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is also a dualizing module. However this is the only way in which the dualizing module fails to be unique: given any two dualizing modules, one is isomorphic to the tensor product of the other with a rank 1 projective module. In particular if the ring is local the dualizing module is unique up to
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then it has a dualizing module. In particular any complete local Cohen–Macaulay ring has a dualizing module. For rings without a dualizing module it is sometimes possible to use the
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A Noetherian ring does not necessarily have a dualizing module. Any ring with a dualizing module must be
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has two non-isomorphic dualizing modules, corresponding to the two classes of invertible ideals.
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199:(the injective hull of the residue field) is the dualizing module.
277:, Éléments de mathématique (in French), Springer-Verlag, Berlin,
222:) has a unique dualizing module, but it is not isomorphic to
248:) is not Cohen–Macaulay so does not have a dualizing module.
305:, Cambridge Studies in Advanced Mathematics, vol. 39,
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considered as a module over itself is a dualizing module.
153:. Conversely if a Cohen–Macaulay ring is a quotient of a
137:A dualizing module need not be unique because the
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299:Bruns, Winfried; Herzog, JĂĽrgen (1993),
141:of any dualizing module with a rank 1
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275:Algèbre commutative. Chapitre 10
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173:is a Gorenstein ring, then
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307:Cambridge University Press
46:Grothendieck local duality
65:finitely generated module
56:A dualizing module for a
36:that is analogous to the
202:The Artinian local ring
302:Cohen-Macaulay rings
130: = height(
118: ≠height(
341:Commutative algebra
70:such that for any
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284:978-3-540-34394-3
161:as a substitute.
159:dualizing complex
143:projective module
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44:. It is used in
38:canonical bundle
34:commutative ring
26:canonical module
24:, also called a
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236:The local ring
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264:References
189:local ring
52:Definition
229:The ring
191:then the
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186:Artinian
165:Examples
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