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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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for a simple point of an algebraic variety is not equivalent to the condition that the local ring is regular.
244:"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie"
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gave the following two examples of local rings that are regular but not geometrically regular.
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280:(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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283:Transactions of the American Mathematical Society
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129:A Noetherian local ring containing a field
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180:th power. Then every point of the curve
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122:) that, over non-perfect fields, the
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192:is regular. However over the field
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297:10.1090/s0002-9947-1947-0021694-1
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133:is geometrically regular over
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164:is a field of characteristic
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100:absolutely simple points
236:Grothendieck, Alexandre
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137:if and only if it is
18:Geometrically regular
330:Commutative algebra
335:Algebraic geometry
262:10.1007/bf02684322
124:Jacobian criterion
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47:regular local ring
172:is an element of
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290:(1): 1–52,
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324:Categories
229:References
141:over
112:André Weil
242:(1965).
217:See also
149:Examples
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104:perfect
88:schemes
78:over a
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