Direct-sum decompositions over local rings
... fashion. We give two separate constructions to realize these monoids: In §2 we show that every expanded submonoid of Nn can be realized by a torsion-free module over a suitable one-dimensional ring, and in §3 we treat the (more general) full submonoids. The first construction is straightforward: All ...
... fashion. We give two separate constructions to realize these monoids: In §2 we show that every expanded submonoid of Nn can be realized by a torsion-free module over a suitable one-dimensional ring, and in §3 we treat the (more general) full submonoids. The first construction is straightforward: All ...
On the Universal Space for Group Actions with Compact Isotropy
... for the family F is a G-CW -complex E(G, F) such that the fixed point set E(G, F)H is weakly contractible for H ∈ F and all its isotropy groups belong to F. Recall that a map f : X → Y of spaces is a weak homotopy equivalence if and only if the induced map f∗ : πn (X, x) → πn (Y, f (x)) is an isomo ...
... for the family F is a G-CW -complex E(G, F) such that the fixed point set E(G, F)H is weakly contractible for H ∈ F and all its isotropy groups belong to F. Recall that a map f : X → Y of spaces is a weak homotopy equivalence if and only if the induced map f∗ : πn (X, x) → πn (Y, f (x)) is an isomo ...
Usha - IIT Guwahati
... These affine varieties are our first objects of study. But before we can go further, in fact before we can even give any interesting examples, we need to explore the relationship between subsets of An and ideals in A more deeply. So for any subset Y ⊆ An , let us define the ideal of Y in A by I(Y ) ...
... These affine varieties are our first objects of study. But before we can go further, in fact before we can even give any interesting examples, we need to explore the relationship between subsets of An and ideals in A more deeply. So for any subset Y ⊆ An , let us define the ideal of Y in A by I(Y ) ...
Spectra of Small Categories and Infinite Loop Space Machines
... between them which arise from the subdivision functors between the interval categories. Definition 2.3 Let α, β ∈ N with β ≥ α. A functor t: Iβ → Iα such that t (0) = 0 and t (β) = α will be called a subdivision functor. A subdivision functor will also be called a transformation. If t: Iβ → Iα is a ...
... between them which arise from the subdivision functors between the interval categories. Definition 2.3 Let α, β ∈ N with β ≥ α. A functor t: Iβ → Iα such that t (0) = 0 and t (β) = α will be called a subdivision functor. A subdivision functor will also be called a transformation. If t: Iβ → Iα is a ...
The Functor Category in Relation to the Model Theory of Modules
... this category into a larger category and the extra objects give us more information. ...
... this category into a larger category and the extra objects give us more information. ...
1 Valuations of the field of rational numbers
... Q[x]/(P ) is a field, finite extension of Q. All finite extensions of Q are of this form since it is known that all finite extensions of a field in characteristic zero can be generated by a single element [?, V,4.6]. Let L be a finite extension of degree n of Q. For every element α ∈ L, the multipl ...
... Q[x]/(P ) is a field, finite extension of Q. All finite extensions of Q are of this form since it is known that all finite extensions of a field in characteristic zero can be generated by a single element [?, V,4.6]. Let L be a finite extension of degree n of Q. For every element α ∈ L, the multipl ...
The Type of the Classifying Space of a Topological Group for the
... Recall from the introduction the G-CW -complex E(G, F ). In particular, notice that we do not work with the stronger condition that E(G, F )H is contractible but only weakly contractible. If G is discrete, then each fixed point set E(G, F )H has the homotopy type of a CW -complex and is contractible ...
... Recall from the introduction the G-CW -complex E(G, F ). In particular, notice that we do not work with the stronger condition that E(G, F )H is contractible but only weakly contractible. If G is discrete, then each fixed point set E(G, F )H has the homotopy type of a CW -complex and is contractible ...
Which spheres admit a topological group structure?
... So deg(Θg ) = deg(Θg−1 ) = ±1, because the degree must be an integer. We can now construct a degree function d : G → Z2 = {±1} as d(g) = deg(Θg ), which is an isomorphism. Indeed, d is an homomorphism by (ii) and because G acts freely on S n we know that Θg has no fixed points for any g 6= e. So (vi ...
... So deg(Θg ) = deg(Θg−1 ) = ±1, because the degree must be an integer. We can now construct a degree function d : G → Z2 = {±1} as d(g) = deg(Θg ), which is an isomorphism. Indeed, d is an homomorphism by (ii) and because G acts freely on S n we know that Θg has no fixed points for any g 6= e. So (vi ...
