---
title: Torsion-Freeness in Borel–Moore Homology
url: https://www.emergentmind.com/topics/torsion-freeness-in-borel-moore-homology
type: topic
---

# Torsion-Freeness in Borel–Moore Homology

Torsion freeness in Borel-Moore homology is a property of substantial structural and computational significance across singularity theory, the topology of algebraic varieties, and geometric representation theory. In Borel-Moore homology, “torsion freeness” asserts that the homology groups are free abelian over their coefficient ring (typically $\mathbb{Z}$ or $H^*_G$), implying the absence of elements annihilated by non-units. This property enables canonical bases, integral presentations (not requiring field extension or denominators), and powerful duality isomorphisms with de Rham or compactly supported cohomology. Torsion-freeness has been explicitly demonstrated in classical arrangements, quiver varieties, Hall algebra contexts, and stacks of algebraic representations, revealing fundamental links between geometry, algebra, and combinatorics.

## 1. Borel-Moore Homology: Definition and Context

Borel-Moore homology $H_*^{\mathrm{BM}}(X; R)$ is the covariant homology theory for locally compact, possibly non-compact, real or complex algebraic varieties or stacks, dual to compactly supported cohomology. For a complex algebraic (or analytic) variety $X$, $H_i^{\mathrm{BM}}(X; R)=H^{-i}(a_{X,*} R_X)$, where $a_X$ is the structure morphism, aligning with the dual space to $H_c^*(X; R)$ and equipping the homology with a canonical mixed Hodge structure when $R=\mathbb{Q}$. In equivariant contexts, equivariant Borel-Moore homology $H_{G,*}^{\mathrm{BM}}(X)$ is constructed as a direct limit over finite-dimensional approximations of the classifying stack $BG$.

Torsion freeness means that, as a module over $R$, $H_*^{\mathrm{BM}}(X; R)$ is free—there are no nonzero elements $x$ and non-unit $r\in R$ with $r x=0$. In equivariant settings, the freeness is often over $H^*_G(\mathrm{pt})$, the equivariant cohomology ring. This property is essential for computations, homological dualities, and the integrity of algebraic structures such as Hall algebras and shuffle algebra embeddings [2601.00124], [1602.02110].

## 2. Torsion-Freeness in Hyperplane Arrangement Complements

For real hyperplane arrangements, the complement 
$$
M(\mathcal{A}) = \mathbb{C}^\ell \setminus \bigcup_{i=1}^n H_i^\mathbb{C}
$$
admits a canonical, torsion-free Borel-Moore homology structure. By constructing a semi-algebraic, real-analytic partition indexed by chambers $C$ in the real arrangement and a sufficiently generic flag, Ito–Yoshinaga established a cell decomposition into contractible, pairwise disjoint pieces $S(C)$ such that ([1102.2039], Thms. 3.10, 4.4):
- The closures $\overline{S(C)}$ are smooth, oriented real submanifolds, providing explicit cycles.
- For each degree, the classes $[\overline{S(C)}]$ (for $C$ running over a partition of the chambers) form a $\mathbb{Z}$-basis of $H_*^{\mathrm{BM}}(M(\mathcal{A});\mathbb{Z})$, making the group free abelian.
- The intersection matrix with canonical cohomology generators is triangular with units on the diagonal, ensuring linear independence and spanning.
- All Borel-Moore homology groups are torsion-free—no nontrivial torsion elements arise in any step or intersection computation.

This explicit basis enables a concrete duality with de Rham cohomology, given by Poincaré–Alexander duality, and a “combinatorial Morse theory” adapted to hyperplane complements ([1102.2039], §5).

## 3. Torsion-Freeness in Stacks of Representations and Hall Algebras

In the context of preprojective algebras $\Pi_Q$ of quivers, the Borel–Moore homology of the stack of representations $\mathfrak{M}_{\Pi_Q,d}=\mu^{-1}(0)/G_d$ and its equivariant version exhibit freeness as modules over the coefficient ring ([1602.02110], Main Theorem):

- The CoHA (cohomological Hall algebra) $\mathcal{H}_{T,\Pi_Q} = \bigoplus_d H^{\mathrm{BM}}_{T\times G_d}(\mu^{-1}(0),\mathbb{Q})\otimes \mathcal{L}^{-(d,d)}$ is a free $H^*_T$-module for any torus $T$ acting compatibly (preserving preprojective relations).
- The localization map to the fraction field of $H^*_T$ is injective, confirming torsion-freeness at both the equivariant and nonequivariant level.
- There exists an integral, denominator-free embedding into a shuffle algebra of symmetric polynomials, confirming the absence of “hidden $q$-torsion” in the Hall algebra realization.

