Cohomological Integrality Theorem

• Cohomological Integrality Theorem is a framework that decomposes large, complicated cohomological objects into finite-dimensional, primitive constituents.
• It leverages parabolic induction, Weyl group techniques, and fixed-point formulae to reorganize cohomology into BPS-type or intersection cohomological pieces.
• This structural decomposition underpins refined Donaldson–Thomas invariants and connects representation theory with the geometry of quotient stacks.

The cohomological integrality theorem is not a single theorem with one universal formulation, but a family of structurally related results in which a cohomological object that is a priori large, stacky, or non-proper is shown to be generated by more elementary “primitive” pieces, or is shown to satisfy an integrality constraint stronger than mere rationality. In recent literature this phrase is used for quotient stacks of weakly symmetric reductive representations, for cohomological Donaldson–Thomas and Hall-algebra formalisms, for foliated fixed-point formulas, and for integral or torsion-sensitive refinements of classical cohomological decompositions (Hennecart, 2024, Davison et al., 2016, Benameur et al., 2024, Zheng, 2013). Across these settings, the common feature is that cohomology is reorganized into finite-dimensional BPS-type pieces, intersection-cohomological summands, or representation-theoretically integral fixed-point contributions.

1. Conceptual pattern and terminology

In the representation-theoretic and Donaldson–Thomas literature, “integrality” typically means that an infinite-dimensional or plethystically defined cohomological object is determined by finitely many finite-dimensional primitive constituents. In the quotient-stack theorem of Hennecart, this takes the form of a direct-sum decomposition of $H^*(V/G)$ into summands indexed by fixed-point data of cocharacters, with finite-dimensional graded vector spaces $P_\lambda$ playing the role of BPS spaces (Hennecart, 2024). In quiver DT theory, the same idea appears as a categorical upgrade of the integrality conjecture: the full cohomological Hall algebra or vanishing-cycle direct image becomes a free supercommutative or symmetric algebra on BPS cohomology (Davison et al., 2016), and analogous statements hold for preprojective stacks, orientifold DT theory, and symmetric quotient stacks (Davison, 2016, Young, 2016, Hennecart, 2024).

A distinct but related usage occurs in index-theoretic and arithmetic settings. For foliations, the cohomological integrality theorem asserts that fixed-point characteristic numbers obtained from Haefliger cohomology and Connes–Moscovici residue cocycles lie in the evaluated representation ring $R(H)(h)$, and for finite-order $h$ become honest integers or cyclotomic integers (Benameur et al., 2024). In Lie-theoretic and algebro-geometric contexts, “integrality” can instead refer to control over integral or mod-$p$ cohomology, as in the generalization of Kostant’s theorem to integral cohomology of positive root systems and the theorem that $K$-equivalent smooth projective varieties have isomorphic integral cohomology groups (Zheng, 2013, Satriano et al., 29 Jan 2026).

This plurality of meanings is not accidental. A plausible implication is that “cohomological integrality” has become a unifying label for results where cohomology admits a rigid structural decomposition, a torsion-sensitive refinement, or a representation-theoretic integrality constraint that is invisible at the level of Euler characteristics or rational classes alone.

2. Weakly symmetric reductive representations and quotient stacks

A central modern formulation concerns quotient stacks $V/G$, where $G$ is a connected reductive group and $V$ is a finite-dimensional weakly symmetric representation (Hennecart, 2024). A representation $V$ is symmetric if $P_\lambda$0 and $P_\lambda$1 have the same weights with the same multiplicities, and weakly symmetric if

$P_\lambda$2

in the quotient of characters by the relation $P_\lambda$3 iff $P_\lambda$4 for some positive integers $P_\lambda$5. This hypothesis includes the adjoint representation $P_\lambda$6, cotangent representations $P_\lambda$7, symmetric quiver representations, and certain representations of groups of types $P_\lambda$8 (Hennecart, 2024).

