In
number theory, a Parshin chain is a higher-dimensional analogue of a
place of an
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
. They were introduced by in order to define an analogue of the
idele class group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the ...
for 2-dimensional
schemes.
A Parshin chain of dimension ''s'' on a scheme is a finite sequence of
points
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
''p''
0, ''p''
1, ..., ''p''
''s'' such that ''p''
''i'' has dimension ''i'' and each point is contained in the closure of the next one.
References
*
*
Algebraic number theory
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