Lê Modules: Linking Singularity & Module Theory

• Lê Modules are algebraic structures that capture local topological invariants of analytic hypersurfaces and generalize module theory through lattice-ordered semigroups.
• They refine classical vanishing cycle techniques by providing explicit cohomological resolutions and Betti number bounds, thus offering precise topological insights.
• Lê Modules facilitate generalized primary decomposition in commutative algebra, ensuring unique factorization and robust structural theorems.

Lê modules are algebraic structures arising both in singularity theory, where they encapsulate local topological invariants of analytic hypersurfaces with one-dimensional singular sets, and in module theory over commutative rings, where they generalize modules to the setting of lattice-ordered semigroups with distributivity and completeness properties. In singularity theory, the Lê module formalism refines classical vanishing cycle methods, yielding explicit cohomological resolutions and Betti number bounds for Milnor fibers, while in algebra it mediates generalized primary decomposition and uniqueness properties analogous to those in classical module theory.

1. Geometric Origins: Lê Numbers and Cycles

Given a reduced analytic function $f: (U,0) \to (\mathbb{C},0)$ on a small open neighborhood $U \subset \mathbb{C}^{n+1}$, suppose the critical locus $\Sigma f$ has dimension one at $0$. For a generic choice of $z_0$ among local coordinates $(z_0, ..., z_n)$, the condition $\dim_{0} \Sigma(f|_{V(z_0)}) = 0$ ensures proper intersection of each irreducible component $C$ of $\Sigma f$ with the hyperplane $V(z_0)$. The analytic 1-cycle $$\displaystyle V(\partial f/\partial z_1, ..., \partial f/\partial z_n)$$ near $0$ decomposes as $\Gamma^1_{f,z} + \Lambda^1_{f,z}$, where $\Gamma^1_{f,z}$ (relative polar curve) has components not contained in $\Sigma f$, and $\Lambda^1_{f,z}$ (1-dimensional Lê cycle) collects those contained in $\Sigma f$.

The Lê-numbers at $0$, which are intersection multiplicities,

$$\lambda^0 = (\Gamma^1_{f,z} \cdot V(\partial f/\partial z_0))_0, \qquad \lambda^1 = (\Lambda^1_{f,z} \cdot V(z_0))_0,$$

provide crucial local invariants. If $\Sigma f = \bigcup C_i$ with $\mu_i^\circ$ the Milnor number of the slice $f|_{V(z_0)}$ at a nearby smooth point of $C_i$, then

$$\Lambda^1_{f,z} = \sum \mu_i^\circ \cdot C_i, \qquad \lambda^1 = \sum (C_i \cdot V(z_0))_0 \cdot \mu_i^\circ.$$

This encoding links geometric intersection phenomena to local topological data (Massey, 28 Dec 2025).

2. Sheaf-Theoretic Construction and Definition of Lê Modules

Fix a principal ideal domain $R$, e.g., $\mathbb{Z}$ or a field. The ambient space $U$ carries the shifted constant sheaf complex $R_U[n+1]$, perverse on $U$. The shifted vanishing cycle complex

$\mathcal{P} := \phi_f[-1] R_U[n+1]$

is again perverse. Applying the nearby and vanishing cycle functors in $z_0$ yields, after shifting, $\psi_{z_0}[-1]\mathcal{P}$ and $\phi_{z_0}[-1]\mathcal{P}$, each supported only at $0$. The canonical morphism

$$\operatorname{can}_{z_0}: \psi_{z_0}[-1]\mathcal{P} \longrightarrow \phi_{z_0}[-1]\mathcal{P}$$

induces $R$-module identities in degree zero: $$H^0(\psi_{z_0}[-1]\mathcal{P})_0 \cong R^{\lambda^1}, \quad H^0(\phi_{z_0}[-1]\mathcal{P})_0 \cong R^{\lambda^0},$$ and a differential

$\partial: R^{\lambda^1} \to R^{\lambda^0}$

on these free modules. These objects are termed the Lê modules (Massey, 28 Dec 2025).

3. The Lê-Module Exact Sequence and Monodromy

The perverse sheaf construction yields a short exact sequence

$$0 \to \ker \partial \to R^{\lambda^1} \xrightarrow{\partial} R^{\lambda^0} \to \operatorname{coker} \partial \to 0,$$

where

$$\ker\partial \cong \widetilde{H}^{n-1}(F_{f,0}; R), \quad \operatorname{coker} \partial \cong \widetilde{H}^{n}(F_{f,0}; R),$$

with $F_{f,0}$ the Milnor fiber at $0$. The Milnor monodromy induces automorphisms $\alpha_1$ on $R^{\lambda^1}$ and $\alpha_0$ on $R^{\lambda^0}$, commuting with $\partial$, i.e., $\partial\circ\alpha_1 = \alpha_0\circ\partial$. The eigenvalues of $\alpha_j$ are all roots of unity, with characteristic polynomials admitting cyclotomic factorization.

Through A'Campo's trace formula, the traces satisfy: $$\operatorname{tr}\, \alpha_0 = (-1)^n(-1 + \operatorname{mult}_0|\Sigma f|), \quad \operatorname{tr}\, \alpha_1 = (-1)^n \operatorname{mult}_0|\Sigma f|,$$ where $\operatorname{mult}_0|\Sigma f|$ is the sum of multiplicities of the reduced critical curve at $0$. The resulting inequality

$\operatorname{mult}_0|\Sigma f| \leq \lambda^0 + 1$

provides an explicit bound on the number of local branches of $\Sigma f$ (Massey, 28 Dec 2025).

