Intersection Cohomology with Torsion Coefficients

• Intersection Cohomology with Torsion Coefficients is a theory that generalizes classical intersection cohomology by incorporating torsion-sensitive perversities and refined duality principles.
• The approach leverages geometric tools such as Bott–Samelson resolutions and blown-up cochain complexes to explicitly detect and analyze torsion phenomena in singular varieties.
• It provides robust methods for ensuring Poincaré duality and stratification invariance, impacting studies in topology, singular algebraic varieties, and modular representation theory.

Intersection cohomology with torsion coefficients generalizes classical intersection cohomology to coefficient rings or fields where torsion phenomena can appear, enabling a unified treatment of duality, stratification invariance, and singular varieties across algebraic, topological, and representation-theoretic contexts. These extensions incorporate torsion-sensitive perversities, geometric models such as Bott–Samelson resolutions, and cochain complexes allowing for arbitrary coefficient rings, including those with prime torsion. Applications span the topology of stratified spaces, singular algebraic varieties, and modular representation theory of algebraic groups.

1. Fundamental Concepts and Definitions

Intersection cohomology replaces singular cohomology for singular spaces, ensuring properties like Poincaré duality persist. For a stratified space $X$, one defines a perversity function $\bar p$ that determines allowability of chains and cochains relative to strata. The classical Goresky–MacPherson theory operates over fields, where stalks of intersection complexes are torsion-free and duality holds without obstruction. When the coefficient ring $R$ has torsion (e.g., $\mathbb{Z}$, $\mathbb{F}_p$), new phenomena arise that are sensitive both to the ring and the stratification, requiring refined algebraic structures and comprehensive notions such as torsion-sensitive perversities (Friedman, 2019), blown-up cochain complexes (Chataur et al., 2016), and perverse sheaves with modular coefficients (Achar et al., 23 May 2025).

2. Torsion in Intersection Cohomology of Singular Varieties

Intersection cohomology over $\mathbb{Z}$ or other torsion rings can exhibit exotic torsion phenomena at singular points. For Schubert varieties $X_w \subset G/B$, Williamson's geometric proof establishes that the local integral intersection cohomology groups $\operatorname{IH}^*_x(X; \mathbb{Z})$ can contain prime torsion summands, with the largest torsion prime growing exponentially in the rank of the group. The mechanism involves explicit computation of the Euler class in a Bott–Samelson resolution, identification of highly singular points, and localization at smooth "miracle fibers," where the intersection form's failure of unimodularity directly realizes p-torsion. The combinatorial structure can be made explicit via divided-difference operators and reduces to counting prime divisors of explicit polynomials, making the subject rich in both geometry and combinatorics (Williamson, 2015).

Phenomenon	Variety	Torsion Behavior
Exponential prime torsion	Schubert varieties	Max prime divisor grows
Miracle fiber	Bott–Samelson	Single smooth half-fiber

3. Modular and Torsion-Sensitive Theories

Intersection cohomology with coefficients in positive characteristic fields or over PIDs with torsion employs refined frameworks:

• Torsion-sensitive perversities $$\bar p: \text{Strata} \to \mathbb{Z} \times 2^{P(R)}$$ record both truncation degree and specific allowed torsion primes. The dual perversity inverts both truncation and allowed primes (Friedman, 2019).
• Chain and sheaf models: The ts-Deligne sheaf $\mathcal{P}^{\bar p}_{X,\mathcal{E}}$ is built using torsion-tipped truncation functors, with explicit control over the torsion components of stalk and costalk cohomology, unifying classical GM-duality with torsion pairings (Cappell–Shaneson linking) (Friedman, 2019).
• Blown-up intersection cohomology: The geometric blown-up cochain complex $N^*_{\mathrm{bl},c}(X;R)$, defined via filtered simplices and cones, detects torsion correctly without requiring field coefficients or special hypotheses. Poincaré duality via cap-products holds for any commutative ring and general perversities (Chataur et al., 2016).

