In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, Matlis duality is a
duality between
Artinian and
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
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 s ...
over a complete Noetherian
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 n ...
. In the special case when the local ring has a field mapping to the
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
it is closely related to earlier work by
Francis Sowerby Macaulay
Francis Sowerby Macaulay FRS (11 February 1862, Witney – 9 February 1937, Cambridge) was an English mathematician who made significant contributions to algebraic geometry. He is known for his 1916 book ''The Algebraic Theory of Modular Systems ...
on
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
s and is sometimes called Macaulay duality, and the general case was introduced by .
Statement
Suppose that ''R'' is a Noetherian complete local ring with residue field ''k'', and choose ''E'' to be 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'' (sometimes called a Matlis module). The dual ''D''
''R''(''M'') of a module ''M'' is defined to be Hom
''R''(''M'',''E''). Then Matlis duality states that the duality functor ''D''
''R'' gives an anti-equivalence between the categories of Artinian and Noetherian ''R''-modules. In particular the duality functor gives an anti-equivalence from the category of finite-length modules to itself.
Examples
Suppose that the Noetherian complete local ring ''R'' has a subfield ''k'' that maps onto a subfield of finite index of its residue field ''R''/''m''. Then the Matlis dual of any ''R''-module is just its dual as a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
over ''k'', if the module is given its ''m''-adic topology. In particular the dual of ''R'' as a topological vector space over ''k'' is a Matlis module. This case is closely related to work of Macaulay on graded polynomial rings and is sometimes called Macaulay duality.
If ''R'' is a
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R'' i ...
with
quotient field
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
''K'' then the Matlis module is ''K''/''R''. In the special case when ''R'' is the ring of
''p''-adic numbers, the Matlis dual of a
finitely-generated module
In mathematics, a finitely generated module is a module (mathematics), module that has a Finite set, finite generating set. A finitely generated module over a Ring (mathematics), ring ''R'' may also be called a finite ''R''-module, finite over ''R' ...
is the
Pontryagin dual
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...
of it considered as a
locally compact abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
.
If ''R'' is a Cohen–Macaulay local ring of dimension ''d'' with
dualizing module Ω, then the Matlis module is given by the
local cohomology In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by , and in 1961-2 at IHES written up as SGA2 - , republished as . Given a fu ...
group H(Ω). In particular if ''R'' is an Artinian local ring then the Matlis module is the same as the dualizing module.
Explanation using adjoint functors
Matlis duality can be conceptually explained using the language of
adjoint functor
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
s and
derived categories
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
:
[ Paul Balmer, Ivo Dell'Ambrogio, and Beren Sanders]
''Grothendieck-Neeman duality and the Wirthmüller isomorphism''
2015. Example 7.2. the functor between the derived categories of ''R''- and ''k''-modules induced by regarding a ''k''-module as an ''R''-module, admits a right adjoint
(derived
internal Hom
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory ...
)
:
This right adjoint sends the injective hull
mentioned above to ''k'', which is a
dualizing object in
. This abstract fact then gives rise to the above-mentioned equivalence.
See also
*
Grothendieck local duality In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves.
Statement
Suppose that ''R'' is a Cohen–Macaulay local ring of dimension ''d'' ...
References
*
*{{Citation , last1=Matlis , first1=Eben , author1-link=Eben Matlis , title=Injective modules over Noetherian rings , url=https://projecteuclid.org/euclid.pjm/1103039896 , archive-url=https://web.archive.org/web/20140503194835/http://projecteuclid.org/euclid.pjm/1103039896 , url-status=dead , archive-date=2014-05-03 , mr=0099360 , year=1958 , journal=
Pacific Journal of Mathematics
The Pacific Journal of Mathematics is a mathematics research journal supported by several universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisat ...
, issn=0030-8730 , volume=8 , pages=511–528 , doi=10.2140/pjm.1958.8.511 , doi-access=free
Commutative algebra
Theorems in algebra