Pure Minimal Injective Coresolution

• Pure minimal injective coresolution is a homological construction that synthesizes minimality, injectivity, and purity to yield a unique resolution of modules.
• It plays a crucial role in computing cosupport in affine algebraic geometry and classifying incidence algebras via lattice-theoretic properties.
• The methodology leverages purity conditions, invariant grades, and functorial techniques to bridge ring spectra, posets, and module structures.

A pure minimal injective coresolution is a homological object in module theory and the representation theory of rings, synthesizing the concepts of minimality, injectivity, and purity in resolutions or coresolutions of modules. These objects play a key role in the structure theory of rings, cosupport computations in affine algebraic geometry, and the homological classification of certain classes of incidence algebras associated to lattices. The existence, uniqueness, and form of pure minimal injective coresolutions are tightly governed by the module-theoretic and combinatorial properties of the underlying ring or algebra, with rich connections to the structure of the spectrum or the associated poset.

1. Definitions and Core Constructions

For a commutative noetherian ring $R$, a pure-injective module $P$ is one which is injective relative to all pure exact sequences, i.e., for any pure exact sequence $0 \to A \to B \to C \to 0$, the induced map $\Hom_R(B,P) \to \Hom_R(A,P)$ is surjective. Equivalently, pure-injective modules are direct summands of products of finitely presented modules, or of completions of free modules at primes.

A pure-injective coresolution of a module $M$ over $R$ is an exact complex

$$0 \longrightarrow M \xrightarrow{\varepsilon^0} P^0 \xrightarrow{d^0} P^1 \xrightarrow{d^1} P^2 \longrightarrow \cdots$$

where every $P^i$ is pure-injective. The minimality condition requires that each map $\Ker d^i \hookrightarrow P^i$ exhibits the domain as the pure-injective envelope of the cycle, or equivalently that no nonzero direct summand of $P^i$ can be discarded. Over noetherian rings, such minimal resolutions exist and are unique up to isomorphism of complexes.

In the more general algebraic setting (e.g., for a two-sided noetherian Iwanaga–Gorenstein $K$-algebra $A$ of self-injective dimension $d$), a pure minimal injective coresolution of the module $_A A$ is an exact sequence

$$0 \longrightarrow A \xrightarrow{\iota} I^0 \xrightarrow{d^0} I^1 \to \cdots \xrightarrow{d^{d-1}} I^d \longrightarrow 0$$

such that each $I^n$ is pure (i.e., every nonzero submodule has grade equal to $\grade(I^n)=n$), and each map $d^n$ is minimal in the sense above (Gottesman et al., 5 Nov 2025).

2. Structural Results for Affine Rings and Cosupport

A fundamental result for affine commutative noetherian $k$-algebras $R$ is that the cosupport of $R$ equals $\operatorname{Spec} R$, i.e.,

$\cosupp_R R = \operatorname{Spec} R,$

where for $X \in D(R)$,

$\cosupp_R X = \{ \mathfrak{p} \in \operatorname{Spec} R \mid \RHom_R(K(\mathfrak{p}), X) \not\simeq 0 \}$

and $$K(\mathfrak{p}) = R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}}$$ (Nakamura, 2018).

Given this, the minimal pure-injective coresolution of $R$ as an $R$-module exhibits the following explicit structure for suitable cardinal data and dimensional constraints: $$0 \longrightarrow R \xrightarrow{\varepsilon^0} \prod_{\mathfrak{p} \in W_0} \widehat{R_{\mathfrak{p}}} \xrightarrow{d^0} \prod_{\mathfrak{p} \in W_1} T_{\mathfrak{p}}^{(1)} \xrightarrow{d^1} \prod_{\mathfrak{p} \in W_2} T_{\mathfrak{p}}^{(2)} \longrightarrow 0,$$ where $$W_i = \{ \mathfrak{p} \in \operatorname{Spec} R : \operatorname{ht}(\mathfrak{p}) = i \}$$, and $$T_{\mathfrak{p}}^{(i)} = (\widehat{R_{\mathfrak{p}}})^{B_{\mathfrak{p}}^{(i)}}$$ for suitable cardinalities $$B_{\mathfrak{p}}^{(i)} = \dim_{k(\mathfrak{p})} H^i(K(\mathfrak{p}) \otimes^{\mathbf L}_R R_{\mathfrak{p}}, R)$$. Each $T_{\mathfrak{p}}^{(i)}$ is nonzero, and the differentials $d^i$ are assembled from canonical maps induced by localization and universal properties of completions (Nakamura, 2018).

