Derived Regulator Maps to Deligne Cohomology

• The paper demonstrates that derived regulator maps establish explicit, functorial morphisms from motivic cohomology theories to Deligne–Beilinson cohomology.
• It employs derived-category techniques, including distinguished triangles and moving lemmas, to capture torsion, product compatibility, and geometric invariants.
• These constructions generalize classical Abel–Jacobi and Lefschetz theorems, offering profound implications for arithmetic, geometric, and homotopical applications.

A derived regulator map to Deligne cohomology is a functorial, homotopically enhanced morphism from a "motivic" cohomology theory—such as Lichtenbaum, higher Chow, or K-theory (possibly relative or with modulus)—into the Deligne–Beilinson cohomology of algebraic varieties. These maps realize deep arithmetic and Hodge-theoretic cycle relations by constructing explicit derived-category morphisms between motivic complexes and Deligne complexes, capturing torsion and product compatibility, geometric invariants, and universal properties of intermediate Jacobians. Such regulators underpin both the formal comparison between motivic and Hodge-theoretic invariants and concrete cycle-class calculus, generalizing the classical Abel–Jacobi and Lefschetz theorems and enabling arithmetic, geometric, and homotopical applications.

1. Motivic and Deligne–Beilinson Cohomology: Foundational Complexes

Motivic cohomology theories model cycles and their relations using derived-category realizations of complexes. Lichtenbaum cohomology $H_L^m(X, \mathbb Z(n))$ is defined via hypercohomology of the Suslin–Friedlander motivic complex $Z(n)^{SF}$ on the eh-topology:

${\mathbb H}^m_{eh}\bigl(X, Z(n)^{SF}_{eh}\bigr)$

with a parallel version with compact supports. For Deligne–Beilinson cohomology, the target is the hypercohomology of a Zariski sheafified cone:

${\mathbb H}_{eh}^m\bigl(X, DB(n)_{eh}\bigr)$

where $DB(n)$ integrates the analytic de Rham complex, Hodge truncations, and Betti components via a mapping cone structure. These complexes satisfy descent for a wide range of Grothendieck topologies, enabling the manipulation of singular and non-complete varieties (Kohrita, 2017).

Relative motivic complexes---modulus type---involve cubical cycle complexes constrained by strict intersection moduli, generalizing additive higher Chow groups to the setting $(X, D)$ with a Cartier divisor $D$ (Binda et al., 2014).

K-theory and its spectral avatars (e.g. $K(X)$, $BGL$) provide a parallel source for regulator maps. The Deligne cohomology spectrum $H_D$ or $IDR(X)$ encapsulates filtered differential form data, Hodge structures, and mixed motives (Scholbach, 2012, Bunke et al., 2015, Pridham, 2014).

2. Construction of Derived Regulator Maps

Regulator maps are constructed at the level of complexes or spectra using derived-category morphisms, often in $D(\mathrm{Sh})$ for a suitable site. The archetype is the Bloch–Suslin moving lemma, producing at the Zariski or eh-site a canonical morphism

$c_{DB}(n): Z(n)^{SF} \to DB(n)_{Zar}$

which sheafifies and passes to global sections to define

$$R_X^{m,n}: H^m_L(X, \mathbb Z(n)) \to H^m_D(X, \mathbb Z(n))$$

and, for compact supports,

$$R_{c,X}^{m,n}: H^m_{c,L}(X, \mathbb Z(n)) \to H^m_{c,D}(X, \mathbb Z(n))$$

These are functorial in $X$ and compatible with pullbacks and pushforwards, as well as with products via distinguished triangles and cup-product structures (Kohrita, 2017).

Relative modulus regulators from $z^r(X|D, \star)$ to relative Deligne complexes $𝔻(r)_{X|D}$ are synthesized using variants of El Zein's fundamental cycle classes and explicit local Čech constructions (Binda et al., 2014).

In the higher Chow context, chain-level regulators are defined by current transforms---integration of currents against algebraic correspondences, using the fundamental triple $(A_n, O_n, W_n)$ on $\mathbb{P}^n$, and compatibility with differential and product structures is established via explicit formulas and Eilenberg–Zilber maps (Lima-Filho, 2023, Santos et al., 2019, Weißschuh, 2014). For K-theory, the regulator arises from spectrum-level Chern character maps

$c: BGL \to \bigoplus_p H_D\{p\}$

and refinements as $E_\infty$-ring morphisms (Scholbach, 2012, Bunke et al., 2015).

3. Distinguished Triangles and Derived Functoriality

A defining feature of derived regulator maps is their fit into distinguished triangles—mapping cone constructions that control torsion, weights, and filtrations:

• $(7)$ $DR/Fil^n \to DR \to DR/Fil^n[+1]$
• $(8)$ $DR/Fil^n[-1] \to DB(n) \to B(n) \to [+1]$
• $(9)$ $$DB(n) \xrightarrow{\times a} DB(n) \to B(\mathbb{Z}/a) \to [+1]$$

These triangles yield exact sequences for motivic, Deligne, and Betti invariants and guarantee descent for the complexes involved. Isomorphism criteria for torsion and functorial surjectivity/cokernel-freeness are extracted from these derived structures (Kohrita, 2017).

Homotopical formalism is equally essential. The regulator map in the stable homotopy category ($SH$) lifts via spectrum-level techniques, with uniqueness and multiplicativity established through Milnor limit arguments and model-category localization (Scholbach, 2012).

4. Core Theorems: Surjectivity, Torsion, Isomorphism

The central structural theorems for derived regulator maps to Deligne cohomology establish surjectivity on torsion and torsion-freeness of cokernels for all $(m,n)$:

• $R_X^{m,n}$ is surjective on torsion; its cokernel is always torsion-free.
• For $m \leq 2n$, $R_{c,X}^{m,n}$ is an isomorphism on the torsion subgroup.
• If $\min\{2m-1, 2 \dim X +1\} \leq 2n$, $R_X^{m,n}$ is an isomorphism on torsion.

