Intersection numberIn mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem. The intersection number is obvious in certain cases, such as the intersection of the x- and y-axes in a plane, which should be one.
Six operationsIn mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism. It originally sprang from the relations in étale cohomology that arise from a morphism of schemes f : X → Y. The basic insight was that many of the elementary facts relating cohomology on X and Y were formal consequences of a small number of axioms. These axioms hold in many cases completely unrelated to the original context, and therefore the formal consequences also hold.
Éléments de géométrie algébriqueThe Éléments de géométrie algébrique ("Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut des Hautes Études Scientifiques. In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes, which he defined. The work is now considered the foundation stone and basic reference of modern algebraic geometry.
Nerve complexIn topology, the nerve complex of a set family is an abstract complex that records the pattern of intersections between the sets in the family. It was introduced by Pavel Alexandrov and now has many variants and generalisations, among them the Čech nerve of a cover, which in turn is generalised by hypercoverings. It captures many of the interesting topological properties in an algorithmic or combinatorial way. Let be a set of indices and be a family of sets . The nerve of is a set of finite subsets of the index set .
