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are G-rings, and it is quite hard to construct examples of
Noetherian rings that are not G-rings. The concept is named after
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are local G-rings. (It is enough to check this just for the maximal ideals, so in particular local G-rings are G-rings.)
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A ring is called a local G-ring if it is a
Noetherian local ring and the map to its completion (with respect to its
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is regular (defined below). Almost all
Noetherian rings that occur naturally in
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A ring is called a G-ring if it is
Noetherian and all its
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over the generic point is not geometrically regular so
302:Here is an example of a discrete valuation ring
250:Every complete Noetherian local ring is a G-ring
342:such that is finite then the formal fiber of
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287:Every localization of a G-ring is a G-ring
384:Publ. Math. IHÉS 24 (1965), section 7
294:over a G-ring is a G-ring. This is a
257:in a finite number of variables over
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196:) is geometrically regular over the
381:Eléments de géométrie algébrique IV
76:A ring that is both a G-ring and a
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276:0, and in particular the ring of
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310:>0 which is not a G-ring. If
53:such that the map of any of its
378:A. Grothendieck, J. Dieudonné,
314:is any field of characteristic
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425:. You can help Knowledge by
170:if it is flat and for every
84:, and if in addition it is
27:For the Saturn G ring, see
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477:Commutative algebra stubs
290:Every finitely generated
253:Every ring of convergent
282:discrete valuation rings
350:is not a G-ring. Here
284:) that are not G-rings.
421:-related article is a
71:Alexander Grothendieck
354:denotes the image of
115:geometrically regular
298:due to Grothendieck.
101:A (Noetherian) ring
86:universally catenary
82:quasi-excellent ring
472:Commutative algebra
419:commutative algebra
389:Commutative algebra
36:commutative algebra
360:Frobenius morphism
306:of characteristic
174: ∈ Spec(
63:algebraic geometry
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334:of power series Σ
213:Popescu's theorem
44:Grothendieck ring
16:(Redirected from
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124:finite extension
88:it is called an
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18:Regular morphism
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247:is a G-ring
122:if for any
96:Definitions
55:local rings
466:Categories
373:References
358:under the
166:is called
112:is called
59:completion
48:Noetherian
133:the ring
278:integers
238:Examples
78:J-2 ring
332:subring
330:is the
296:theorem
292:algebra
182: ⊗
168:regular
137: ⊗
57:to the
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268:Every
243:Every
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40:G-ring
417:This
245:field
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147:is a
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46:is a
423:stub
392:ISBN
51:ring
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272:in
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154:A
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