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étale Spectrum
In algebraic geometry, a branch of mathematics, the étale spectrum of a commutative ring or an E-infinity ring, E∞-ring, denoted by Specét or Spét, is an analog of the prime spectrum Spec of a commutative ring that is obtained by replacing Zariski topology with étale topology. The precise definition depends on one's formalism. But the idea of the definition itself is simple. The usual prime spectrum Spec enjoys the relation: for a scheme (mathematics), scheme (''S'', ''O''''S'') and a commutative ring ''A'', :\operatorname(S, \operatorname(A)) \simeq \operatorname(A, \Gamma(S, \mathcal_S)) where Hom on the left is for morphism of schemes, morphisms of schemes and Hom on the right ring homomorphisms. This is to say Spec is the right adjoint to the global section functor (S, \mathcal_S) \mapsto \Gamma(S, \mathcal_S). So, roughly, one can (and typically does) simply define the étale spectrum Spét to be the right adjoint to the global section functor on the category of "spaces" wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
