Constructible de Rham Cohomology

• Constructible de Rham cohomology is a de Rham-type framework where differential forms are adapted to stratified and support-sensitive data on real analytic manifolds.
• It employs Whitney tensor products and subanalytic sheaf theory to produce quasi-isomorphic resolutions of constructible sheaves, ensuring accurate cohomological recovery.
• The theory extends to globally subanalytic, p‑adic, and derived contexts, offering refined comparisons with classical topological and Betti cohomologies.

Searching arXiv for recent and foundational papers on constructible de Rham cohomology and related variants. Constructible de Rham cohomology denotes a family of de Rham-type formalisms in which differential forms are adapted to constructible, stratified, or support-sensitive data rather than only to the constant sheaf. In the most precise sheaf-theoretic form represented here, the basic statement is Prelli’s constructible de Rham theorem: if $M$ is a real analytic manifold and $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$, then the Whitney-de Rham complex of $F$ is quasi-isomorphic to $F$,

$$F \xrightarrow{\sim} F\overset{\mathrm w}{\otimes}\mathcal C_M^{\infty,\bullet},$$

so a bounded $\mathbb R$-constructible complex is recovered from a differential-form model with Whitney coefficients (Prelli, 2015).

1. Canonical sheaf-theoretic formulation on real analytic manifolds

The foundational setting is a real analytic manifold $M$, with $\mathcal C_M^\infty$ the sheaf of smooth functions, $\mathcal A_M$ the sheaf of analytic functions, $\Omega_M^p$ the sheaf of analytic $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$0-forms, and $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$1 the sheaf of differential operators with analytic coefficients. The relevant coefficient category is the bounded derived category $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$2 of bounded complexes with $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$3-constructible cohomology sheaves, meaning locally constant of finite rank on strata of a subanalytic stratification. In this context, the Whitney tensor product

$F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$4

is exact, extends to derived categories, and produces soft sheaves for constructible inputs. On basic constructible sheaves it is characterized by

$F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$5

for $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$6 closed subanalytic and $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$7 open subanalytic. Here $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$8 is the sheaf of Whitney $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$9-functions on $F$0, while $F$1 is the sheaf of smooth functions vanishing to infinite order on $F$2 (Prelli, 2015).

The associated Whitney-de Rham complex is

$F$3

that is,

$F$4

Prelli’s Theorem 5.2 identifies this complex with the original constructible object in $F$5: $F$6 For a closed subanalytic subset $F$7, this specializes to

$F$8

hence

$F$9

For an open subanalytic $F$0 with complement $F$1, the corresponding flat-coefficient complex computes compactly supported cohomology,

$F$2

Constructible de Rham cohomology is therefore not merely a comparison of numerical cohomology groups: it is a derived-category resolution statement for constructible sheaves themselves (Prelli, 2015).

2. Local mechanisms: Whitney conditions, subanalytic sites, and simplicial currents

A decisive technical point is that Whitney functions do not form an ordinary sheaf on the usual topology of $F$3. The remedy is the subanalytic site $F$4, whose objects are open subanalytic subsets and whose coverings are locally finite on compact subsets. On this site, Whitney coefficients become sheaf-theoretically well behaved, and the morphism of sites

$F$5

enters the proof through the comparison

$F$6

and similarly for the differential-graded version. The local analytic input is a Whitney version of the Poincaré lemma: on suitable $F$7-regular opens, Whitney functions coincide with smooth functions with bounded derivatives, and the de Rham complex with bounded derivatives satisfies the homotopy axiom. This yields

$F$8

which, combined with constructible biduality $F$9, gives the global theorem. The proof is thus a blend of local bounded-derivative de Rham theory, subanalytic sheaf theory, formal cohomology, and constructible duality rather than a purely formal restatement of classical smooth de Rham theory (Prelli, 2015).

A current-theoretic analogue appears in Melrose’s treatment of distributional de Rham cohomology on a triangulated compact oriented manifold. For a $$F \xrightarrow{\sim} F\overset{\mathrm w}{\otimes}\mathcal C_M^{\infty,\bullet},$$0-simplex $$F \xrightarrow{\sim} F\overset{\mathrm w}{\otimes}\mathcal C_M^{\infty,\bullet},$$1, the Poincaré dual current $$F \xrightarrow{\sim} F\overset{\mathrm w}{\otimes}\mathcal C_M^{\infty,\bullet},$$2 satisfies

$$F \xrightarrow{\sim} F\overset{\mathrm w}{\otimes}\mathcal C_M^{\infty,\bullet},$$3

and the chain map

$$F \xrightarrow{\sim} F\overset{\mathrm w}{\otimes}\mathcal C_M^{\infty,\bullet},$$4

induces an isomorphism between simplicial $$F \xrightarrow{\sim} F\overset{\mathrm w}{\otimes}\mathcal C_M^{\infty,\bullet},$$5-homology and distributional $$F \xrightarrow{\sim} F\overset{\mathrm w}{\otimes}\mathcal C_M^{\infty,\bullet},$$6-de Rham cohomology. The proof reduces arbitrary closed currents to simplex-supported currents, then pushes residual terms onto lower skeleta. Although Melrose does not use the language of constructible sheaves, the resulting complex is explicitly stratified by simplices and behaves like a triangulation-adapted constructible de Rham model in complementary degree (Melrose, 2011).