Let $P_\lambda$9 be a maximal torus and $R(H)(h)$0 a cocharacter. Then $R(H)(h)$1 is the Levi subgroup centralizing $R(H)(h)$2, and $R(H)(h)$3 is the $R(H)(h)$4-fixed subrepresentation. The theorem is organized by a finite quotient $R(H)(h)$5, where $R(H)(h)$6 is defined from the partial order

$R(H)(h)$7

and $R(H)(h)$8 is the Weyl group. For each $R(H)(h)$9, there is a parabolic induction morphism

$h$0

The cohomological integrality theorem states that after choosing suitable finite-dimensional complements $h$1, the sum of the induction morphisms yields an isomorphism

$h$2

Here $h$3 is the subgroup of the center of $h$4 acting trivially on $h$5, $h$6 with $h$7 placed in degree $h$8, and $h$9 is a character of the relative Weyl group

$p$0

The spaces $p$1 are finite-dimensional graded vector spaces characterized as complements to the images of induction from “smaller” cocharacters (Hennecart, 2024).

Conceptually, this theorem upgrades an integrality or plethystic factorization picture into an actual direct-sum decomposition of graded vector spaces. The cohomology $p$2 is generally infinite-dimensional, but it is reconstructed from finitely many finite-dimensional pieces together with tautological polynomial factors $p$3. In the terminology of the paper, the $p$4 are the BPS spaces, and their graded dimensions define refined Donaldson–Thomas-type invariants

$p$5

3. Induction kernels, Weyl-group signs, and structural consequences

The induction maps are defined geometrically from the standard parabolic correspondence involving the parabolic $p$6, the fixed locus $p$7, and the attracting locus $p$8 (Hennecart, 2024). At the cohomological level, the key explicit formula is a shuffle-type expression

$p$9

with kernel

$K$0

where $K$1 denotes the multiset of $K$2-weights of $K$3 pairing positively with $K$4, and similarly for $K$5. The character $K$6 is defined by

$K$7

so the sign twist familiar from the symmetric quiver case is replaced by a more general relative Weyl-group character.

The theorem comes with strong finiteness and vanishing results. If the connected kernel of $K$8 is not contained in the center of $K$9, then $V/G$0. There is also the degree bound

$V/G$1

These constraints sharply restrict the possible BPS spaces (Hennecart, 2024).

The examples clarify the theorem’s scope. For symmetric quivers it recovers Efimov’s cohomological integrality theorem. For $V/G$2 acting on $V/G$3, the decomposition is explicit and the relative Weyl character becomes the sign representation. For $V/G$4-representations $V/G$5, the theorem predicts the dimensions of the relevant BPS spaces and is compared with intersection cohomology of GIT quotients. In the adjoint case $V/G$6, the induction maps are surjective and only generic cocharacters contribute nontrivially, producing a particularly simple integrality statement (Hennecart, 2024).

The paper also formulates a stronger conjecture: under a stability hypothesis on the generic stabilizers of the action of $V/G$7 on $V/G$8, the space $V/G$9 should identify canonically with the intersection cohomology $G$0. This suggests an algorithmic route from the cohomological integrality theorem to intersection-cohomology computations for GIT quotients (Hennecart, 2024).

4. Hall-algebra and Donaldson–Thomas forms of integrality

The modern use of the theorem is strongly shaped by cohomological Hall algebra and Donaldson–Thomas theory. For quivers with potential, Davison and Meinhardt proved a Hodge-theoretic categorification of the integrality conjecture by showing that the vanishing-cycle direct image is a symmetric or free supercommutative algebra generated by BPS cohomology (Davison et al., 2016). In one form, for a generic stability condition,

$G$1

and the associated graded of the perverse filtration on the CoHA is likewise free supercommutative on the BPS pieces. The theorem is realized as a Poincaré–Birkhoff–Witt-type statement for the cohomological Hall algebra (Davison et al., 2016).

This paradigm was transported to preprojective stacks by reducing to the tripled quiver with potential. For the preprojective algebra $G$2, one obtains a cohomological integrality isomorphism in which the direct image or CoHA is the symmetric algebra on a 2d BPS sheaf tensored with $G$3, and this yields PBW-type statements, purity, and positivity of restricted Kac polynomials (Davison, 2016). In orientifold Donaldson–Thomas theory, the analogous theorem for $G$4-symmetric quivers asserts integrality of orientifold DT invariants $G$5, defined from the primitive quotient of the cohomological Hall module, and the proof uses a signed shuffle description adapted to orthogonal and symplectic structure groups (Young, 2016).

