BASIC DEFINITIONS IN CATEGORY THEORY MATH 250B 1
... A covariant functor F : C → Set said to be representable by A ∈ C if one has an isomorphism of functors: F ∼ = HomC (A, •). Similarly, a contravariant functor G : C → Set is said to be representable by A ∈ C if one has an isomorphism of functors: G∼ = HomC (•, A). It is more-or-less standard to call ...
... A covariant functor F : C → Set said to be representable by A ∈ C if one has an isomorphism of functors: F ∼ = HomC (A, •). Similarly, a contravariant functor G : C → Set is said to be representable by A ∈ C if one has an isomorphism of functors: G∼ = HomC (•, A). It is more-or-less standard to call ...
Essential dimension and algebraic stacks
... group G introduced by J. Buhler and the second author to study the complexity of G-torsors over a field K. It has since been studied by several other authors in a variety of contexts. In this paper, we extend this notion to algebraic stacks. This allows us to answer the following type of question: g ...
... group G introduced by J. Buhler and the second author to study the complexity of G-torsors over a field K. It has since been studied by several other authors in a variety of contexts. In this paper, we extend this notion to algebraic stacks. This allows us to answer the following type of question: g ...
廖寶珊紀念書院Liu Po Shan Memorial College
... • Recognize angles associated with a transversal: corresponding angles, alternate angles and interior angles on the same side of the transversal. • Explore and use the properties of angles associated with parallel lines and their ...
... • Recognize angles associated with a transversal: corresponding angles, alternate angles and interior angles on the same side of the transversal. • Explore and use the properties of angles associated with parallel lines and their ...
Preliminary version
... We have transformed this algorithm in order to obtain a new and simpler one, as above, without using straight-line programs anymore, neither for multivariate polynomials nor for integer numbers. We give a new estimate of the exponents of the complexity of Theorem [37] above improving the results of ...
... We have transformed this algorithm in order to obtain a new and simpler one, as above, without using straight-line programs anymore, neither for multivariate polynomials nor for integer numbers. We give a new estimate of the exponents of the complexity of Theorem [37] above improving the results of ...
A refinement of the Artin conductor and the base change conductor
... A refinement of the Artin conductor without making any assumptions on the dimension or the ramification behavior. Let us assume that K has positive residue characteristic p. To generalize (1.1), we extend the classes of objects on both sides of that equation. On the left-hand side, we consider anal ...
... A refinement of the Artin conductor without making any assumptions on the dimension or the ramification behavior. Let us assume that K has positive residue characteristic p. To generalize (1.1), we extend the classes of objects on both sides of that equation. On the left-hand side, we consider anal ...
West Windsor-Plainsboro Regional School District Geometry Honors
... Geometry (Honors and Accelerated) is a course for mathematically gifted ninth‐grade students who have completed an enriched Advanced Algebra II curriculum. It consists of a college preparatory course in Euclidean plane and solid geometry, considered mostly from a traditional ...
... Geometry (Honors and Accelerated) is a course for mathematically gifted ninth‐grade students who have completed an enriched Advanced Algebra II curriculum. It consists of a college preparatory course in Euclidean plane and solid geometry, considered mostly from a traditional ...
Chapter III. Basic theory of group schemes. As we have seen in the
... give a group law on an object X means that for each object T in C we have to specify a group law on the set hX (T ) = HomC (T, X), such that for every morphism f : T1 → T2 the induced map hX (f ): hX (T2 ) → hX (T1 ) is a homomorphism of groups. An object of C together with a C-group law on it is c ...
... give a group law on an object X means that for each object T in C we have to specify a group law on the set hX (T ) = HomC (T, X), such that for every morphism f : T1 → T2 the induced map hX (f ): hX (T2 ) → hX (T1 ) is a homomorphism of groups. An object of C together with a C-group law on it is c ...
Several approaches to non-archimedean geometry
... By Theorem 1.1.5, every k-affinoid algebra A is noetherian and Jacobson with finite Krull dimension, and A/m is a finite extension of k for every m ∈ M (A). For a point x ∈ M (A) we write k(x) to denote this associated finite extension of k and we write a(x) ∈ k(x) to denote the image of a ∈ A in k( ...
... By Theorem 1.1.5, every k-affinoid algebra A is noetherian and Jacobson with finite Krull dimension, and A/m is a finite extension of k for every m ∈ M (A). For a point x ∈ M (A) we write k(x) to denote this associated finite extension of k and we write a(x) ∈ k(x) to denote the image of a ∈ A in k( ...
Flatness
... Using the LES of Tor, this immediately implies: Proposition 1.2 If 0 → M 0 → M → M 00 → 0 is an exact sequence of A-modules, and M 0 and M 00 are flat, then so is M . If M and M 00 are both flat, so is M 0 . Another useful consequence is the following, which again is immediate upon passing to the LE ...