The Abel–Jacobi map from the homological part of motivic cohomology to the intermediate Jacobian

$$AJ_X^{m,n}: H_{L,\mathrm{hom}}^m(X, \mathbb Z(n)) \to J^{m,n}(X)$$

is an isomorphism on torsion in these ranges, and $J^{m,n}(X)$ becomes a generalized complex torus whenever the mixed Hodge structure has top weight $\leq m-1$ (Kohrita, 2017).

5. Explicit Models: Complexes, Currents, and Cocycles

Chain-level and analytic models underpin explicit implementations and computations:

• The path complex $P_{\mathcal{D}}^*(X, D, \mathbb{Z}(p))$ and related cone complexes serve as strictly commutative multiplicative models for Deligne cohomology (Weißschuh, 2014).
• Current-theoretic maps, triple $(A_n, O_n, W_n)$, and Eilenberg–Zilber morphisms encode group-like and cup-product behavior; convex combinations describe interior intersection products at the level of currents (Lima-Filho, 2023, Santos et al., 2019).
• For exponential complexes, period morphisms provide derived-level maps from motivic cobar complexes to explicit Deligne cocycles built of polylogarithms and local functions; this yields explicit Chern class formulas for weights $n \leq 4$ (Goncharov, 2015).

Table: Regulator Map Realizations

Source Complex/Theory	Target Complex (Deligne-type)	Regulator Construction
$Z(n)^{SF}$, $z^p(X, \star)$	$DB(n)$, $P_{\mathcal{D}}^*$, $IDR(X)$	Moving lemma, currents, spectra
Relative $z^r(X|D, \star)$	$𝔻(r)_{X|D}$	El Zein cycle, Čech classes
$BGL$, $K(X)$	$H_D$, $IDR(X)$	Chern character spectra

6. Special Cases: Abel–Jacobi, Lefschetz, Universal Properties

For $(m,n)=(2,1)$, the compact-support regulator generalizes the Abel–Jacobi theorem:

• Canonical isomorphism $$H^2_{c,cdh}(X, \mathbb Z(1)) \cong H^2_{c,L}(X, \mathbb Z(1))$$
• Abel–Jacobi isomorphism $$AJ_{c,X}^{2,1}: H^2_{c,L}(X, \mathbb Z(1))_{\rm hom} \to J^{2,1}(X)$$, with $J^{2,1}(X)$ Carlson's mixed-Hodge intermediate Jacobian.
• Surjective cycle map to $H^2_B(X, \mathbb Z(1)) \cap F^1 H^2_B(X,\mathbb C)$ with kernel the maximal divisible subgroup.

These statements recover classical results--Abel–Jacobi, Lefschetz $(1,1)$--for the smooth proper case and extend to arbitrary separated finite-type $\mathbb{C}$-schemes (Kohrita, 2017).

A universal algebraic construction of the "algebraic part" of Griffiths's intermediate Jacobian is achieved by identifying algebraically trivial cycles in Lichtenbaum cohomology and constructing a universal regular homomorphism to an abelian variety representing the image of Abel–Jacobi maps (Kohrita, 2017).

7. Arithmetic, Homotopic, and Functorial Reflections

Spectrum-level and homotopical enhancements (e.g., $E_\infty$-ring refinements (Bunke et al., 2015), pro-Banach completions (Pridham, 2014)) guarantee compatibility with products, Adams operations, and functoriality. In derived categories, regulators can be crafted as ring morphisms:

• Multiplicativity up to canonical homotopy through explicit Eilenberg–Zilber morphisms and permutation Hopf algebra characters (Lima-Filho, 2023).
• Uniqueness and coherence ensured by model-category arguments, detecting the regulator map as the unique compatible (up to contractible choice) ring morphism in $SH(S)$.

Regulator values on Shimura varieties, expressed via tempered currents, specialize to automorphic period integrals and connect directly to the special values of $L$-functions, thus providing evidence for Beilinson-type conjectures in arithmetic geometry (Gil et al., 2022).

• Kohrita, T. "Deligne–Beilinson cycle maps for Lichtenbaum cohomology" (Kohrita, 2017)
• Binda, F., Saito, S. "Relative cycles with moduli and regulator maps" (Binda et al., 2014)
• Scholbach, J. "Arakelov motivic cohomology II" (Scholbach, 2012)
• Burgos Gil, J., et al. "Tempered currents and Deligne cohomology..." (Gil et al., 2022)
• Weißschuh, P. "A commutative regulator map into Deligne–Beilinson cohomology" (Weißschuh, 2014)
• Pridham, J.P. "Semiregularity as a consequence of Goodwillie's theorem" (Pridham, 2012)
• Goncharov, A.B. "Exponential complexes, period morphisms, and characteristic classes" (Goncharov, 2015)
• Riou, J. "The Beilinson regulator is a map of ring spectra" (Bunke et al., 2015)
• Müller-Stach, S., et al., "Regulator Maps for Higher Chow Groups via Current Transforms" (Santos et al., 2019)
• Burgos Gil et al., "Multiplicative properties of the current transform regulator" (Lima-Filho, 2023)
• Pridham, J.P. "A K-theoretic interpretation of real Deligne cohomology" (Pridham, 2014)

This compendium establishes the regulator maps as canonical transformations in the derived/homotopical setting, capturing torsion, geometric, and algebraic cycle phenomena, universal Jacobian properties, and strict functorial and multiplicative behavior through explicit models and derived categorical apparatus.