SCHEMES 01H8 Contents 1. Introduction 1 2. Locally ringed spaces
... Definition 2.1. Locally ringed spaces. (1) A locally ringed space (X, OX ) is a pair consisting of a topological space X and a sheaf of rings OX all of whose stalks are local rings. (2) Given a locally ringed space (X, OX ) we say that OX,x is the local ring of X at x. We denote mX,x or simply mx th ...
... Definition 2.1. Locally ringed spaces. (1) A locally ringed space (X, OX ) is a pair consisting of a topological space X and a sheaf of rings OX all of whose stalks are local rings. (2) Given a locally ringed space (X, OX ) we say that OX,x is the local ring of X at x. We denote mX,x or simply mx th ...
Vector Bundles And F Theory
... general polynomial in x and y with at most an nth order pole at infinity, and (modulo the Weierstrass equation) at most a linear dependence on y. To allow for a completely general set of Qi , one restricts the ak only by requiring that they are not all identically zero. (For example, an vanishes if ...
... general polynomial in x and y with at most an nth order pole at infinity, and (modulo the Weierstrass equation) at most a linear dependence on y. To allow for a completely general set of Qi , one restricts the ak only by requiring that they are not all identically zero. (For example, an vanishes if ...
Lectures on Etale Cohomology
... such theorems as the Lefschetz fixed point formula are available. For a variety X over an arbitrary algebraically closed field k, there is only the Zariski topology, which is too coarse (i.e., has too few open subsets) for the methods of algebraic topology to be useful. For example, if X is irreduci ...
... such theorems as the Lefschetz fixed point formula are available. For a variety X over an arbitrary algebraically closed field k, there is only the Zariski topology, which is too coarse (i.e., has too few open subsets) for the methods of algebraic topology to be useful. For example, if X is irreduci ...
On Klein`s So-called Non
... geometers like Poncelet8 had prepared the ground for Klein’s general idea that a geometry is a transformation group. The fact that the three constant curvature geometries (hyperbolic, Euclidean and spherical) can be developed in the realm of projective geometry is expressed by the fact that the tran ...
... geometers like Poncelet8 had prepared the ground for Klein’s general idea that a geometry is a transformation group. The fact that the three constant curvature geometries (hyperbolic, Euclidean and spherical) can be developed in the realm of projective geometry is expressed by the fact that the tran ...
Abelian Varieties
... connected projective2 variety with a group structure defined by regular maps. Definition (d) does generalize, but with a caution. If A is an abelian variety over C, then A.C/ Cg = for some lattice in Cg (isomorphism simultaneously of complex manifolds and of groups). However, when g > 1, the qu ...
... connected projective2 variety with a group structure defined by regular maps. Definition (d) does generalize, but with a caution. If A is an abelian variety over C, then A.C/ Cg = for some lattice in Cg (isomorphism simultaneously of complex manifolds and of groups). However, when g > 1, the qu ...
Some structure theorems for algebraic groups
... The above theorems have a long history. Theorem 1 was first obtained by Rosenlicht in 1956 for smooth connected algebraic groups, see [48, Sec. 5]. The version presented here is due to Demazure and Gabriel, see [22, III.3.8]. In the setting of smooth connected algebraic groups again, Theorem 2 was a ...
... The above theorems have a long history. Theorem 1 was first obtained by Rosenlicht in 1956 for smooth connected algebraic groups, see [48, Sec. 5]. The version presented here is due to Demazure and Gabriel, see [22, III.3.8]. In the setting of smooth connected algebraic groups again, Theorem 2 was a ...
Non-euclidean shadows of classical projective
... Beyond the Euler line, we construct a nine-point conic which is a noneuclidean version of the nine-point circle. Menelaus’ Theorem and non-euclidean trigonometry Hyperbolic trigonometry is a recurrent topic in most treatments on non-euclidean geometry since the early works of N. I. Lobachevsky and J ...
... Beyond the Euler line, we construct a nine-point conic which is a noneuclidean version of the nine-point circle. Menelaus’ Theorem and non-euclidean trigonometry Hyperbolic trigonometry is a recurrent topic in most treatments on non-euclidean geometry since the early works of N. I. Lobachevsky and J ...
Examples - Stacks Project
... chain). Then blow up the resulting surface at the two singularities of X1 , and let X2 be the reduced preimage of X1 (which has five rational components), etc. Take X to be the inverse limit. The only problem with this construction is that blow-ups glue in a projective line, so X1 is not affine. Let ...
... chain). Then blow up the resulting surface at the two singularities of X1 , and let X2 be the reduced preimage of X1 (which has five rational components), etc. Take X to be the inverse limit. The only problem with this construction is that blow-ups glue in a projective line, so X1 is not affine. Let ...
What is a generic point? - Emory Math/CS Department
... and every open subset of an irreducible scheme contains that scheme’s unique generic point. All of this material is standard, and [Liu] is a great reference. Let X be a scheme. Recall X is irreducible if its underlying topological space is irreducible. A (nonempty) topological space is irreducible i ...
