In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
, Grothendieck local duality is a
duality theorem
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involutio ...
for
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
of
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
over
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...
s, analogous to
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexa ...
of
coherent sheaves.
Statement
Suppose that ''R'' is a
Cohen–Macaulay local ring of dimension ''d'' with maximal ideal ''m'' and residue field ''k'' = ''R''/''m''. Let ''E''(''k'') be a
Matlis module
In algebra, Matlis duality is a duality between Artinian and Noetherian modules over a complete Noetherian local ring. In the special case when the local ring has a field mapping to the residue field it is closely related to earlier work by Franc ...
, an
injective hull
In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in .
Definition
...
of ''k'', and let be the completion of its
dualizing module 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.
Definition
A dualizing module for ...
. Then for any ''R''-module ''M'' there is an isomorphism of modules over the completion of ''R'':
:
where ''H''
''m'' is a
local cohomology group.
There is a generalization to Noetherian local rings that are not Cohen–Macaulay, that replaces the dualizing module with a
dualizing complex.
See also
*
Matlis duality
In algebra, Matlis duality is a duality (mathematics), duality between Artinian module, Artinian and Noetherian module, Noetherian module (mathematics), modules over a complete Noetherian local ring. In the special case when the local ring has a fi ...
References
*
Commutative algebra
Duality theories
{{abstract-algebra-stub