In mathematics, the Séminaire de Géométrie Algébrique du Bois Marie (SGA) was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris. The seminar notes were eventually published in twelve volumes, all except one in the Springer Lecture Notes in Mathematics series.
In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.
In algebraic geometry, a finite morphism between two affine varieties is a dense regular map which induces isomorphic inclusion between their coordinate rings, such that is integral over . This definition can be extended to the quasi-projective varieties, such that a regular map between quasiprojective varieties is finite if any point like has an affine neighbourhood V such that is affine and is a finite map.
In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is of finite type and satisfies any of the following equivalent conditions:
In algebraic geometry, an étale morphism is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.
In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale topology was originally introduced by Grothendieck to define étale cohomology, and this is still the étale topology's most well-known use.
In topology, constructible sets are a class of subsets of a topological space that have a relatively "simple" structure. They are used particularly in algebraic geometry and related fields. A key result known as Chevalley's theorem in algebraic geometry shows that the image of a constructible set is constructible for an important class of mappings (more specifically morphisms) of algebraic varieties . In addition, a large number of "local" geometric properties of schemes, morphisms and sheaves are (locally) constructible. Constructible sets also feature in the definition of various types of constructible sheaves in algebraic geometry and intersection cohomology.
In algebraic geometry, a local ring A is said to be unibranch if the reduced ring Ared is an integral domain, and the integral closure B of Ared is also a local ring. A unibranch local ring is said to be geometrically unibranch if the residue field of B is a purely inseparable extension of the residue field of Ared. A complex variety X is called topologically unibranch at a point x if for all complements Y of closed algebraic subsets of X there is a fundamental system of neighborhoods of x whose intersection with Y is connected.
In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free. They are due to Alexander Grothendieck.
In algebraic geometry, dévissage is a technique introduced by Alexander Grothendieck for proving statements about coherent sheaves on noetherian schemes. Dévissage is an adaptation of a certain kind of noetherian induction. It has many applications, including the proof of generic flatness and the proof that higher direct images of coherent sheaves under proper morphisms are coherent.
In algebraic geometry, a morphism of schemes f from X to Y is called quasi-separated if the diagonal map from X to X × YX is quasi-compact. A scheme X is called quasi-separated if the morphism to Spec Z is quasi-separated. Quasi-separated algebraic spaces and algebraic stacks and morphisms between them are defined in a similar way, though some authors include the condition that X is quasi-separated as part of the definition of an algebraic space or algebraic stack X. Quasi-separated morphisms were introduced by Grothendieck as a generalization of separated morphisms.
In commutative algebra and algebraic geometry, a morphism is called formally étale if it has a lifting property that is analogous to being a local diffeomorphism.
In algebraic geometry and commutative algebra, a ring homomorphism is called formally smooth if it satisfies the following infinitesimal lifting property:
In mathematics, the Grothendieck existence theorem, introduced by Grothendieck, gives conditions that enable one to lift infinitesimal deformations of a scheme to a deformation, and to lift schemes over infinitesimal neighborhoods over a subscheme of a scheme S to schemes over S.
In algebra, a Cohen ring is a field or a complete discrete valuation ring of mixed characteristic whose maximal ideal is generated by p. Cohen rings are used in the Cohen structure theorem for complete Noetherian local rings.
In algebraic geometry, a Noetherian local ring R is called parafactorial if it has depth at least 2 and the Picard group Pic(Spec − m) of its spectrum with the closed point m removed is trivial.
In algebraic geometry, the Ramanujam–Samuel theorem gives conditions for a divisor of a local ring to be principal.
In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field. Geometrically regular schemes are defined in a similar way. In older terminology, points with regular local rings were called simple points, and points with geometrically regular local rings were called absolutely simple points. Over fields that are of characteristic 0, or algebraically closed, or more generally perfect, geometrically regular rings are the same as regular rings. Geometric regularity originated when Claude Chevalley and André Weil pointed out to Oscar Zariski (1947) that, over non-perfect fields, the Jacobian criterion for a simple point of an algebraic variety is not equivalent to the condition that the local ring is regular.
In algebraic geometry, a universal homeomorphism is a morphism of schemes such that, for each morphism , the base change is a homeomorphism of topological spaces.
In algebra, specifically in the theory of commutative rings, a quasi-unmixed ring is a Noetherian ring such that for each prime ideal p, the completion of the localization Ap is equidimensional, i.e. for each minimal prime ideal q in the completion , = the Krull dimension of Ap.