... Using the LES of Tor, this immediately implies: Proposition 1.2 If 0 → M 0 → M → M 00 → 0 is an exact sequence of A-modules, and M 0 and M 00 are flat, then so is M . If M and M 00 are both flat, so is M 0 . Another useful consequence is the following, which again is immediate upon passing to the LE ...
Notes - Math Berkeley
... Definition 1.9. An abelian scheme over a base scheme S is a group scheme A/S such that A ! S is of finite presentation, smooth, and all fibers are geometrically connected. Remark 1.10. Note the distinction between “abelian scheme” and “abelian group scheme”. Proposition 1.11. (i) Abelian schemes are ...
... Definition 1.9. An abelian scheme over a base scheme S is a group scheme A/S such that A ! S is of finite presentation, smooth, and all fibers are geometrically connected. Remark 1.10. Note the distinction between “abelian scheme” and “abelian group scheme”. Proposition 1.11. (i) Abelian schemes are ...
arXiv:0706.3441v1 [math.AG] 25 Jun 2007
... spaces in Example 3.1.1, we do not think it is possible to prove this theorem via the methods of rigid geometry. The key to our success is to study quotient problems in the category of k-analytic spaces in the sense of Berkovich, and only a posteriori translating such results back into classical rig ...
... spaces in Example 3.1.1, we do not think it is possible to prove this theorem via the methods of rigid geometry. The key to our success is to study quotient problems in the category of k-analytic spaces in the sense of Berkovich, and only a posteriori translating such results back into classical rig ...
Miles Reid's notes
... where g(x) is a polynomial of degree n − 1 and c is a constant. Moreover, c = f (α). In particular, α is a root of f if and only if x − α divides f (x). Proof The “moreover” clause follows trivially from the first part on substituting x = α. For the first part, we use induction on n. Suppose that f ...
... where g(x) is a polynomial of degree n − 1 and c is a constant. Moreover, c = f (α). In particular, α is a root of f if and only if x − α divides f (x). Proof The “moreover” clause follows trivially from the first part on substituting x = α. For the first part, we use induction on n. Suppose that f ...
Operations with Polynomials
... The product is obtained by direct application of the Distributive Property. For instance, to multiply the monomial 3x by the polynomial 2x2 5x 3, multiply each term of the polynomial by 3x. ...
... The product is obtained by direct application of the Distributive Property. For instance, to multiply the monomial 3x by the polynomial 2x2 5x 3, multiply each term of the polynomial by 3x. ...
Moduli Problems for Ring Spectra - International Mathematical Union
... (a) Allowing R to range over the category Ring of commutative rings, we obtain the notion of a classical moduli problem (Definition 1.3). We will discuss this notion and give several examples in §1. (b) To understand the deformation theory of a moduli space X, it is often useful to extend the defini ...
... (a) Allowing R to range over the category Ring of commutative rings, we obtain the notion of a classical moduli problem (Definition 1.3). We will discuss this notion and give several examples in §1. (b) To understand the deformation theory of a moduli space X, it is often useful to extend the defini ...
4 Ideals in commutative rings
... by assumption, must stop at some In . But the process can stop only if In is maximal in I, as required. [By the way, the proof of (ii)⇒(i) is redundant, given the proofs of the other implications, but it is instructive, hence included.] Examples 4.2. (1) Any field (e.g. Q, R, Zp p prime) is noether ...
... by assumption, must stop at some In . But the process can stop only if In is maximal in I, as required. [By the way, the proof of (ii)⇒(i) is redundant, given the proofs of the other implications, but it is instructive, hence included.] Examples 4.2. (1) Any field (e.g. Q, R, Zp p prime) is noether ...
LOGIC AND INCIDENCE GEOMETRY
... most obvious. Although it is "obvious" that two points determine a unique line, Euclid stated this as his first postulate. So ifin some proof we want to say that every line has points lying on it, we should list this statement as another postulate (or prove it. but we can't). In other words, all our ...
... most obvious. Although it is "obvious" that two points determine a unique line, Euclid stated this as his first postulate. So ifin some proof we want to say that every line has points lying on it, we should list this statement as another postulate (or prove it. but we can't). In other words, all our ...
STRATIFICATION BY THE LOCAL HILBERT
... In section 3, we propose a general algorithm for a stratification by a constant set of exponents (or initial ideal) with three variants: one uses primary decomposition and the two others do not. We apply this to our initial problem on Hilbert-Samuel functions. We end this section with 3.4 where addi ...
