. They can be ...
... A common feature of these examples is that they can be described in terms of algebras over operads or, in some cases, algebras over coloured operads. Coloured operads first appeared in the book of Boardman and Vogt [BV73] on homotopy invariant algebraic structures on topological spaces. They can be ...
HOMOTOPY THEORY 1. Homotopy Let X and Y be two topological
... Let X and Y be two topological spaces and f0 , f1 : X → Y be two continuous maps. (In the following, ‘map’ will mean ‘continuous map’.) 1.1. Definition. The map f0 is homotopic to the map f1 , f0 ' f1 , if there exists a map (a homotopy) f : X × I → Y such that f0 (x) = f (x, 0) and f1 (x) = f (x, 1 ...
... Let X and Y be two topological spaces and f0 , f1 : X → Y be two continuous maps. (In the following, ‘map’ will mean ‘continuous map’.) 1.1. Definition. The map f0 is homotopic to the map f1 , f0 ' f1 , if there exists a map (a homotopy) f : X × I → Y such that f0 (x) = f (x, 0) and f1 (x) = f (x, 1 ...
pdf
... Hida proves this through a series of group cohomological calculations combined with his theory of the ordinary part of the p-adic Hecke algebra. In this note we present a simple proof of the same result (Theorem 5.3 below) using only the elementary algebraic topology of the Riemann surfaces Y1 (pr ) ...
... Hida proves this through a series of group cohomological calculations combined with his theory of the ordinary part of the p-adic Hecke algebra. In this note we present a simple proof of the same result (Theorem 5.3 below) using only the elementary algebraic topology of the Riemann surfaces Y1 (pr ) ...
Algebraic models for higher categories
... complexes are ’the same’ from the viewpoint of homotopy theory. To make this statement precise Quillen [Qui67] introduced the concept of model categories and equivalence of model categories as an abstract framework for homotopy theory. He endowed the category Top of topological spaces and the catego ...
... complexes are ’the same’ from the viewpoint of homotopy theory. To make this statement precise Quillen [Qui67] introduced the concept of model categories and equivalence of model categories as an abstract framework for homotopy theory. He endowed the category Top of topological spaces and the catego ...
Vector bundles and torsion free sheaves on degenerations of elliptic
... In this section we review some classical results about vector bundles on smooth curves. However, we provide non-classical proofs which, as we think, are simpler and fit well in our approach to coherent sheaves over singular curves. The behavior of the category of vector bundles on a smooth projectiv ...
... In this section we review some classical results about vector bundles on smooth curves. However, we provide non-classical proofs which, as we think, are simpler and fit well in our approach to coherent sheaves over singular curves. The behavior of the category of vector bundles on a smooth projectiv ...
Final Exam Review Ch. 5
... A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side. X ...
... A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side. X ...
Chapter 6 Vocabulary Sheet
... Write the definition and give an example when applicable. Prerequisite Vocabulary congruent circles - ...
... Write the definition and give an example when applicable. Prerequisite Vocabulary congruent circles - ...
Chapter 5 Summary Sheet File
... Theorem 5-1 Opposite sides of a parallelogram are congruent. Theorem 5-2 Opposite angles of a parallelogram are congruent. Theorem 5-3 Diagonals of a parallelogram bisect each other. Theorem 5-4 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogr ...
... Theorem 5-1 Opposite sides of a parallelogram are congruent. Theorem 5-2 Opposite angles of a parallelogram are congruent. Theorem 5-3 Diagonals of a parallelogram bisect each other. Theorem 5-4 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogr ...
Help File
... f 0 is homotopic to gh−1 , this gives f 0 f ' f f 0 , so π1 (X, x0 ) is Abelian. Conversely, suppose that π1 (X, x0 ) is Abelian and let g and h be two paths from x0 to x1 . Then we get two isomorphisms π1 (X, x0 ) → π1 (X, x1 ) via f 7→ g −1 f g and f 7→ h−1 f h, and we wish to show these two maps ...
... f 0 is homotopic to gh−1 , this gives f 0 f ' f f 0 , so π1 (X, x0 ) is Abelian. Conversely, suppose that π1 (X, x0 ) is Abelian and let g and h be two paths from x0 to x1 . Then we get two isomorphisms π1 (X, x0 ) → π1 (X, x1 ) via f 7→ g −1 f g and f 7→ h−1 f h, and we wish to show these two maps ...
rings without a gorenstein analogue of the govorov–lazard theorem
... If R is regular then k has projective dimension 1 by the Auslander–Buchsbaum formula, so (5) shows that m is projective whence the biduality homomorphism δm is an isomorphism as desired. Assume that R is not regular. For reasons of clarity, we start by reproducing, in our notation, part of Takahashi ...
... If R is regular then k has projective dimension 1 by the Auslander–Buchsbaum formula, so (5) shows that m is projective whence the biduality homomorphism δm is an isomorphism as desired. Assume that R is not regular. For reasons of clarity, we start by reproducing, in our notation, part of Takahashi ...
Group Cohomology
... We will continue to let A denote a G-module throughout the section. We remark that C0 (G, A) is taken simply to be A, as G0 is a singleton set. The proof of the following is left to the reader. L EMMA 1.2.2. For any i ≥ 0, one has d i+1 ◦ d i = 0. R EMARK 1.2.3. Lemma 1.2.2 shows that C· (G, A) = (C ...
