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In the previous example, the equation defining the curve becomes reducible over a finite extension of the base field. This is not the real cause of the phenomenon: Chevalley pointed out to
Zariski that the curve
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201:(with the notation of the previous example) is absolutely irreducible but still has a point that is regular but not geometrically regular.
185:, every point of the curve is singular. So the points of this curve are regular but not geometrically regular.
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for a simple point of an algebraic variety is not equivalent to the condition that the local ring is regular.
233:"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie"
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95:, geometrically regular rings are the same as regular rings. Geometric regularity originated when
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gave the following two examples of local rings that are regular but not geometrically regular.
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269:(1947), "The concept of a simple point of an abstract algebraic variety.",
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are defined in a similar way. In older terminology, points with regular
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after any finite extension of the base field. Geometrically regular
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272:Transactions of the American Mathematical Society
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118:A Noetherian local ring containing a field
29:It has been suggested that this article be
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169:th power. Then every point of the curve
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111:) that, over non-perfect fields, the
238:Publications Mathématiques de l'IHÉS
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286:10.1090/s0002-9947-1947-0021694-1
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61:geometrically regular ring
46:Proposed since July 2024.
89:absolutely simple points
225:Grothendieck, Alexandre
157: > 0 and
126:if and only if it is
319:Commutative algebra
324:Algebraic geometry
251:10.1007/bf02684322
113:Jacobian criterion
57:algebraic geometry
36:regular local ring
161:is an element of
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267:Zariski, Oscar
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279:(1): 1–52,
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313:Categories
218:References
130:over
101:André Weil
231:(1965).
206:See also
138:Examples
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93:perfect
77:schemes
67:over a
41:Discuss
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291:JSTOR
69:field
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