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In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
Schemes can also be categorised as flagship schemes. [10] 10 flagship schemes were allocated ₹ 1.5 lakh crore (equivalent to ₹ 1.7 trillion or US$20 billion in 2023) in the 2021 Union budget of India. [10] The subsidy for kerosene, started in the 1950s, was slowly decreased since 2009 and eliminated in 2022. [11] [12] [13]
Hom(−,X) : (Affine schemes) op Sets. sending an affine scheme Y to the set of scheme maps. [4] A scheme is determined up to isomorphism by its functor of points. This is a stronger version of the Yoneda lemma, which says that a X is determined by the map Hom(−,X) : Schemes op → Sets.
Fix a scheme S, called a base scheme. Then a morphism : is called a scheme over S or an S-scheme; the idea of the terminology is that it is a scheme X together with a map to the base scheme S. For example, a vector bundle E → S over a scheme S is an S-scheme.
A rhyme scheme is the pattern of rhymes at the end of each line of a poem or song. It is usually referred to by using letters to indicate which lines rhyme; lines ...
An example of a singular (non-smooth) scheme over a field k is the closed subscheme x 2 = 0 in the affine line A 1 over k. An example of a singular (non-smooth) variety over k is the cuspidal cubic curve x 2 = y 3 in the affine plane A 2 , which is smooth outside the origin ( x , y ) = (0,0).
The universal examples of flat morphisms of schemes are given by Hilbert schemes. This is because Hilbert schemes parameterize universal classes of flat morphisms, and every flat morphism is the pullback from some Hilbert scheme. I.e., if : is flat, there exists a commutative diagram
For a locally Noetherian scheme X, the local rings, are also Noetherian rings. A Noetherian scheme is a Noetherian topological space. But the converse is false in general; consider, for example, the spectrum of a non-Noetherian valuation ring. The definitions extend to formal schemes.