4. Cup and Cap Products; Poincaré Duality

Intersection cohomology admits well-defined cup and cap products in blown-up cochain complexes and ts-Deligne sheaves, enabling a canonical graded ring structure and duality pairings:

• Cup product: $$IH^i_{\mathrm{bl}}(X; R) \otimes IH^j_{\mathrm{bl}}(X; R) \to IH^{i+j}_{\mathrm{bl}}(X; R)$$, with compatibility rules ensuring that $\bar p+\bar q$-allowable cochains result (Chataur et al., 2016).
• Cap product and duality: Cap product with the intersection homology fundamental class $[X] \in IH_n^{D\bar p}(X; R)$ induces Poincaré duality $$IH^i_{\mathrm{bl},c}(X; R) \cong IH_{n-i}^{\bar p}(X; R)$$ without restriction to principal ideal domains or field coefficients, provided certain topological conditions on $X$ (no codimension one strata, paracompactness) (Chataur et al., 2016).

5. Topological and Stratification Invariance

Torsion-sensitive intersection cohomology theories exhibit strong invariance properties:

• For ts-Deligne sheaves, quasi-isomorphism under stratification refinement is established for compatible torsion-sensitive perversities, ensuring invariance despite torsion and arbitrary coefficient rings. This generalizes King's topological invariance theorem without requiring support/cosupport axioms or local cohomology vanishing (Friedman, 2019).
• The blown-up cochain complex construction is independent of PL-triangulation. The intrinsic aggregation (King's method) yields invariance of $IH^*_{\mathrm{bl},c}(X; R)$ under stratified homeomorphisms for spaces without codimension-one strata (Chataur et al., 2016).

Model/Theory	Topological Invariance	Coefficient Restrictions
ts-Deligne sheaf	Quasi-isomorphism/refinement	Any PID
Blown-up intersection cohom.	Stratified homeo. invariance	Any comm. ring

6. Torsion Phenomena in Modular and Geometric Contexts

Recent work in geometric representation theory, including "Modular intersection cohomology of Drinfeld's compactifications" (Achar et al., 23 May 2025), demonstrates that for good characteristic fields $\Bbbk$, intersection cohomology stalk dimensions are independent of $\mathrm{char}(\Bbbk)$, reflecting the absence of modular torsion in these settings. Dimensions are computed by Lusztig–Kostant partition polynomials, indicating that purity and parity sheaf properties persist, and supporting a general principle: when such gradings survive in positive characteristic, intersection cohomology stalks exhibit no unexpected torsion.

Examples establish nuanced torsion phenomena:

• Suspensions: Extreme perversities outside the classical range force the collapse of intersection cohomology in certain degrees, revealing the limits of duality unless the blown-up complex is used (Chataur et al., 2016).
• Cones: For spaces like $cL$, torsion in $IH^k_{\mathrm{bl},c}(cL; R)$ matches torsion in the cohomology of the link $IH^{k-1}(L; R)$, showcasing precise torsion detection.
• Thom spaces: Poincaré duality is obstructed by torsion (e.g., Euler class in $S^2$), but the blown-up cochain approach recovers nondegenerate pairings on the torsion-free parts (Chataur et al., 2016).

7. Implications, Open Questions, and Generalizations

Intersection cohomology with torsion coefficients provides a highly flexible and robust framework for duality and stratification-invariant analysis of singular spaces, even in arithmetic and modular settings. Williamson's results suggest that torsion is intrinsic to singularities and can be quantifiably large, while geometric and modular approaches demonstrate regimes where torsion does not arise. The frameworks unify classical intersection homology, torsion pairings, and cap/cup products into a single formalism, and allow for explicit computational models suitable for complex singularity theory, representation theory, and arithmetic geometry.

Emerging directions include factorization structures on representation-theoretic moduli spaces, extension of blown-up techniques to more general perversities and coefficient rings, and further elucidation of torsion phenomena in modular geometric representation theory.

Cited works:

• G. Williamson, "On torsion in the intersection cohomology of Schubert varieties" (Williamson, 2015)
• D. Chataur, M. Saralegi–Aranguren, D. Tanré, "Poincaré duality with cap products in intersection homology" (Chataur et al., 2016)
• G. Friedman, "Topological invariance of torsion sensitive intersection homology" (Friedman, 2019)
• P. Achar, G. Dhillon, S. Riche, "Modular intersection cohomology of Drinfeld's compactifications" (Achar et al., 23 May 2025)