Approximation property (ring theory)
In algebra, a commutative Noetherian ring A is said to have the approximation property with respect to an ideal I if each finite system of polynomial equations with coefficients in A has a solution in A if and only if it has a solution in the I-adic completion of A.[1][2] The notion of the approximation property is due to Michael Artin.
See also
Notes
- ^ Rotthaus, Christel (1997). "Excellent Rings, Henselian Rings, and the Approximation Property". Rocky Mountain Journal of Mathematics. 27 (1): 317–334. doi:10.1216/rmjm/1181071964. JSTOR 44238106.
- ^ "Tag 07BW: Smoothing Ring Maps". The Stacks Project. Columbia University, Department of Mathematics. Retrieved 2018-02-19.
References
- Popescu, Dorin (1986). "General Néron desingularization and approximation". Nagoya Mathematical Journal. 104: 85–115. doi:10.1017/S0027763000022698.
- Rotthaus, Christel (1987). "On the approximation property of excellent rings". Inventiones Mathematicae. 88: 39–63. Bibcode:1987InMat..88...39R. doi:10.1007/BF01405090.
- Artin, M (1969). "Algebraic approximation of structures over complete local rings". Publications Mathématiques de l'IHÉS. 36: 23–58. doi:10.1007/BF02684596. ISSN 0073-8301.
- Artin, M (1968). "On the solutions of analytic equations". Inventiones Mathematicae. 5 (4): 277–291. Bibcode:1968InMat...5..277A. doi:10.1007/BF01389777. ISSN 0020-9910.
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