Dualizing module

In 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.

Definition

A 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 Extⁿ
_R(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.

Examples

If 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.

References

• Bourbaki, N. (2007), Algèbre commutative. Chapitre 10, Éléments de mathématique (in French), Springer-Verlag, Berlin, ISBN 978-3-540-34394-3; 3-540-34394-6, MR 2333539 {{citation}}: Check |isbn= value: invalid character (help)
• Bruns, Winfried; Herzog, Jürgen (1993), Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, ISBN 978-0-521-41068-7, MR 1251956