3. Purity, Minimality, and Uniqueness

The purity of a module $M$ within this context is characterized by the invariance of grade: for any nonzero submodule $N\subseteq M$, $\grade(N) = \grade(M)$, with $\grade(M) = \inf\{ i \geq 0 \mid \Ext_A^i(M, A) \neq 0 \}$. A pure minimal injective coresolution thus requires each component $I^n$ to be pure with $\grade(I^n) = n$, and each exact sequence $$0 \to \operatorname{Im} d^{n-1} \to I^n \to \operatorname{Im} d^n \to 0$$ is grade-split (every submodule of $I^n$ has grade $n$) (Gottesman et al., 5 Nov 2025).

Minimality ensures that each embedding of cycles into their pure-injective covers is "as small as possible," i.e., no nonzero direct summand can be omitted from any $P^i$. This property guarantees uniqueness up to isomorphism of complexes over noetherian rings (Nakamura, 2018).

4. Pure Minimal Injective Coresolutions and Lattice-Theoretic Classification

For a finite distributive lattice $L$ and its incidence algebra $A = K L$, the existence of a pure minimal injective coresolution is characterized in terms of the lattice structure. Specifically, $A$ has a pure minimal injective coresolution if and only if $L$ is isomorphic to the lattice of order ideals of an upward-linear poset (one in which every element is covered by at most one element), or equivalently, if the Hasse diagram of its poset of join-irreducibles is a disjoint union of rooted trees (Gottesman et al., 5 Nov 2025).

This equivalence is captured by several lattice-theoretic and module-theoretic statements:

Lattice/Module Condition	Equivalent Characterization	Reference
$L$ is order-ideal lattice of upward-linear poset	Hasse diagram of join-irreducibles is a forest of rooted trees	(Gottesman et al., 5 Nov 2025), Thm 3.5
$A$ is Auslander regular and $A$ has a pure minimal injective coresolution	Every indecomposable injective $A$-module is perfect	(Gottesman et al., 5 Nov 2025), Thm 3.5
For every cover $a \lessdot b$ in $L$: $$|\operatorname{cov}(a)| \geq |\operatorname{cov}(b)|$$	—	(Gottesman et al., 5 Nov 2025)

The only distributive lattices whose incidence algebras admit pure minimal injective coresolutions are those whose join-irreducible poset is a forest of rooted trees.

5. Antichain Modules, Perfection, and Canonical Resolutions

Within the context of incidence algebras of distributive lattices, antichain modules serve as building blocks for all indecomposable injective modules. Given an antichain $C = \{x_1, \ldots, x_\ell\}$ in $L$, define $N_C = \sum_{i=1}^\ell p_{x_i}^m A \subseteq P(m)$ where $m = \min L$, and set $M_C = P(m)/N_C$. The canonical antichain resolution has the form

$$0 \to P_\ell \xrightarrow{\partial_\ell} \cdots \xrightarrow{\partial_1} P_0 \xrightarrow{\partial_0} M_C \to 0$$

with $P_r = \bigoplus_{S\subseteq C, |S|=r} P(\vee S)$.

Two combinatorial properties dictate the homological properties of $M_C$:

• $C$ is strong if $\vee S \leq \vee S'$ implies $S \subseteq S'$ for all $S, S' \subseteq C$.
• $C$ is Boolean if for all $S, S' \subseteq C$, $\vee S \wedge \vee S' = \vee(S \cap S')$.

The minimality of the resolution corresponds to $C$ being strong, while perfection (i.e., $\grade(M_C) = \pdim(M_C)$) occurs if and only if $C$ is Boolean whenever the resolution is minimal. This delineation leads to the equivalence of the lattice-theoretic and module-theoretic criteria for the existence of pure minimal injective coresolutions (Gottesman et al., 5 Nov 2025).

6. Functoriality, Dimension Bounds, and Applications

For affine $k$-algebras $R$, the construction of minimal pure-injective coresolutions is functorial with respect to finite morphisms: a minimal pure-injective resolution of $R$ is mapped to a minimal resolution of $S$ under $R \to S$ finite, with restriction of scalars preserving minimality under suitable finiteness conditions (Nakamura, 2018).

The length of the minimal pure-injective resolution is constrained by both the dimension of $R$ and the cardinality of the base field $k$; specifically, for $|k| = \aleph_1$ and $\dim R \geq 2$ or $|k| \geq \aleph_1$ and $\dim R = 2$, one obtains three-term resolutions, with each term described explicitly as a product indexed by primes of a given height (Nakamura, 2018).

These results link the algebraic and homological structure of rings to lattice theory, module-theoretic purity and perfection, and the geometry of spectra, offering a unified perspective on the minimal decomposition and cohomological detection of modules in various algebraic settings.