The quotient-stack theorem of (Hennecart, 2024) is explicitly in direct analogy with integrality and PBW-type theorems in cohomological Hall algebra theory, but later work strengthened this relation. Hennecart established a sheafified version for stacks $G$6 with $G$7 a smooth affine symmetric algebraic variety, replacing graded-vector-space decompositions by isomorphisms in the category of monodromic mixed Hodge modules on the good moduli space (Hennecart, 2024). The central statement is

$G$8

with $G$9. Applying vanishing cycles gives a critical version for invariant functions on symmetric quotient stacks, and in the weakly symplectic weak-moment-map setting this leads to purity of the BPS sheaf and to Halpern-Leistner’s purity conjecture for $V$0-shifted symplectic stacks with proper good moduli spaces (Hennecart, 2024).

A still stronger result was obtained for smooth symmetric stacks. The strong cohomological integrality theorem identifies the summands geometrically with intersection cohomology of the good moduli spaces of graded pieces: $V$1 This confirms the conjecture that the algebraic BPS cohomology of the quotient stack of a symmetric representation matches the intersection cohomology group whenever it is nonzero, and it extends to $V$2- and $V$3-shifted symplectic stacks and to character stacks of compact oriented $V$4-manifolds with reductive gauge groups (Hennecart et al., 25 Sep 2025).

A parallel application appears in cohomological DT theory for $V$5-local systems on the topological $V$6-torus. For $V$7-local systems on $V$8, cohomological integrality is obtained via an exponential map that compares the local-system stack with the tripled Jordan quiver, thereby importing the Davison–Meinhardt theorem. The paper then proves versions for $V$9 and $V$0 when $V$1 is prime, and derives a Langlands duality statement for the corresponding cohomological DT invariants (Kaubrys, 2024).

5. Fixed-point and index-theoretic integrality for foliations

In foliation theory, the phrase denotes a different but structurally analogous phenomenon. For a closed foliated manifold $V$2 with a compact Lie group $V$3 acting by leaf-preserving isometries, the higher Lefschetz formalism produces fixed-point expressions in Haefliger cohomology paired with closed holonomy invariant currents (Benameur et al., 2024). The cohomological fixed-point formula is

$V$4

for an $V$5-equivariant leafwise elliptic complex $V$6.

When the foliation is Riemannian and transversely spin, the Connes–Moscovici residue cocycle identifies the relevant cyclic-cohomology class with the Haefliger class of $V$7, and the paper proves the integrality statement

$V$8

Equivalently, the associated characteristic number lies in the value set of the representation ring at $V$9; for finite-order $P_\lambda$00 of order $P_\lambda$01, it lies in $P_\lambda$02, and for an involution it is an integer (Benameur et al., 2024).

Here integrality is not a direct-sum decomposition but a fixed-point evaluation theorem: a residue or characteristic-number expression built from Haefliger cohomology, transverse Dirac operators, and cyclic cohomology is forced into a representation-theoretic integer lattice. The paper uses this to derive rigidity results: if $P_\lambda$03 is even-dimensional spin and there exists a closed holonomy invariant current $P_\lambda$04 with

$P_\lambda$05

then no compact connected Lie group can act nontrivially by leaf-preserving isometries preserving the spin structure (Benameur et al., 2024).

6. Integral, torsion-sensitive, and arithmetic variants

Another line of work uses “cohomological integrality” to emphasize integral rather than rational control of cohomology. For the positive root system $P_\lambda$06 of a semisimple Lie algebra, Zheng decomposes the combinatorial chain complex $P_\lambda$07 into weight subcomplexes

$P_\lambda$08

and proves that over a field $P_\lambda$09 of characteristic $P_\lambda$10, if the rank $P_\lambda$11 of a weight component is not divisible by $P_\lambda$12, then

$P_\lambda$13

The key identity is

$P_\lambda$14

which makes the weight subcomplex contractible whenever $P_\lambda$15 in the ground field (Zheng, 2013). This is an integral refinement of the weight-space structure underlying Kostant’s theorem.

