Perfect Antichain Modules

• Perfect antichain modules are modules defined via an antichain in a finite lattice, capturing key combinatorial and algebraic structures.
• Their construction utilizes a canonical projective resolution whose minimality relies on strong and Boolean antichain conditions.
• They are crucial in homological characterizations of purity in incidence algebras, aiding the classification of distributive lattice structures.

A perfect antichain module is a module, constructed from an antichain in a finite lattice, whose algebraic and homological properties are tightly regulated by combinatorial data of the lattice. Perfect antichain modules arise in the paper of minimal injective and pure minimal injective coresolutions, especially for incidence algebras of distributive lattices, where their classification is governed by Boolean and strong antichain conditions. Their role is prominent in the homological characterisation of purity and in lattice-theoretic classifications of when an incidence algebra admits a pure minimal injective coresolution.

1. Antichain Modules: Definitions and Construction

Let $L$ be a finite lattice with minimum element $m$ and incidence algebra $A = KL$ over a field $K$. An antichain $C \subseteq L$ is a set of mutually incomparable elements. For such $C$, define the right $A$-submodule

$N_C = \sum_{x \in C} p_x^m A \subseteq P(m),$

where $p_x^m$ denotes the unique path in $L$ from $m$ to $x$, and $P(m) = e_m A$ is the projective-injective module associated to $m$. The associated antichain module is

$M_C = P(m)/N_C.$

Every indecomposable injective $I(y)$ in $KL$ can be realised as an antichain module by taking $C = \min([m,y]^c)$, the minimal elements outside $[m,y]$.

2. Canonical Resolution and Minimality Conditions

Antichain modules $M_C$ admit a canonical projective resolution of length $|C|$ (where $\ell=|C|$): $$0 \longrightarrow P_\ell \xrightarrow{\partial_\ell} \cdots \xrightarrow{\partial_2} P_1 \xrightarrow{\partial_1} P_0 \xrightarrow{\partial_0} M_C \longrightarrow 0,$$ with $P_0 = P(m)$, and for $1 \leq r \leq \ell$,

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

where $P(\vee S)$ denotes the projective corresponding to the join of $S \subseteq C$. The differentials are defined via standard inclusions and Koszul signs.

This resolution is minimal if and only if $C$ is a strong antichain, i.e.,

$$\forall S, S' \subseteq C:~ (\vee S \leq \vee S') \implies (S \subseteq S').$$

This ensures that in each degree, the boundary maps do not have direct summands mapping isomorphically, enforcing minimality.

3. Perfection: Boolean Antichains and Homological Criteria

A module $M$ of finite projective dimension is called perfect if

$$\operatorname{grade}_A M = \operatorname{pdim}_A M,$$

where $$\operatorname{grade}_A M = \inf\{i \geq 0 \mid \operatorname{Ext}^i_A(M, A) \ne 0\}$$. For antichain modules, if $C$ is a strong antichain so that the canonical resolution is minimal, then $M_C$ is perfect if and only if $C$ is Boolean: $$\forall S, S' \subseteq C:\quad \vee S \wedge \vee S' = \vee (S \cap S').$$ This Boolean condition means that sublattice intervals under joins and meets correspond to those in a Boolean algebra.

For distributive lattices, all indecomposable injective modules $I(x)$ admit minimal antichain resolutions, and $I(x)$ is perfect precisely when its defining antichain $C_x$ is Boolean.

4. Lattice-Theoretic Classification via Perfect Antichain Modules

The classification of incidence algebras of distributive lattices with pure minimal injective coresolutions reduces to the perfection of indecomposable injective antichain modules. For a distributive lattice $L \cong \mathcal{O}(P)$, perfection is governed by the structure of the poset $P$ of join-irreducibles.

Key lattice-theoretic equivalence:

• $KL$ has a pure minimal injective coresolution if and only if every indecomposable injective $I(x)$ is perfect, if and only if $L \cong \mathcal{O}(P)$ for an upward-linear poset $P$, i.e., every element of $P$ has at most one cover.
• This lattice-theoretic property is further encoded combinatorially: For any cover relation $a \lessdot b$ in $L$,

$$|\operatorname{cov}(a)| \geq |\operatorname{cov}(b)|,$$

where $\operatorname{cov}(y)$ denotes the set of covers of $y$.

The two-sided version—requiring both left and right purity—occurs if and only if $P$ is a disjoint union of chains, so $L$ is a divisor lattice.

5. Homological Characterisation of Purity for Incidence Algebras

For a finite-dimensional, Auslander-regular algebra $A$ (i.e., finite global dimension with $\operatorname{pdim} I^i \leq i$ for the successive injective terms), $A$ has a pure minimal injective coresolution if and only if:

1. $A$ is right diagonal: every indecomposable summand of $I^i$ has projective dimension exactly $i$,
2. Every indecomposable injective right module is perfect.

Applying this to the incidence algebra of a finite lattice, these conditions are met precisely when every antichain module realising an indecomposable injective is perfect, i.e., its defining antichain is Boolean and strong.

6. Examples and Computational Aspects

Examples clarifying the role of perfect antichain modules include:

• Chains of any length (single-chain poset): every antichain module is perfect, so $KC_n$ is pure.
• Boolean lattices $2^{[k]}$ (antichain poset): again, all antichain modules are perfect.
• The pentagon lattice ($N_5$) is not distributive, so its incidence algebra fails the purity criterion.
• Distributive lattices with upward-linear join-irreducibles which are not disjoint unions of chains are pure for right and left modules but not two-sided unless the poset is a union of chains.

Computationally, verification can be performed with tools such as the GAP–QPA package, specifically via an implementation of “IsAuslanderregularwithPMIC.”

7. Applications and Directions for Further Study

The classification of perfect antichain modules provides a homological and combinatorial bridge between the structure theory of distributive lattices and properties of their incidence algebras. Directions for further research include:

• Classification of all finite posets whose incidence algebras admit a pure minimal injective coresolution.
• Extending the framework to geometric lattices and exploring implications for posets with additional topological properties (e.g., Cohen–Macaulayness).
• Understanding the interplay of Boolean-antichain conditions with order-complexes and combinatorial topology.

A plausible implication is that in classes beyond distributive lattices, perfection of antichain modules may signal deeper structural properties of incidence algebras and associated module categories, potentially generalizing the relationship between lattice-theoretical combinatorics and Auslander-regularity.