In mathematics, a schematic diagram is an important concept that links the mathematical subdivisions of algebraic geometry, commutative algebra and number theory. Schemes were introduced in mathematics by Alexander Grothendieck, with the aim of generalizing the notion of algebraic variety; Some consider schemas as the research object of excellence in modern algebraic geometry. Formally, a scheme is a topological space along with commutative rings for all open sub-assemblies of this topological space. A scheme is created by the "glueing" of ring spectra (spaces of prime ideals) of commutative rings along their open collections. Formal definition
An affiniation scheme X is a topological space provided with a local Oops of local rings (the structural sheath), such that X is isomorphic with the ringer spectrum Spec R of a commutative ring R, provided with the associated spectrum shear.
A scheme X is a topological space provided with a sketch OX of local rings (again referred to as the structural schematic), such that X is local isomorphic with an affiency scheme, that is, X allows open cover consisting of subcategories Ui such that each Ui is isomorphic with a affiliate scheme. Accountability
They are an affine algebraic variety, i.e. the zero point collection of a prime ideal p in a polynomial k [X1, ..., Xn]. The quotient ring R = k [X1, ..., Xn] / p, which arises from multiplexing modulo elements of the prime ideal, can be construed as a ring of k-value functions on V, the so-called regular functions.
The topological space Spec R strongly resembles V. The points of Spec R are the prime dimensions of R while the V points correspond to the maximum ideals of R. The spectrum shift of R explains the relationship between points and other irreducible V collections, on the one hand, and the regular functions in their environments.
For a projective algebraic variety, such a unique spectrum shovel does not exist, but a projective variety can be covered with open subdivisions that form affine varieties.
The concept scheme thus includes the classic algebraic varieties, in addition to a number of abstract objects that do not fall under the classic definition.
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