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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).
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).
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 ...
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 locally Noetherian scheme is a locally Noetherian formal scheme in the canonical way: the formal completion along itself. In other words, the category of locally Noetherian formal schemes contains all locally Noetherian schemes. Formal schemes were motivated by and generalize Zariski's theory of formal holomorphic functions.
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
In May 2012, Joseph Blimline was sentenced to 20 years in federal prison for operating two oil and gas Ponzi schemes. He operated a Ponzi scheme from 2003 to 2005 in Michigan, netting over $28 million. He then operated a Ponzi scheme in Texas, using a company called Provident Royalties, that lasted from 2006 to 2009 and netted over $400 million ...
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.