Concept
Spectrum of a ring
Related concepts (26)
Algebraic geometry
Algebraic geometry is a branch of mathematics which classically studies 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.
Generic point
In algebraic geometry, a generic point P of an algebraic variety X is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic geometry, a generic point of an affine or projective algebraic variety of dimension d is a point such that the field generated by its coordinates has transcendence degree d over the field generated by the coefficients of the equations of the variety.
Projective variety
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .
Equivalence of categories
In , a branch of abstract mathematics, an equivalence of categories is a relation between two that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned.
Kolmogorov space
In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them has a neighborhood not containing the other. In a T0 space, all points are topologically distinguishable. This condition, called the T0 condition, is the weakest of the separation axioms. Nearly all topological spaces normally studied in mathematics are T0 spaces. In particular, all T1 spaces, i.e.
Sheaf (mathematics)
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts).
Ringed space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous (scalar-valued) functions on open subsets. Among ringed spaces, especially important and prominent is a locally ringed space: a ringed space in which the analogy between the stalk at a point and the ring of germs of functions at a point is valid.
Irreducible component
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (for set inclusion) for this property. For example, the set of solutions of the equation xy = 0 is not irreducible, and its irreducible components are the two lines of equations x = 0 and y =0.
Stalk (sheaf)
The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point. Sheaves are defined on open sets, but the underlying topological space consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a single fixed point of . Conceptually speaking, we do this by looking at small neighborhoods of the point. If we look at a sufficiently small neighborhood of , the behavior of the sheaf on that small neighborhood should be the same as the behavior of at that point.
Sober space
In mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every irreducible closed subset has a unique generic point. Sober spaces have a variety of cryptomorphic definitions, which are documented in this section. All except the definition in terms of nets are described in. In each case below, replacing "unique" with "at most one" gives an equivalent formulation of the T0 axiom.