3. Globally subanalytic and o-minimal forms

A distinct but closely related development occurs for globally subanalytic manifolds. In that setting, the naïve complex of globally subanalytic differential forms does not satisfy a de Rham theorem on noncompact spaces: on the interval $$F \xrightarrow{\sim} F\overset{\mathrm w}{\otimes}\mathcal C_M^{\infty,\bullet},$$7, the closed form

$$F \xrightarrow{\sim} F\overset{\mathrm w}{\otimes}\mathcal C_M^{\infty,\bullet},$$8

defines a nonzero class in $$F \xrightarrow{\sim} F\overset{\mathrm w}{\otimes}\mathcal C_M^{\infty,\bullet},$$9 for every $\mathbb R$0, because its primitive $\mathbb R$1 is analytic on $\mathbb R$2 but not globally subanalytic. The successful replacement is the complex of constructible differential forms, where coefficients lie in the Cluckers–Miller class generated by globally subanalytic functions and logarithms, and where the differential in regularity $\mathbb R$3 is defined by extension from a dense $\mathbb R$4-zone. For a globally subanalytic $\mathbb R$5-manifold $\mathbb R$6 with $\mathbb R$7, the constructible de Rham theorem states that the period pairing with globally subanalytic singular homology is perfect, equivalently

$\mathbb R$8

In sheaf form,

$\mathbb R$9

This theory is notable for already working in the $M$0-setting, with local Poincaré lemmas proved on open globally subanalytic $M$1-cells and homotopy invariance restored by parametric integration of constructible coefficients (Huber et al., 5 Aug 2025).

This o-minimal variant clarifies a common misconception. The obstruction in noncompact tame geometry is not the de Rham method itself, but the coefficient class. Globally subanalytic coefficients are too small because primitives naturally introduce logarithms; constructible coefficients restore closure under the homotopy operators required for de Rham comparison. In that sense, “constructible” here refers to coefficient functions rather than to sheaf-theoretic $M$2-constructibility, but the structural role is parallel (Huber et al., 5 Aug 2025).

4. Rigid, $M$3-adic, and coefficient-supported variants

In non-archimedean geometry, a comparable issue arises from boundary phenomena in rigid analytic spaces. Grosse-Klönne defines de Rham cohomology for a rigid space $M$4 over a complete non-archimedean field $M$5 of characteristic $M$6 by choosing a closed immersion

$M$7

into the rigid space associated with a smooth dagger space $M$8, and setting

$M$9

This is independent of the embedding, contravariantly functorial, satisfies exact sequences for decompositions by Zariski closed subspaces and localization off closed subsets, admits a Gysin sequence for smooth closed immersions, and agrees with algebraic de Rham cohomology on analytifications of finite-type $\mathcal C_M^\infty$0-schemes. It also identifies Berthelot’s rigid cohomology with de Rham cohomology of tubes,

$\mathcal C_M^\infty$1

for $\mathcal C_M^\infty$2 closed in a smooth formal model $\mathcal C_M^\infty$3 (Grosse-Klönne, 2014).

Berthelot provides a characteristic-$\mathcal C_M^\infty$4 coefficient-supported analogue using de Rham–Witt complexes. For a closed immersion $\mathcal C_M^\infty$5 into a smooth $\mathcal C_M^\infty$6-scheme over a perfect field $\mathcal C_M^\infty$7 of characteristic $\mathcal C_M^\infty$8, he constructs a $\mathcal C_M^\infty$9-algebra $\mathcal A_M$0 supported on $\mathcal A_M$1 such that

$\mathcal A_M$2

and, with $\mathcal A_M$3,

$\mathcal A_M$4

Thus rigid cohomology and rigid cohomology with compact supports are computed by de Rham–Witt complexes on a smooth ambient space, equipped with coefficients supported on the singular or nonproper locus. The same paper extends Bloch–Illusie comparison from trivial coefficients to crystals flat over $\mathcal A_M$5 or $\mathcal A_M$6, thereby producing a concrete coefficient-enhanced de Rham theory in the $\mathcal A_M$7-adic setting (Berthelot, 2012).

