Parshin chain

In number theory, a Parshin chain is a higher-dimensional analogue of a place of an algebraic number field. They were introduced by Parshin (1978) in order to define an analogue of the idele class group for 2-dimensional schemes.

A Parshin chain of dimension s on a scheme is a finite sequence of points p₀, p₁, ..., p_s such that p_i has dimension i and each point is contained in the closure of the next one.

References

• Kerz, Moritz (2011), "Ideles in higher dimension", Mathematical Research Letters, 18 (4): 699–713, arXiv:0907.5337, doi:10.4310/mrl.2011.v18.n4.a9, ISSN 1073-2780, MR 2831836
• Parshin, A. N. (1978), "Abelian coverings of arithmetic schemes", Doklady Akademii Nauk SSSR, 243 (4): 855–858, ISSN 0002-3264, MR 0514485