2-Reflexive Modules

• 2-reflexive modules are modules that are isomorphic to their Matlis double dual via a canonical evaluation map, establishing a key duality in commutative algebra.
• They possess the Krull–Schmidt property, allowing unique finite decompositions into indecomposable modules with local endomorphism rings.
• Finiteness conditions in Noetherian rings classify these modules as artinian-by-noetherian, underpinning applications in singularity theory and algebraic geometry.

A 2-reflexive module, also known as a Matlis reflexive module, occupies a central role in the representation theory of modules over commutative rings, commutative algebra, and singularity theory. Such a module is defined by the property that the canonical evaluation map to its Matlis double dual is an isomorphism. This notion underlies classifications of module-theoretic structure, dualities, and decomposition theory, particularly in categories exhibiting Krull–Schmidt properties and in the context of Noetherian and artinian modules.

1. Matlis Duality and the Evaluation Map

Let $A$ be a commutative ring. Matlis duality is constructed via the injective cogenerator

$E = \bigoplus_{m \in \operatorname{Max} A} E(A/m)$

where $E(A/m)$ denotes the injective hull of $A/m$. The associated Matlis duality functor is

$D(-) := \operatorname{Hom}_A(-, E)$

which is an exact, contravariant functor on the category $A$–Mod. For any $A$-modules $M, N$, this functor yields the adjunction isomorphism:

$$\operatorname{Hom}_A(M, D N) \cong \operatorname{Hom}_A(N, D M)$$

The unit of this adjunction is the natural evaluation map $\varphi_M: M \rightarrow D D M$, defined by $\varphi_M(x)(f) = f(x)$ for all $f\in D M$. The transformation $\varphi$ is natural and satisfies compatibility under further dualization (Krause, 2024).

2. Definition and Fundamental Properties of 2-Reflexive Modules

An $A$-module $M$ is Matlis reflexive (or 2-reflexive) if the canonical map

$$\varphi_M: M \stackrel{\sim}{\longrightarrow} D D M$$

is an isomorphism. That is, $M$ coincides with its Matlis double dual via the specified injective cogenerator. Modules of finite length are primary examples: if $\ell(M) < \infty$, then $M \cong D D M$.

The reflexive property translates into closure properties: reflexive modules form a Serre subcategory of $A$–Mod, being closed under submodules, quotient modules, and extensions. This follows from exactness of $D$ and the injectivity of $\varphi_M$.

3. Structural Decomposition and the Krull–Schmidt Property

The category of Matlis reflexive modules, denoted $\mathrm{refl}\,A$, is Krull–Schmidt: every reflexive module decomposes uniquely (up to isomorphism and ordering) as a finite direct sum of indecomposable reflexive modules, each possessing a local endomorphism ring [(Krause, 2024), Theorem 2.1]. This decomposition is enforced by the pure-injectivity of reflexive modules. Specifically, each reflexive module is a direct summand of some $D N$, and the continuous direct summand part must vanish in a reflexive module due to the absence of infinite direct sum decompositions [(Krause, 2024), Lemmas 2.2, 2.3].

Structural Property	Description
Serre subcategory	Closed under submodules, quotients, extensions
Krull–Schmidt property	Unique (finite) direct sum decomposition into indecomposables
Local endomorphism ring	Each indecomposable’s endomorphism algebra is local

4. Finiteness Conditions in the Noetherian Setting

In the Noetherian context, Matlis reflexivity is equivalent to several finiteness properties. For an $A$-module $M$, the following are equivalent [(Krause, 2024), Proposition 6.1]:

• $M$ is Matlis reflexive.
• $M$ is linearly compact (Zelinsky).
• No subquotient of $M$ is isomorphic to an infinite direct sum of nonzero modules.
• $M$ fits into an exact sequence

$0 \to U \to M \to V \to 0$

with $U$ Noetherian and $V$ artinian.

Thus, over a Noetherian ring, the 2-reflexive modules are precisely the "artinian-by-noetherian" modules of finite support. Absence of infinite direct summands is both necessary and sufficient for reflexivity in this case.

5. Matlis Duality on Artinian and Noetherian Modules

Restricting to $A$-modules of finite support, Matlis duality gives quasi-inverse contravariant equivalences:

• $D: \mathrm{art}\,A \rightarrow \mathrm{noeth}\,A$
• $D: \mathrm{noeth}\,A \rightarrow \mathrm{art}\,A$

Every artinian or noetherian module is 2-reflexive. Matlis duality thus establishes categorical equivalence between these classes under the functor $D = \operatorname{Hom}_A(-, E)$ [(Krause, 2024), Proposition 3.2].

6. Explicit Classification in Key Examples

In the Krull–Schmidt paradigm, classifying indecomposable 2-reflexive modules suffices. Every indecomposable lives over the completed localization $A_\mathfrak{m}$ at some maximal ideal $\mathfrak{m}$.

Principal Examples

Ring $A$	Indecomposable 2-Reflexive Modules
$\mathbb{Z}$	Finite-length abelian groups; direct sums of $\mathbb{Z}/p^n$
Dedekind domain	Completed ring $A^\wedge$, injective hull $E(A/\mathfrak{m})$, field of fractions $Q(A)$, and $A/\mathfrak{m}^n$ for $n \geq 1$
$A = k[x, y]/(xy)$ (local-complete)	String modules $M(C)$: finite or stabilizing words in $\{ x, y, x^{-1}, y^{-1} \}$; band modules $M(C, V)$: periodic word $C$, finite-length indecomposable $k[t, t^{-1}]$-module $V$

In $k[x, y]/(xy)$, such reflexives are either string modules associated with words encoding quiver-shapes or band modules determined by periodic words and local systems $V$. Duality properties map string modules to those attached to the inverse word, and band modules are self-dual up to applying $V \mapsto \operatorname{Hom}_k(V, k)$.

7. Reflexive Modules and Geometric Applications

Within the context of singularity theory, particularly rational surface singularities, the concept of reflexive modules generalizes to the geometric setting: a coherent sheaf $M$ on a normal surface $X$ is reflexive if the double dual map

$$M \to M^{**} := \operatorname{Hom}_X(\operatorname{Hom}_X(M,\mathcal{O}_X), \mathcal{O}_X)$$

is an isomorphism. On surfaces, reflexive modules correspond to maximal Cohen–Macaulay sheaves (Gustavsen et al., 2016). Blowing up such modules on singularities yields partial resolutions dominated by the minimal resolution. These constructions play essential roles in the McKay–Wunram correspondence and in the structure of flops and small resolutions in threefold geometry, relating deformations of singularities and their associated modules to deformations of corresponding resolutions (Gustavsen et al., 2016).

8. Summary

2-reflexive modules (Matlis reflexive modules) are characterized by the isomorphism between a module and its Matlis double dual. They form a Serre subcategory with the Krull–Schmidt property, admitting unique finite decompositions into indecomposables with local endomorphism rings. Over Noetherian rings, they are precisely the artinian-by-noetherian modules, equivalently the linearly compact modules or those without infinite direct sum subquotients. Explicit classifications are feasible in key classes of rings, notably $\mathbb{Z}$, Dedekind domains, and certain string algebras. Reflexive modules further underpin geometric operations in singularity theory, particularly in the study of rational surface singularities and their resolutions (Krause, 2024, Gustavsen et al., 2016).