The proof relies on dimensional reduction to vanishing-cycle cohomology on the 3-Calabi–Yau completion, the PBW theorem for vanishing-cycle complexes, and the purity of BPS perverse sheaves, ensuring that Leray spectral sequences degenerate and no torsion is introduced ([1602.02110], §§1–3).

## 4. Torsion-Freeness for Cotangent Representations and Hall Induction

For cotangent stacks of the form $T^*(V/G)\cong \mu_V^{-1}(0)/G$ for $V$ a finite-dimensional $G$-representation, torsion-freeness in Borel–Moore homology is established under mild assumptions on extra torus actions ([2601.00124], Theorem):

- With $T_s$ a torus acting on $T^*V\times \mathfrak{g}$, commuting with $G$ and preserving the null-fiber of the moment map, if $T_s$ contains two specified one-parameter subgroups, then $H^{\mathrm{BM}}_{G\times T_s}(\mu_V^{-1}(0))$ is a torsion-free $H^*_{G\times T_s}(\mathrm{pt})$-module.
- Equivalently, the restriction homomorphism to the equivariant cohomology ring of the point is injective.
- The proof utilizes the Atiyah–Bott localization theorem, reduction to homology over subtori, spectral sequence arguments, and the purity of Borel–Moore homology for nilpotent orbits.

Stratification by nilpotent orbits of $\mathfrak{g}$ and injectivity on stabilizer homology ensures the absence of torsion classes. Concrete examples (e.g., $G = GL_2$, $V = \mathrm{End}(\mathbb{C}^2)$) confirm these results without torsion appearing outside explicitly characterized loci ([2601.00124], §5).

## 5. Conceptual Mechanisms and Structural Implications

The proofs of torsion-freeness across the above settings share key mechanisms:
- **Explicit cell decompositions**: semi-algebraic partitions or CW-complex models provide cycle classes with canonical bases, revealing freeness ([1102.2039]).
- **Filtration and stratification**: stratified spaces (e.g., nilpotent cones, quiver representation spaces) are filtered so that associated graded pieces are themselves known to be free.
- **Degeneration of spectral sequences**: Purity of cohomology (often informed by mixed Hodge module theory) forces degeneration, thus precluding torsion extension classes ([1602.02110]).
- **Integrality in Hall algebra structures**: The shuffle algebra embeddings are integral—no denominators are required—mirroring torsion-freeness at the level of algebraic presentations ([1602.02110]).
- **Equivariant methods and localizations**: Applying localization theorems (Atiyah–Bott/Borel–Weil–Bott) reduces proofs to computations over polynomial rings and their localizations, allowing exact control over possible sources of torsion ([2601.00124]).

The upshot is that torsion-freeness implies well-behaved Poincaré and Alexander dualities, canonical combinatorial or geometric bases, and computational tractability in both algebraic and topological settings.

## 6. Connections to Positivity, Combinatorics, and Representation Theory

Torsion-freeness in Borel–Moore homology underlies and implies further positivity and combinatorial phenomena:
- For preprojective algebras, torsion-freeness and purity translate into the positivity of restricted Kac polynomials and the factorial plethystic factorization of generating series ([1602.02110]).
- For hyperplane arrangements, the combinatorial nature of the basis links to matroid structures, and a plausible implication is that similar torsion-free bases may exist for a broader class of oriented matroid complements ([1102.2039]).
- In geometric representation theory, torsion-freeness ensures that weight spaces, character formulas, and convolution algebras are supported on integral, canonical generators, not requiring denominators or field extensions for their presentations ([2601.00124]).

## 7. Generalizations and Open Directions

The results indicate, and in special cases confirm, that semi-algebraic decompositions yielding torsion-free Borel–Moore homology should extend to complements of more general real algebraic hypersurfaces and to further classes of moduli stacks, particularly those admitting cellular decompositions or deep filtrations ([1102.2039], [1602.02110]). The explicit absence of torsion, both in homological and Hall-algebraic incarnations, points to broader patterns in the topology and representation theory of algebraic stacks, and suggests potential for further applications in cohomological wall-crossing, positivity results, and categorified representation theory.

**Key References:**  
"Ito–Yoshinaga, Semi-algebraic partition and basis of Borel–Moore homology of hyperplane arrangements" [1102.2039]  
"Hall induction for cotangent representations and wheel conditions" [2601.00124]  
"The integrality conjecture and the cohomology of preprojective stacks" [1602.02110]

Source: https://www.emergentmind.com/topics/torsion-freeness-in-borel-moore-homology