... We will continue to let A denote a G-module throughout the section. We remark that C0 (G, A) is taken simply to be A, as G0 is a singleton set. The proof of the following is left to the reader. L EMMA 1.2.2. For any i ≥ 0, one has d i+1 ◦ d i = 0. R EMARK 1.2.3. Lemma 1.2.2 shows that C· (G, A) = (C ...
Notes - Mathematics and Statistics
... only have a natural transformation 1 7→ ∗∗ which is not an equivalence. The problem being that for infinite dimensional vector spaces the map V → V ∗∗ is only an inclusion. Let K be a category. A subcategory C of K is a category whose objects are a subcollection of those of K and such that for every ...
... only have a natural transformation 1 7→ ∗∗ which is not an equivalence. The problem being that for infinite dimensional vector spaces the map V → V ∗∗ is only an inclusion. Let K be a category. A subcategory C of K is a category whose objects are a subcollection of those of K and such that for every ...
REVIEW OF MONOIDAL CONSTRUCTIONS 1. Strict monoidal
... the first factor, the last factor, and taking the sum a1 + a2 . Thus the geometric realization of the Γ-space A coincides with the classifying space of the category A. ...
... the first factor, the last factor, and taking the sum a1 + a2 . Thus the geometric realization of the Γ-space A coincides with the classifying space of the category A. ...
Derived splinters in positive characteristic
... Example 2.2 (D-splinters in characteristic 0). Let X be a variety over C. Then X is a D-splinter if and only if X has rational singularities, i.e. if R f∗ OY ' OX for some (equivalently, every) resolution of singularities f : Y → X. This assertion is contained in [Kov00, Theorem 3]. However, as indi ...
... Example 2.2 (D-splinters in characteristic 0). Let X be a variety over C. Then X is a D-splinter if and only if X has rational singularities, i.e. if R f∗ OY ' OX for some (equivalently, every) resolution of singularities f : Y → X. This assertion is contained in [Kov00, Theorem 3]. However, as indi ...
EXERCISES IN MA 510 : COMMUTATIVE ALGEBRA
... the ring homomorphism f : R → k[t] by f (x) = t9 , f (y) = t6 , f (z) = t4 . Show that the kernel of f is P and hence P is a prime ideal. Show that I(V (P )) = P. Is R integrally closed in its quotient field ? Find f ∈ R/P which is transcendental over k such that R/P is a finite k[f ]-algebra. (26) ...
... the ring homomorphism f : R → k[t] by f (x) = t9 , f (y) = t6 , f (z) = t4 . Show that the kernel of f is P and hence P is a prime ideal. Show that I(V (P )) = P. Is R integrally closed in its quotient field ? Find f ∈ R/P which is transcendental over k such that R/P is a finite k[f ]-algebra. (26) ...
Connes–Karoubi long exact sequence for Fréchet sheaves
... 2. Exact Sequence of K-theories for a Fréchet sheaf Let X be a noetherian scheme of finite type over C. In this paper, we will consider the K-theory of sheaves of Fréchet algebras or ultrametric Banach algebras on X. We recall here the notion of an ultrametric Banach algebra as defined in [14, 5.1 ...
... 2. Exact Sequence of K-theories for a Fréchet sheaf Let X be a noetherian scheme of finite type over C. In this paper, we will consider the K-theory of sheaves of Fréchet algebras or ultrametric Banach algebras on X. We recall here the notion of an ultrametric Banach algebra as defined in [14, 5.1 ...
Elliptic Curves Lecture Notes
... algebraically closed and may have positive characteristic). Definition 1.1. An elliptic curve over k is a nonsingular projective algebraic curve E of genus 1 over k with a chosen base point O ∈ E. Remark. There is a somewhat subtle point here concerning what is meant by a point of a curve over a non ...
... algebraically closed and may have positive characteristic). Definition 1.1. An elliptic curve over k is a nonsingular projective algebraic curve E of genus 1 over k with a chosen base point O ∈ E. Remark. There is a somewhat subtle point here concerning what is meant by a point of a curve over a non ...
DIFFERENTIABLE GROUP ACTIONS ON HOMOTOPY SPHERES. II
... triangulation theorem this category contains all G-homotopy types N/M, where N is a compact differentiable G-manifold and M is an invariant smooth submanifold (an alternate proof may be given using invariant Morse functions). As noted by Matumoto [52], the equivariant CW category satisfies all the a ...
... triangulation theorem this category contains all G-homotopy types N/M, where N is a compact differentiable G-manifold and M is an invariant smooth submanifold (an alternate proof may be given using invariant Morse functions). As noted by Matumoto [52], the equivariant CW category satisfies all the a ...
Commutative ring objects in pro-categories and generalized Moore spectra June 30, 2013
... By its very nature, this requires our category to carry a symmetric monoidal structure, a model structure, and an enrichment in spaces, and all of these must obey compatibility rules. This presents us with a significant number of adjectives to juggle. We study this compatibility in Section 2.2, fina ...
... By its very nature, this requires our category to carry a symmetric monoidal structure, a model structure, and an enrichment in spaces, and all of these must obey compatibility rules. This presents us with a significant number of adjectives to juggle. We study this compatibility in Section 2.2, fina ...