In arithmetic topology, Church, Farb, and Putman prove an “Integrality Theorem” for the Steinberg module of $P_\lambda$16: for a Dedekind domain $P_\lambda$17 with $P_\lambda$18 under the stated arithmetic hypotheses, the Steinberg module

$P_\lambda$19

is generated by integral apartment classes, and if $P_\lambda$20 and $P_\lambda$21, then it is not generated by such classes (Church et al., 2015). Via Borel–Serre duality, this yields vanishing and nonvanishing theorems for top-degree cohomology of $P_\lambda$22, with the outcome controlled by the class group (Church et al., 2015).

In birational geometry, the theorem of (Satriano et al., 29 Jan 2026) introduces an integral version of the Hodge polynomial that encodes the full integral cohomology of a smooth projective variety, including torsion. The invariant is shown to descend to the Grothendieck ring of varieties and to be compatible with motivic equalities arising from $P_\lambda$23-equivalence. The main consequence is: $P_\lambda$24 This is presented as an integral analogue of Kontsevich’s theorem on equality of Hodge numbers (Satriano et al., 29 Jan 2026).

These variants suggest that the term “cohomological integrality theorem” often marks a passage from rational or decategorified invariants to integral, torsion-sensitive, or representation-theoretically rigid structures.

7. Significance, examples, and open directions

The modern importance of cohomological integrality lies in its conversion of difficult global cohomology into tractable primitive data. In the quotient-stack setting, the finite-dimensional spaces $P_\lambda$25 encode refined Donaldson–Thomas-type invariants and sharply constrain the possible cohomology of $P_\lambda$26 (Hennecart, 2024). In Hall-algebra and DT settings, BPS sheaves or BPS cohomology become the primitive generators of the full cohomological Hall algebra, and PBW-type theorems identify the ambient structure as a symmetric or free supercommutative algebra on these generators (Davison et al., 2016, Davison, 2016, Young, 2016).

A major recent trend is the replacement of abstract primitive pieces by intersection cohomology. The sheafified and strong symmetric-stack theorems show that under symmetry assumptions and the existence of good moduli spaces, the BPS summands can often be identified with $P_\lambda$27-complexes or intersection cohomology of graded moduli spaces (Hennecart, 2024, Hennecart et al., 25 Sep 2025). This has immediate consequences for purity: the strong theorem implies a version of cohomological integrality for $P_\lambda$28-shifted symplectic stacks and yields Halpern-Leistner’s conjecture that the Borel–Moore homology of a $P_\lambda$29-shifted symplectic stack with proper good moduli space is pure (Hennecart et al., 25 Sep 2025). The sheafified quotient-stack theory proves the same purity phenomenon for derived stacks with self-dual cotangent complex and gives, as a concrete application, purity of the Borel–Moore homology of the moduli stack of principal Higgs bundles over a smooth projective curve for a reductive group (Hennecart, 2024).

Several examples show that these theorems are not merely formal. Symmetric quivers recover Efimov-type integrality; the adjoint representation yields an especially simple decomposition; preprojective stacks lead to positivity of restricted Kac polynomials and explicit calculations for $P_\lambda$30; and local systems on the $P_\lambda$31-torus exhibit integrality compatible with Langlands duality for prime-rank $P_\lambda$32 and $P_\lambda$33 (Hennecart, 2024, Davison, 2016, Kaubrys, 2024).

Open directions are stated explicitly in the literature. The principal conjectural direction is the identification of BPS spaces with intersection cohomology under stability hypotheses on generic stabilizers (Hennecart, 2024). A broader geometric version has now been confirmed for symmetric stacks (Hennecart et al., 25 Sep 2025), suggesting that the quotient-stack theorem may be one instance of a more general principle in which Hall induction reconstructs stack cohomology from $P_\lambda$34-theoretic data on good moduli spaces. A plausible implication is that future work will continue to blur the distinction between cohomological integrality, PBW decompositions, purity theorems, and algorithmic computation of singular-moduli cohomology.