... and every open subset of an irreducible scheme contains that scheme’s unique generic point. All of this material is standard, and [Liu] is a great reference. Let X be a scheme. Recall X is irreducible if its underlying topological space is irreducible. A (nonempty) topological space is irreducible i ...
3 Hyperbolic Geometry in Klein`s Model
... diagram in proposition 3.2. Proposition 3.3. The definitions of the projective polar of points and lines are consistent with incidence: A point K lies on a hyperbolic line k if and only if the projective polar K proj⊥ goes through the polar k ⊥ . Proof using the development above. As shown in the las ...
... diagram in proposition 3.2. Proposition 3.3. The definitions of the projective polar of points and lines are consistent with incidence: A point K lies on a hyperbolic line k if and only if the projective polar K proj⊥ goes through the polar k ⊥ . Proof using the development above. As shown in the las ...
Vector bundles and torsion free sheaves on degenerations of elliptic
... Theorem 3 (see [BD03, BK3]). Let E be a cuspidal cubic curve over an algebraically closed field k then a stable vector bundle E is completely determined by its rank r, its degree d, that should be coprime, and its determinant det(E) ∈ Picd (E) ∼ = k. The technique of matrix problems is a very conven ...
... Theorem 3 (see [BD03, BK3]). Let E be a cuspidal cubic curve over an algebraically closed field k then a stable vector bundle E is completely determined by its rank r, its degree d, that should be coprime, and its determinant det(E) ∈ Picd (E) ∼ = k. The technique of matrix problems is a very conven ...
Higher regulators and values of L
... mixed Hodge structure on H ~ ( X ) . For example, if X is compact, then ch (Kj(X)) = 0 for j > 0. It turns out that the Hodge conditions can be used, and, untangling them, it is possible to obtain finer analytic invariants of the elements of K.(X) than the usual cohomology classes. For the case of C ...
... mixed Hodge structure on H ~ ( X ) . For example, if X is compact, then ch (Kj(X)) = 0 for j > 0. It turns out that the Hodge conditions can be used, and, untangling them, it is possible to obtain finer analytic invariants of the elements of K.(X) than the usual cohomology classes. For the case of C ...
Basic Arithmetic Geometry Lucien Szpiro
... algebra of finite type over k. Choosing variables X1 , . . . , Xn , A can be regarded as a quotient of the ring of polynomials k [X1 , . . . , Xn ] by an ideal generated by a finite set of polynomials F1 , . . . , Fm . Given a finite set of polynomials Fj , an algebraic variety is the set of all n-t ...
... algebra of finite type over k. Choosing variables X1 , . . . , Xn , A can be regarded as a quotient of the ring of polynomials k [X1 , . . . , Xn ] by an ideal generated by a finite set of polynomials F1 , . . . , Fm . Given a finite set of polynomials Fj , an algebraic variety is the set of all n-t ...
Outline - Durham University
... Homogeneous coordinates: a line though the O is determined by a triple of numbers (ξ1 , ξ2 , ξ3 ), where (ξ1 , ξ2 , ξ3 ) 6= (0, 0, 0); triples (ξ1 , ξ2 , ξ3 ) and (λξ1 , λξ2 , λξ3 ) determine the same line, so are considered equivalent. Projective transformations in homogeneous coordinates: A : (ξ1 ...
... Homogeneous coordinates: a line though the O is determined by a triple of numbers (ξ1 , ξ2 , ξ3 ), where (ξ1 , ξ2 , ξ3 ) 6= (0, 0, 0); triples (ξ1 , ξ2 , ξ3 ) and (λξ1 , λξ2 , λξ3 ) determine the same line, so are considered equivalent. Projective transformations in homogeneous coordinates: A : (ξ1 ...
PROJECTIVE MODULES AND VECTOR BUNDLES The basic
... The basic objects studied in algebraic K-theory are projective modules over a ring, and vector bundles over schemes. In this first chapter we introduce the cast of characters. Much of this information is standard, but collected here for ease of reference in later chapters. Here are a few running con ...
... The basic objects studied in algebraic K-theory are projective modules over a ring, and vector bundles over schemes. In this first chapter we introduce the cast of characters. Much of this information is standard, but collected here for ease of reference in later chapters. Here are a few running con ...
My notes - Harvard Mathematics Department
... complex torus (so that it has a natural group structure), and it also has the structure of a projective variety. These two structures are in fact compatible with each other: the addition law is a morphism between algebraic varieties. This motivates: 1.5 Definition. An abelian variety over C is a pro ...
... complex torus (so that it has a natural group structure), and it also has the structure of a projective variety. These two structures are in fact compatible with each other: the addition law is a morphism between algebraic varieties. This motivates: 1.5 Definition. An abelian variety over C is a pro ...