... In section 3, we propose a general algorithm for a stratification by a constant set of exponents (or initial ideal) with three variants: one uses primary decomposition and the two others do not. We apply this to our initial problem on Hilbert-Samuel functions. We end this section with 3.4 where addi ...
IDEMPOTENT RESIDUATED STRUCTURES: SOME CATEGORY EQUIVALENCES AND THEIR APPLICATIONS
... isomorphic to the identity functors on D and C, respectively. In the concrete category associated with a class of similar algebras, the objects are the members of the class, and the morphisms are all the algebraic homomorphisms between pairs of objects. The set of homomorphisms from A into B is deno ...
... isomorphic to the identity functors on D and C, respectively. In the concrete category associated with a class of similar algebras, the objects are the members of the class, and the morphisms are all the algebraic homomorphisms between pairs of objects. The set of homomorphisms from A into B is deno ...
Standard Monomial Theory and applications
... geometric properties. Here we suppose that G is a reductive algebraic group defined over an algebraically closed field k, B is a fixed Borel subgroup, and Lλ is the line bundle on the flag variety G/B associated to a dominant weight. The purpose of the program is to extend the Hodge-Young standard m ...
... geometric properties. Here we suppose that G is a reductive algebraic group defined over an algebraically closed field k, B is a fixed Borel subgroup, and Lλ is the line bundle on the flag variety G/B associated to a dominant weight. The purpose of the program is to extend the Hodge-Young standard m ...
Solving Problems with Magma
... all aspects of algebra. We welcome comments on this book and submissions of new approaches ...
... all aspects of algebra. We welcome comments on this book and submissions of new approaches ...
Higher regulators and values of L
... Q -+@ff~-J(X,Q(i)). The corresponding constructions are recalled in Sec. 2. Let l-l~-J(X, Q (~))cKj(X)~Q be the eigenspace of weight i relative to the Adams operator [2]; then ch~ defines a r e g u l a t o r - a morphism r~:HJs~(X,Q(i))-+I-I~(X,Q(i)). [It is thought that for any schemes there exists ...
... Q -+@ff~-J(X,Q(i)). The corresponding constructions are recalled in Sec. 2. Let l-l~-J(X, Q (~))cKj(X)~Q be the eigenspace of weight i relative to the Adams operator [2]; then ch~ defines a r e g u l a t o r - a morphism r~:HJs~(X,Q(i))-+I-I~(X,Q(i)). [It is thought that for any schemes there exists ...
Homework assignments
... Please study the following Problems 1-10 by January 18 (Friday). In Problems 1-10, let X be a compact Hausdorff space and let A be the ring of all C-valued continuous functions on X. We consider how X is reflected in A and how A is reflected in X like the relation of a flower and its image in the wa ...
... Please study the following Problems 1-10 by January 18 (Friday). In Problems 1-10, let X be a compact Hausdorff space and let A be the ring of all C-valued continuous functions on X. We consider how X is reflected in A and how A is reflected in X like the relation of a flower and its image in the wa ...
Determine the equation of a line given any of the following
... Obtuse angle, right angle, acute angle, straight angle, vertical angles, overlapping angles, linear pair, adjacent angles, complementary angles, supplementary angles, overlapping ...
... Obtuse angle, right angle, acute angle, straight angle, vertical angles, overlapping angles, linear pair, adjacent angles, complementary angles, supplementary angles, overlapping ...
Math 8211 Homework 1 PJW
... We say that a functor F : C → D is an isomorphism of categories if there is a functor G : D → C so that GF = 1C and F G = 1D . We say that a functor F : C → D is an equivalence of categories if there is a functor G : D → C so that GF is naturally isomorphic to the identity functor 1C and F G is natu ...
... We say that a functor F : C → D is an isomorphism of categories if there is a functor G : D → C so that GF = 1C and F G = 1D . We say that a functor F : C → D is an equivalence of categories if there is a functor G : D → C so that GF is naturally isomorphic to the identity functor 1C and F G is natu ...
Algebraic variety
In mathematics, algebraic varieties (also called varieties) are one of the central objects of study in algebraic geometry. Classically, an algebraic variety was defined to be the set of solutions of a system of polynomial equations, over the real or complex numbers. Modern definitions of an algebraic variety generalize this notion in several different ways, while attempting to preserve the geometric intuition behind the original definition.Conventions regarding the definition of an algebraic variety differ slightly. For example, some authors require that an ""algebraic variety"" is, by definition, irreducible (which means that it is not the union of two smaller sets that are closed in the Zariski topology), while others do not. When the former convention is used, non-irreducible algebraic varieties are called algebraic sets.The notion of variety is similar to that of manifold, the difference being that a variety may have singular points, while a manifold will not. In many languages, both varieties and manifolds are named by the same word.Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specificity of algebraic geometry among the other subareas of geometry.