A further $\mathcal A_M$8-adic direction is the construction of overconvergent relative de Rham cohomology over the Fargues–Fontaine curve. For separated quasi-compact smooth rigid spaces, the theory assigns

$\mathcal A_M$9

via overconvergent $\Omega_M^p$0-modified period-sheaf cohomology, factors through Ayoub’s rigid analytic motives, and under the hypothesis $\Omega_M^p$1 yields vector bundles $\Omega_M^p$2 on the Fargues–Fontaine curve. The paper explicitly states that it is not a treatment of constructible de Rham cohomology in the usual algebraic or complex-analytic sense, but it supplies descent, finiteness, and motivic factorization features that resemble coefficient-style de Rham theories (Bras, 2018).

5. Categorical and singular-geometric generalizations

At a more categorical level, the abstract of “Modules over the de Rham cohomology spectrum” announces that the bounded derived category of regular holonomic $\Omega_M^p$3-modules on a smooth variety is equivalent to the homotopy category of compact, or constructible, modules over the motivic ring spectrum $\Omega_M^p$4 representing algebraic de Rham cohomology, and that this equivalence is compatible with the six functors. It also announces an application producing a motivic $\Omega_M^p$5-structure on $\Omega_M^p$6-modules over not necessarily smooth schemes. Within the present topic, this places constructible de Rham phenomena in a spectral and motivic framework rather than only in complexes of forms (Pavlov et al., 2016).

Derived differential geometry introduces a different distinction. For derived $\Omega_M^p$7-spaces, one de Rham theory is built from the Hodge-completed exterior algebra of the cotangent complex; it detects singular local invariants and can fail to agree with constant-sheaf cohomology. A second theory, defined via a suitably formulated $\Omega_M^p$8-de Rham stack, always recovers the constant-sheaf cohomology of the underlying topological space. The comparison map between these two theories becomes an equivalence under the algebraically locally acyclic condition, while explicit singular counterexamples show extra local cohomology in the cotangent-complex model. This suggests, though only inferentially, that future constructible de Rham theories on singular differentiable spaces may need to separate Betti-like locally constant behavior from singularity-sensitive infinitesimal corrections (Taroyan, 6 May 2025).

6. Specialized extensions, adjacent models, and conceptual boundaries

Several further theories exhibit constructible de Rham behavior in more specialized settings. Eyssidieux develops $\Omega_M^p$9-constructible cohomology for constructible complexes on infinite Galois coverings of compact complex manifolds, together with $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$00-de Rham cohomology for coherent $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$01-modules. For holonomic $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$02, the natural map

$F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$03

is a quasi-isomorphism, yielding an $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$04 comparison

$F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$05

The resulting cohomology objects live in Farber–Lück’s category $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$06, admit von Neumann dimensions, satisfy Atiyah-type $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$07-index and Poincaré–Verdier duality statements, and fit mixed Hodge modules into an $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$08-de Rham/Betti framework (Eyssidieux, 2022).

There are also adjacent singular and stratified models that are not formulated as constructible sheaf theories but are structurally close. Banagl’s intersection-space theory uses the complex

$F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$09

for a manifold with boundary $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$10 resolving a pseudomanifold with isolated singularities, and proves

$F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$11

as rings. The singularity is encoded through link cotruncation rather than via a constructible sheaf complex, but the mechanism is again a de Rham realization of stratified topological data (Schlöder et al., 2019). On the discrete side, the arbitrary-order discrete de Rham complex on a connected polyhedral domain $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$12 has cohomology canonically isomorphic to the continuous de Rham cohomology, with the lowest-order complex identified with the cellular cochain complex of the mesh. This gives a polyhedral, cellwise-constructible realization of de Rham cohomology in a computational setting (Pietro et al., 2022).

A final boundary case appears in characteristic $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$13: de Rham cohomology on smooth $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$14-algebras is shown to be formally étale, in the sense that it has a unique functorial deformation to any Artinian local thickening of $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$15. The mechanism rewrites derived de Rham cohomology on quasiregular semiperfect algebras as the unwinding of the quasi-ideal $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$16 in $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$17, thereby recovering crystalline cohomology as the unique functorial deformation. This is not constructible de Rham cohomology in the usual stratified sense, but it shows that de Rham theories can sometimes be controlled by small coefficient-like algebraic objects rather than directly by differential complexes (Mondal, 2021).

Constructible de Rham cohomology is therefore best viewed not as a single formalism but as a stable pattern. The recurring features are a replacement of naive coefficients by constructible, Whitney, current-theoretic, overconvergent, or support-sensitive ones; a derived or sheaf-theoretic comparison with constant-sheaf or topological cohomology; and localization mechanisms adapted to stratifications, closed supports, or singular loci. In the real analytic setting, Prelli’s Whitney theorem gives the cleanest canonical statement. The surrounding literature shows how that paradigm extends, mutates, or fails across simplicial, o-minimal, $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$18, rigid, $F\in D^b_{\mathbb R\text{-}c}(\mathbb C_M)$19-adic, motivic, and derived contexts (Prelli, 2015).