Finite flat group schemes course
... a morphism Hom(A, B) → Hom(A, C) preserving the product just means that for every ring B the natural maps mB : Hom(A ⊗ A, B) → Hom(A, B) fit into the obvious commutative diagrams: given f : B → C, the two ways of getting from Hom(A ⊗ A, B) to Hom(A, C) must be the same. This looks like a vast amount ...
... a morphism Hom(A, B) → Hom(A, C) preserving the product just means that for every ring B the natural maps mB : Hom(A ⊗ A, B) → Hom(A, B) fit into the obvious commutative diagrams: given f : B → C, the two ways of getting from Hom(A ⊗ A, B) to Hom(A, C) must be the same. This looks like a vast amount ...
A Relative Spectral Sequence for Topological Hochschild Homology
... When C is commutative and M is a C ∧A C op - or a C ∧B C op -algebra, respectively, these spectral sequences are multiplicative with respect to the standard products on Hochschild homology and topological Hochschild homology. The method of constructing these spectral sequences is an elaboration of ...
... When C is commutative and M is a C ∧A C op - or a C ∧B C op -algebra, respectively, these spectral sequences are multiplicative with respect to the standard products on Hochschild homology and topological Hochschild homology. The method of constructing these spectral sequences is an elaboration of ...
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 ...
dmodules ja
... of . The following theorem of Cox [2] indicates the significance of the pair S . We write -Mod for the category of quasi-coherent sheaves on X and S-GrMod for the category of graded S-modules. A graded S-module F is called -torsion if, for all f ∈ F, there exists > 0 such that f = 0. Let ...
... of . The following theorem of Cox [2] indicates the significance of the pair S . We write -Mod for the category of quasi-coherent sheaves on X and S-GrMod for the category of graded S-modules. A graded S-module F is called -torsion if, for all f ∈ F, there exists > 0 such that f = 0. Let ...
(A SOMEWHAT GENTLE INTRODUCTION TO) DIFFERENTIAL
... Theorem 1.1. Let (R, m) → (S, n) be a flat local ring homomorphism, that is, a ring homomorphism making S into a flat R-module such that mS ⊆ n. Then S is Gorenstein if and only if R and S/mS are Gorenstein. Moreover, there is an equality of Bass series I S (t) = I R (t)I S/mS (t). (See Definition A ...
... Theorem 1.1. Let (R, m) → (S, n) be a flat local ring homomorphism, that is, a ring homomorphism making S into a flat R-module such that mS ⊆ n. Then S is Gorenstein if and only if R and S/mS are Gorenstein. Moreover, there is an equality of Bass series I S (t) = I R (t)I S/mS (t). (See Definition A ...
The Fundamental Group
... Definition 2.1. A homotopy of paths is a family of functions ft : I → X, 0 ≤ t ≤ 1 such that ft (0) and ft (1) are constant as functions of t and the map F : I × I → X defined by F (s, t) = ft (s) is continuous. We call ft (0) and ft (1) the fixed endpoints of the homotopy. Two paths α and β are sai ...
... Definition 2.1. A homotopy of paths is a family of functions ft : I → X, 0 ≤ t ≤ 1 such that ft (0) and ft (1) are constant as functions of t and the map F : I × I → X defined by F (s, t) = ft (s) is continuous. We call ft (0) and ft (1) the fixed endpoints of the homotopy. Two paths α and β are sai ...
ON QUILLEN`S THEOREM A FOR POSETS 1. Introduction In his
... We deduce then a stronger version of Dowker’s Theorem [15]. The proof is the same as in [8, Theorem 10.9] but using Theorem 4.3. Theorem 4.4. Let X and Y be two finite sets and let R ⊆ X × Y be a relation. Consider the simplicial complex K whose simplices are the subsets σ of X for which exists an e ...
... We deduce then a stronger version of Dowker’s Theorem [15]. The proof is the same as in [8, Theorem 10.9] but using Theorem 4.3. Theorem 4.4. Let X and Y be two finite sets and let R ⊆ X × Y be a relation. Consider the simplicial complex K whose simplices are the subsets σ of X for which exists an e ...
Math 8211 Homework 2 PJW
... Date due: Monday October 15, 2012. In class on Wednesday September 17 we will grade your answers, so it is important to be present on that day, with your homework. As practice, but not part of the homework, make sure you can do questions in Rotman apart from the ones listed below, such as 2.19a. Ass ...
... Date due: Monday October 15, 2012. In class on Wednesday September 17 we will grade your answers, so it is important to be present on that day, with your homework. As practice, but not part of the homework, make sure you can do questions in Rotman apart from the ones listed below, such as 2.19a. Ass ...