An Introduction to K-theory
... and only if they have the same dimension. Thus, P(F ) ( N and K0 (F ) = Z. Example 1.4. Let K/Q be a finite field extension of the rational numbers (K is said to be a number field) and let OK ⊂ K be the ring of algebraic integers in K. Thus, O is the subring of those elements α ∈ K which satisfy a m ...
... and only if they have the same dimension. Thus, P(F ) ( N and K0 (F ) = Z. Example 1.4. Let K/Q be a finite field extension of the rational numbers (K is said to be a number field) and let OK ⊂ K be the ring of algebraic integers in K. Thus, O is the subring of those elements α ∈ K which satisfy a m ...
Homogeneous coordinates in the plane Homogeneous coordinates
... Homogeneous vectors x = (x1 , x2 , x3 )> with x3 6= 0 correspond to finite points in the real space R2 or “the set of intersections between non-parallel lines”. If we extend R2 with points having x3 = 0 (but (x1 , x2 )> 6= (0, 0)> ) we get the projective space P 2 . Points with x3 = 0 are called ide ...
... Homogeneous vectors x = (x1 , x2 , x3 )> with x3 6= 0 correspond to finite points in the real space R2 or “the set of intersections between non-parallel lines”. If we extend R2 with points having x3 = 0 (but (x1 , x2 )> 6= (0, 0)> ) we get the projective space P 2 . Points with x3 = 0 are called ide ...
Introduction
... lines are geodesics. In other words, d.x; y/ D d.x; z/ C d.z; y/ whenever z 2 Œx; y. Furthermore, if the convex domain is strictly convex, then the affine segment is the unique geodesic joining two points. The fourth Hilbert problem asks for a description of all projective metrics in a convex regio ...
... lines are geodesics. In other words, d.x; y/ D d.x; z/ C d.z; y/ whenever z 2 Œx; y. Furthermore, if the convex domain is strictly convex, then the affine segment is the unique geodesic joining two points. The fourth Hilbert problem asks for a description of all projective metrics in a convex regio ...
Contents - Harvard Mathematics Department
... infinity” are included. Moreover, when endowed with the complex topology, (complex) projective varieties are compact, unlike all but degenerate affine varieties (i.e. finite sets). It is when defining the notion of a “variety” in projective space that one encounters gradedness. Now a variety in Pn m ...
... infinity” are included. Moreover, when endowed with the complex topology, (complex) projective varieties are compact, unlike all but degenerate affine varieties (i.e. finite sets). It is when defining the notion of a “variety” in projective space that one encounters gradedness. Now a variety in Pn m ...
2.1. Functions on affine varieties. After having defined affine
... in real analysis, or holomorphic functions in complex analysis. Of course, in the case of algebraic geometry we want to have algebraic functions, i. e. (quotients of) polynomial functions. Definition 2.1.1. Let X ⊂ An be an affine variety. We call A(X) := k[x1 , . . . , xn ]/I(X) the coordinate ring ...
... in real analysis, or holomorphic functions in complex analysis. Of course, in the case of algebraic geometry we want to have algebraic functions, i. e. (quotients of) polynomial functions. Definition 2.1.1. Let X ⊂ An be an affine variety. We call A(X) := k[x1 , . . . , xn ]/I(X) the coordinate ring ...
Dimension theory of arbitrary modules over finite von Neumann
... Of course Denition 1.6 of dim (P) for a nitely generated projective A-module agrees with the usual von Neumann dimension of the associated Hilbert A-module l (P) after any choice of inner product on P. 2. The generalized dimension function In this section we give the proof of Theorem 0.6 and inves ...
... Of course Denition 1.6 of dim (P) for a nitely generated projective A-module agrees with the usual von Neumann dimension of the associated Hilbert A-module l (P) after any choice of inner product on P. 2. The generalized dimension function In this section we give the proof of Theorem 0.6 and inves ...
Projective ideals in rings of continuous functions
... Finney and Rotman [5] have presented a direct proof of Bkouche's result for locally compact spaces. This paper is concerned with the problem of topologically characterizing projectivity within the class of all ideals in C(X). The remaining paragraphs in this section introduce the terminology and not ...
... Finney and Rotman [5] have presented a direct proof of Bkouche's result for locally compact spaces. This paper is concerned with the problem of topologically characterizing projectivity within the class of all ideals in C(X). The remaining paragraphs in this section introduce the terminology and not ...
MARCH 10 Contents 1. Strongly rational cones 1 2. Normal toric
... An affine toric variety X is normal if and only is X = Spec(C[Sσ ]) where σ is a strongly convex polyhedral cone. Recall that an affine toric variety Spec(C[S]) ⊂ Cs has a fixed point if and only if S ∩ (−S) = {0}. In this case the fixed point is the 0. In the case S = Sσ , this translates to the co ...
... An affine toric variety X is normal if and only is X = Spec(C[Sσ ]) where σ is a strongly convex polyhedral cone. Recall that an affine toric variety Spec(C[S]) ⊂ Cs has a fixed point if and only if S ∩ (−S) = {0}. In this case the fixed point is the 0. In the case S = Sσ , this translates to the co ...