Geometric genusIn algebraic geometry, the geometric genus is a basic birational invariant p_g of algebraic varieties and complex manifolds. The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number h^n,0 (equal to h^0,n by Serre duality), that is, the dimension of the canonical linear system plus one. In other words for a variety V of complex dimension n it is the number of linearly independent holomorphic n-forms to be found on V.
Algèbre graduéevignette|Un organigramme de diverses structures algébriques et leurs relations les unes avec les autres. En mathématiques, en algèbre linéaire, on appelle algèbre graduée une algèbre dotée d'une structure supplémentaire, appelée graduation. Soit A une algèbre sur un corps (ou plus généralement sur un anneau) K. Une graduation sur A est la donnée d’une famille de sous-espaces vectoriels de A vérifiant : c'est-à-dire que . L’algèbre A est alors dite graduée (parfois N-graduée, comme cas particulier de la notion d'algèbre M-graduée pour un monoïde M).
Algebraic geometry of projective spacesThe 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.
Géométrie complexeIn mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.
Base change theoremsIn mathematics, the base change theorems relate the and the of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves: where is a of topological spaces and is a sheaf on X. Such theorems exist in different branches of geometry: for (essentially arbitrary) topological spaces and proper maps f, in algebraic geometry for (quasi-)coherent sheaves and f proper or g flat, similarly in analytic geometry, but also for étale sheaves for f proper or g smooth.
Fibré tangentEn mathématiques, et plus précisément en géométrie différentielle, le fibré tangent TM associé à une variété différentielle M est la somme disjointe de tous les espaces tangents en tous les points de la variété, soit : où est l'espace tangent de M en x. Un élément de TM est donc un couple (x, v) constitué d'un point x de M et d'un vecteur v tangent à M en x. Le fibré tangent peut être muni d'une topologie découlant naturellement de celle de M.
Coherent dualityIn mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory. The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality more apparent.
Resolution (algebra)In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of s of an ), which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions.
Euler sequenceIn 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.
Dual bundleIn mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces. The dual bundle of a vector bundle is the vector bundle whose fibers are the dual spaces to the fibers of . Equivalently, can be defined as the Hom bundle that is, the vector bundle of morphisms from to the trivial line bundle Given a local trivialization of with transition functions a local trivialization of is given by the same open cover of with transition functions (the inverse of the transpose).
D-moduleEn mathématiques, un D-module est un module sur un anneau D d'opérateurs différentiels. L'intérêt principal des D-modules réside en son utilisation dans l'étude d'équations aux dérivées partielles. La théorie générale des D-modules nécessite une variété algébrique lisse X définie sur un corps K algébriquement clos de caractéristique nulle, par exemple K = C. Le faisceau des opérateurs différentiels DX est défini comme la OX-algèbre générée par les champs de vecteurs sur X, interprétés comme des dérivations.
MacOSmacOS (auparavant Mac OS X – , puis OS X) est un système d’exploitation partiellement propriétaire développé et commercialisé par Apple depuis , dont la version la plus récente est macOS Ventura () pour la version actuelle lancée le 24 octobre 2022 et macOS Monterey (12) lancée le 25 octobre 2021 pour le grand public. Avec iOS, iPadOS, watchOS et tvOS, il fait partie des systèmes d'exploitation d'Apple. macOS est le successeur de Mac OS Classic, la principale série des systèmes d'exploitation d'Apple depuis .