Lecture Notes - Mathematics
... Given a collection of polynomials defining an affine algebraic variety, it is not at all obvious, a priori, whether or not the variety is irreducible, or how many components it has.2 For hypersurfaces, it is easy to describe the irreducible components, as well as choose a “nice” defining equation: E ...
... Given a collection of polynomials defining an affine algebraic variety, it is not at all obvious, a priori, whether or not the variety is irreducible, or how many components it has.2 For hypersurfaces, it is easy to describe the irreducible components, as well as choose a “nice” defining equation: E ...
Semisimplicity - UC Davis Mathematics
... We begin with some reminders and remarks on non-commutative rings. First, recall that if R is a not necessarily commutative ring, an element x ∈ R is invertible if it has both a left (multiplicative) inverse and a right (multiplicative) inverse; it then follows that the two are necessarily equal. No ...
... We begin with some reminders and remarks on non-commutative rings. First, recall that if R is a not necessarily commutative ring, an element x ∈ R is invertible if it has both a left (multiplicative) inverse and a right (multiplicative) inverse; it then follows that the two are necessarily equal. No ...
An introduction to schemes - University of Chicago Math
... Example 3.6. Let M be a smooth manifold. Then for each open set U of M , we have C(U ), the set of real-valued continuous functions on U . Under point-wise addition and multiplication, this is a ring. If V ⊆ U then we have the restriction homomorphism C(U ) → C(V ) given by actually restricting func ...
... Example 3.6. Let M be a smooth manifold. Then for each open set U of M , we have C(U ), the set of real-valued continuous functions on U . Under point-wise addition and multiplication, this is a ring. If V ⊆ U then we have the restriction homomorphism C(U ) → C(V ) given by actually restricting func ...
SOME ALGEBRAIC DEFINITIONS AND CONSTRUCTIONS
... For an ideal I, the quotient ring R/I is the set of equivalence classes of elements of R, where x is equivalent to y if x − y is in I. It inherits an addition and multiplication from R that makes it a commutative ring such that the quotient map R −→ R/I is a homomorphism of rings. Proposition 13. If ...
... For an ideal I, the quotient ring R/I is the set of equivalence classes of elements of R, where x is equivalent to y if x − y is in I. It inherits an addition and multiplication from R that makes it a commutative ring such that the quotient map R −→ R/I is a homomorphism of rings. Proposition 13. If ...
Topology in the 20th century
... consisting of homotopy classes of closed paths starting and ending at one common point – and constructed the topological theory of covering spaces. He discovered the Poincaré duality law, which asserts that for a closed n-dimensional manifold the Betti numbers of some definite type in dimensions k a ...
... consisting of homotopy classes of closed paths starting and ending at one common point – and constructed the topological theory of covering spaces. He discovered the Poincaré duality law, which asserts that for a closed n-dimensional manifold the Betti numbers of some definite type in dimensions k a ...
String topology and the based loop space.
... and HH∗ (C ∗ X ) when X is simply connected, taking a cohomological version of the ∆ operator to B [21]. With the introduction of string topology, similar isomorphisms relating the loop homology H∗ ( LM) of M to the Hochschild cohomologies HH ∗ (C ∗ M) and HH ∗ (C∗ ΩM ) were developed. One such fami ...
... and HH∗ (C ∗ X ) when X is simply connected, taking a cohomological version of the ∆ operator to B [21]. With the introduction of string topology, similar isomorphisms relating the loop homology H∗ ( LM) of M to the Hochschild cohomologies HH ∗ (C ∗ M) and HH ∗ (C∗ ΩM ) were developed. One such fami ...
Topological Models for Arithmetic William G. Dwyer and Eric M
... to Fω (BUn ) and π1 Xi acts trivially on the mod ` (co)-homology of this pro-space (cf. [4, p. 260]). It follows from the fibre lemma [2] that F is equivalent to the homotopy fibre of the completed map Fω (BGLn,Xi ) → Fω (Xi ), or to the homotopy fibre of the composite map Fω (BGLn,Xi ) → Fω (Xi ) → ...
... to Fω (BUn ) and π1 Xi acts trivially on the mod ` (co)-homology of this pro-space (cf. [4, p. 260]). It follows from the fibre lemma [2] that F is equivalent to the homotopy fibre of the completed map Fω (BGLn,Xi ) → Fω (Xi ), or to the homotopy fibre of the composite map Fω (BGLn,Xi ) → Fω (Xi ) → ...
THEOREM 4-3 – Isosceles Triangle Theorem THEOREM 4
... If a triangle is __________________________, then it is also __________________________. ...
... If a triangle is __________________________, then it is also __________________________. ...