Euler sequence

In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an -fold sum of the dual of the Serre twisting sheaf. The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.) Let be the n-dimensional projective space over a commutative ring A. Let be the sheaf of 1-differentials on this space, and so on. The Euler sequence is the following exact sequence of sheaves on : The sequence can be constructed by defining a homomorphism with and in degree 1, surjective in degrees , and checking that locally on the standard charts, the kernel is isomorphic to the relative differential module. We assume that A is a field k. The exact sequence above is dual to the sequence where is the tangent sheaf of . Let us explain the coordinate-free version of this sequence, on for an -dimensional vector space V over k: This sequence is most easily understood by interpreting sections of the central term as 1-homogeneous vector fields on V. One such section, the Euler vector field, associates to each point of the variety the tangent vector . This vector field is radial in the sense that it vanishes uniformly on 0-homogeneous functions, that is, the functions that are invariant by homothetic rescaling, or "independent of the radial coordinate". A function (defined on some open set) on gives rise by pull-back to a 0-homogeneous function on V (again partially defined). We obtain 1-homogeneous vector fields by multiplying the Euler vector field by such functions. This is the definition of the first map, and its injectivity is immediate. The second map is related to the notion of derivation, equivalent to that of vector field. Recall that a vector field on an open set U of the projective space can be defined as a derivation of the functions defined on this open set. Pulled-back in V, this is equivalent to a derivation on the preimage of U that preserves 0-homogeneous functions.

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (3)

Concepts associés (7)

Séances de cours associées (5)

The cohomology of semi-simple Lie groups, viewed from infinity

Nicolas Monod

We prove that the real cohomology of semi-simple Lie groups admits boundary values, which are measurable cocycles on the Furstenberg boundary. This generalises known invariants such as the Maslov index on Shilov boundaries, the Euler class on projective sp ...

2022

Two-dimensional systems with C2T (PT) symmetry exhibit the Euler class topology E is an element of Z in each two-band subspace realizing a fragile topology beyond the symmetry indicators. By systematically studying the energy levels of Euler insulating pha ...

AMER PHYSICAL SOC2022

The norm of the Euler class

Nicolas Monod

We prove that the norm of the Euler class

\mathcal{E}

for flat vector bundles is

2^{-n}

(in even dimension

n

, since it vanishes in odd dimension). This shows that the Sullivan-Smillie bound considered by Gromov and Ivanov-Turaev is sharp. We construc ...

Springer Verlag2012

Tautological bundle

In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of -dimensional subspaces of , given a point in the Grassmannian corresponding to a -dimensional vector subspace , the fiber over is the subspace itself. In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles.

Algebraic geometry of projective spaces

The concept of a Projective space plays a central role in algebraic geometry. This article aims to define the notion in terms of abstract algebraic geometry and to describe some basic uses of projective spaces. Let k be an algebraically closed field, and V be a finite-dimensional vector space over k. The symmetric algebra of the dual vector space V* is called the polynomial ring on V and denoted by k[V]. It is a naturally graded algebra by the degree of polynomials.

Complex projective space

In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a complex Euclidean space (see below for an intuitive account). Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space.

De nombreuses gerbes inversées MATH-510: Algebraic geometry II - schemes and sheaves

Introduit de larges gerbes invertibles et leurs propriétés en géométrie algébrique.

Ensembles vectoriaux et gerbes locales gratuites MATH-510: Algebraic geometry II - schemes and sheaves

Couvre les faisceaux vectoriels, les gerbes locales libres, la profondeur, Serre torsion de la gerbe, et les modules gradués en géométrie algébrique.

Théorème algébrique de la Kunneth MATH-506: Topology IV.b - cohomology rings

Couvre le théorème algébrique de la Kunneth, expliquant les complexes de chaîne et les calculs de cohomologie.