A dualizing module need not be unique because the tensor product of any dualizing module with a rank 1 projective module 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 isomorphism.
A Noetherian ring does not necessarily have a dualizing module. Any ring with a dualizing module must be Cohen–Macaulay. Conversely if a Cohen–Macaulay ring is a quotient of a Gorenstein ring 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 dualizing complex as a substitute.
Examples
If R is a Gorenstein ring, then R considered as a module over itself is a dualizing module.
If R is an Artinianlocal ring then the Matlis module of R (the injective hull of the residue field) is the dualizing module.
The Artinian local ring R = k[x,y]/(x2,y2,xy) has a unique dualizing module, but it is not isomorphic to R.
The ring Z[√–5] has two non-isomorphic dualizing modules, corresponding to the two classes of invertible ideals.
The local ring k[x,y]/(y2,xy) is not Cohen–Macaulay so does not have a dualizing module.
dualizing, module, abstract, algebra, dualizing, module, also, called, canonical, module, module, over, commutative, ring, that, analogous, canonical, bundle, smooth, variety, used, grothendieck, local, duality, contents, definition, examples, also, references. In abstract algebra a dualizing module also called a canonical module is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety It is used in Grothendieck local duality Contents 1 Definition 2 Examples 3 See also 4 ReferencesDefinition EditA dualizing module for a Noetherian ring R is a finitely generated module M such that for any maximal ideal m the R m vector space ExtnR R m M vanishes if n height m and is 1 dimensional if n height m A dualizing module need not be unique because the tensor product of any dualizing module with a rank 1 projective module 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 isomorphism A Noetherian ring does not necessarily have a dualizing module Any ring with a dualizing module must be Cohen Macaulay Conversely if a Cohen Macaulay ring is a quotient of a Gorenstein ring 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 dualizing complex as a substitute Examples EditIf R is a Gorenstein ring then R considered as a module over itself is a dualizing module If R is an Artinian local ring then the Matlis module of R the injective hull of the residue field is the dualizing module The Artinian local ring R k x y x2 y2 xy has a unique dualizing module but it is not isomorphic to R The ring Z 5 has two non isomorphic dualizing modules corresponding to the two classes of invertible ideals The local ring k x y y2 xy is not Cohen Macaulay so does not have a dualizing module See also Editdualizing sheafReferences EditBourbaki N 2007 Algebre commutative Chapitre 10 Elements de mathematique in French Springer Verlag Berlin ISBN 978 3 540 34394 3 MR 2333539 Bruns Winfried Herzog Jurgen 1993 Cohen Macaulay rings Cambridge Studies in Advanced Mathematics vol 39 Cambridge University Press ISBN 978 0 521 41068 7 MR 1251956 Retrieved from https en wikipedia org w index php title Dualizing module amp oldid 830998207, wikipedia, wiki, book, books, library,
