|
Quasi-compact Morphism
In algebraic geometry, a morphism f: X \to Y between schemes is said to be quasi-compact if ''Y'' can be covered by open affine subschemes V_i such that the pre-images f^(V_i) are quasi-compact (as topological space). If ''f'' is quasi-compact, then the pre-image of a quasi-compact open subscheme (e.g., open affine subscheme) under ''f'' is quasi-compact. It is not enough that ''Y'' admits a covering by quasi-compact open subschemes whose pre-images are quasi-compact. To give an example, let ''A'' be a ring that does not satisfy the ascending chain conditions on radical ideals, and put X = \operatorname A. ''X'' contains an open subset ''U'' that is not quasi-compact. Let ''Y'' be the scheme obtained by gluing two ''Xs along ''U''. ''X'', ''Y'' are both quasi-compact. If f: X \to Y is the inclusion of one of the copies of ''X'', then the pre-image of the other ''X'', open affine in ''Y'', is ''U'', not quasi-compact. Hence, ''f'' is not quasi-compact. A morphism from a quasi-compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fpqc Morphism
In algebraic geometry, there are two slightly different definitions of an fpqc morphism, both variations of faithfully flat morphisms. Sometimes an fpqc morphism means one that is faithfully flat and quasicompact. This is where the abbreviation fpqc comes from: fpqc stands for the French phrase "fidèlement plat et quasi-compact", meaning "faithfully flat and quasi-compact". However it is more common to define an fpqc morphism f: X \to Y of schemes to be a faithfully flat morphism that satisfies the following equivalent conditions: # Every quasi-compact open subset of Y is the image of a quasi-compact open subset of ''X''. # There exists a covering V_i of Y by open affine subschemes such that each V_i is the image of a quasi-compact open subset of X. # Each point x \in X has a neighborhood U such that f(U) is open and f: U \to f(U) is quasi-compact. # Each point x \in X has a quasi-compact neighborhood such that f(U) is open affine. Examples: An open faithfully flat morphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