4. Betti Number Bounds via Lê Modules

For $R = \mathbb{Z}$, reduced Milnor-fiber cohomology is

$$\widetilde{H}^{n-1}(F_{f,0}; \mathbb{Z}) \cong \mathbb{Z}^{\tilde{b}_{n-1}}, \quad \widetilde{H}^{n}(F_{f,0}; \mathbb{Z}) \cong \mathbb{Z}^{\tilde{b}_n} \oplus T,$$

with $T$ a torsion group, and for prime $p$ let $\tau_p$ be the number of $p$-power cyclic summands in $T$. By the universal coefficient theorem,

$$\widetilde{H}^*(F_{f,0}; \mathbb{Z}/p) \cong (\mathbb{Z}/p)^{\tilde{b}_* + \tau_p}.$$

The sharp Betti-bound theorem distinguishes the isolated singularity case ($\lambda^0=0$, $\Sigma f$ smooth at $0$), for which

$$\widetilde{H}^{n-1}(F_{f,0};\mathbb{Z})\cong\mathbb{Z}^{\mu},\qquad \widetilde{H}^n(F_{f,0}; \mathbb{Z})=0,$$

from cases where $\lambda^0 \ne 0$, and for all $p$: $$\tilde{b}_{n-1} + \tau_p < \lambda^1, \qquad \tilde{b}_n + \tau_p < \lambda^0.$$ This provides universal bounds for ranks and torsion in Milnor fiber cohomology in terms of intersection-theoretic Lê numbers (Massey, 28 Dec 2025).

5. Example: Isolated Line Singularity

For $f(z_0, z_1, z_2) = z_2^2 - z_1^3 - z_0 z_1^2$, the critical set $\Sigma f$ is the $z_0$-axis, smooth at $0$, and the plane slice $z_2^2 - z_1^3$ has Milnor number $\mu^\circ = 1$. Thus, $\Lambda^1_{f,z} = 1 \cdot C$ (where $C$ is the axis), so $\lambda^1 = 1$; $\Gamma^1_{f,z} \neq \emptyset$ so $\lambda^0 \geq 1$. Explicit calculation yields $\lambda^0 = 2$. The differential $\partial: R \to R^2$ is injective, so $\ker \partial = 0$, $\operatorname{coker} \partial \cong R$, and the Milnor fiber has the homotopy type of a bouquet of $n$-spheres, with

$$\widetilde{H}^{n-1}(F; R) = 0, \qquad \widetilde{H}^n(F; R) \cong R^{\lambda^0 - 1}.$$

This agrees with general bouquet theorems for line singularities and demonstrates sharp realization of the Betti-bound (Massey, 28 Dec 2025).

6. Lê Modules in Lattice-Ordered Module Theory

An $R$–le-module $(M, +, \le, e)$, as developed in the context of commutative algebra, is a lattice-ordered semigroup enriched with an $R$-action satisfying five compatibility axioms:

• (M1) $r(m_1 + m_2) = r m_1 + r m_2$,
• (M2) $(r_1 + r_2)m \leq r_1 m + r_2 m$,
• (M3) $(r_1 r_2) m = r_1 (r_2 m)$,
• (M4) $1_R m = m$, $0_R m = 0_M$, $r 0_M = 0_M$,
• (M5) $$r(\bigwedge_{i \in I} m_i) = \bigwedge_{i \in I} (r m_i)$$.

Submodule elements $n \in M$ are those with $n + n \leq n$ and $r n \leq n$ for all $r \in R$; these are idempotent and satisfy $0_M \leq n \leq e$. Classical examples include the complete lattice of submodules of a module $N$, and the lattice of ideals of $R$ itself (Bhuniya et al., 2018).

7. Primary Decomposition and Uniqueness in Laskerian le-Modules

A submodule element $q \neq e$ is called primary if for all $a \in R$, $x \in M$,

$$a x \leq q \implies x \leq q \ \text{or}\ a^n e \leq q \ \text{for some} \ n \in \mathbb{N}.$$

If $\operatorname{Rad}(q) = P$ (the radical of the ideal $(q:e)$), then $q$ is $P$-primary. Similarly, a prime submodule element $p$ satisfies $r n \leq p \implies r \in (p:e)$ or $n \leq p$.

A Laskerian le-module is one in which every submodule element admits a reduced primary decomposition into a meet of primary elements with distinct radicals. Associated primes and isolated components are determined canonically. Uniqueness theorems assert that the set of associated primes is independent of the decomposition, and the meet of isolated components associated to a subset of primes is canonical (i.e., independent of the reduced decomposition). Minimal components are unique, and the primeness of the radical is equivalent to having a unique isolated prime divisor.

The explicit characterization of annihilators states that for submodule element $n$ and $r \in R$,

$$(n:r) = n \iff r \notin P \ \text{for every associated prime} \ P \ \text{of} \ n.$$

This recovers and generalizes classical primary decomposition theory in module settings (Bhuniya et al., 2018).

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A plausible implication is that Lê modules serve as a unifying formalism connecting topological invariants of singularities and the algebraic structure of subobjects in module theory, with their exact sequences, Betti-number bounds, and decomposition theorems providing robust tools for both singularity theory and commutative algebra. Open questions persist concerning the existence of torsion phenomena and the realization of certain Lê-number pairs, as well as deeper connections to perverse sheaf theory and vanishing cycle techniques in modern singularity analysis (Massey, 28 Dec 2025, Bhuniya et al., 2018).