Relative Chern Character Isomorphism

• Relative Chern character is a framework linking algebraic K-theory with negative cyclic homology through regulator comparisons and explicit transgression constructions.
• It emphasizes regulator compatibility by integrating varying approaches including Deligne–Beilinson, syntomic, and p-adic regulators in commutative diagrams.
• Under nilpotent and split conditions, the relative Chern character yields degreewise isomorphisms between K-theory and cyclic homology, offering effective infinitesimal insights.

Searching arXiv for the cited works to ground the article in the literature. The expression relative Chern character isomorphism does not denote a single uniform theorem. In the literature represented here, it refers to several distinct but related patterns: an actual nilpotent relative isomorphism between relative algebraic $K$-theory and relative negative cyclic homology; regulator comparison theorems in which a relative Chern character fits into commutative diagrams with Deligne–Beilinson, syntomic, Borel, or $p$-adic regulators; categorical uniqueness statements characterizing Chern character maps by universal properties; and explicit transgression or cocycle constructions that supply the form-level data underlying relative theories. A central terminological point is that Tamme’s dissertation and its $p$-adic sequel do not prove a general theorem asserting that $\operatorname{ch}^{\mathrm{rel}}$ is itself an isomorphism; their strongest central results are comparison theorems (Tamme, 2010).

1. Relative $K$-theory targets and the basic shape of the map

In the complex-geometric setting of smooth affine schemes $X=\operatorname{Spec}(A)$ over $\mathbf C$, algebraic and topological $K$-theory are modeled by

$$K_i(X)=\pi_i(BGL(A)^+), \qquad K^{-i}_{\mathrm{top}}(X)=\pi_i(BU^X), \qquad i>0.$$

The relative $K$-group is defined as the homotopy group of the homotopy fiber of the map from algebraic to topological $p$0-theory; concretely,

$p$1

and it fits into the long exact sequence

$p$2

For simplicial algebraic varieties $p$3, the corresponding cohomological target is defined by a cone construction. After choosing a good compactification $p$4 with boundary $p$5,

$p$6

and Tamme explicitly notes the identification

$p$7

Using the Hurewicz map and the decomposition of relative cohomology for simplicial objects $p$8, the relative Chern character in the complex theory is

$p$9

This is the exact map constructed in Tamme’s complex theory, and it is the basic object around which the later comparison statements are organized (Tamme, 2010).

At the form level, the relative class is built from secondary characteristic forms. Given topological bundles $p$0 on a simplicial manifold $p$1, with connections $p$2 and a bundle morphism $p$3, one sets on $p$4

$p$5

and defines

$p$6

Its boundary formula is

$p$7

so the construction is intrinsically transgressive.

2. Comparison theorems versus actual isomorphism theorems

A major source of confusion is that several papers construct relative Chern characters but do not prove that those maps are isomorphisms. In Tamme’s dissertation, the principal complex-case theorem for smooth affine $p$8 is the commutative diagram

$p$9

and by Jouanolou’s trick the same statement extends from smooth affine schemes to all smooth separated finite-type $\operatorname{ch}^{\mathrm{rel}}$0-schemes. Likewise, for a simplicial algebraic variety $\operatorname{ch}^{\mathrm{rel}}$1, an algebraic $\operatorname{ch}^{\mathrm{rel}}$2-bundle $\operatorname{ch}^{\mathrm{rel}}$3, and a trivialization $\operatorname{ch}^{\mathrm{rel}}$4 of the associated topological bundle, the refined class

$\operatorname{ch}^{\mathrm{rel}}$5

is mapped to

$\operatorname{ch}^{\mathrm{rel}}$6

under the natural map to Deligne–Beilinson cohomology. These are regulator comparison statements, not isomorphism statements for $\operatorname{ch}^{\mathrm{rel}}$7 itself (Tamme, 2010).

The same distinction persists in the $\operatorname{ch}^{\mathrm{rel}}$8-adic sequel. For $\operatorname{ch}^{\mathrm{rel}}$9, with $K$0 a complete discrete valuation ring and

$K$1

one obtains

$K$2

The main theorem is again a commutative comparison diagram

$K$3

showing compatibility with the rigid syntomic regulator. In the proper case, $K$4, and the comparison map from relative cohomology to syntomic cohomology is an isomorphism in many degrees, but the paper still does not assert that $K$5 itself is an isomorphism in general (Tamme, 2011).

This suggests a useful conceptual rule: in this literature, the words relative Chern character and relative Chern character isomorphism must be separated carefully. Construction and regulator compatibility are common; general bijectivity is not.

3. The nilpotent relative isomorphism

The strongest actual theorem in the supplied corpus that matches the phrase relative Chern character isomorphism is the nilpotent relative statement used in the infinitesimal theory of Chow groups. For split nilpotent pairs $K$6, the relative algebraic Chern character

$K$7

extends to an isomorphism of functors from relative algebraic $K$8-theory to relative negative cyclic homology, viewed as functors on the category of split nilpotent pairs $K$9. At spectrum level, the cited result is a homotopy equivalence

$X=\operatorname{Spec}(A)$0

where $X=\operatorname{Spec}(A)$1 is a sheaf of nilpotent ideals on $X=\operatorname{Spec}(A)$2. Passing to homotopy groups yields degreewise isomorphisms

$X=\operatorname{Spec}(A)$3

The paper emphasizing this framework is explicit that the isomorphism is used in ring-level, spectrum-level, group-level, with-supports, sheafified, coniveau-complex, and Adams-eigenspace forms (Dribus et al., 2015).

In the main geometric application, $X=\operatorname{Spec}(A)$4 is a nonsingular quasiprojective variety over a field $X=\operatorname{Spec}(A)$5 of characteristic zero, $X=\operatorname{Spec}(A)$6 is an Artinian local $X=\operatorname{Spec}(A)$7-algebra with maximal ideal $X=\operatorname{Spec}(A)$8, and $X=\operatorname{Spec}(A)$9 is the infinitesimal thickening of $\mathbf C$0 with respect to $\mathbf C$1. The coniveau machine has four columns built from $\mathbf C$2, $\mathbf C$3, $\mathbf C$4, and $\mathbf C$5, with the first three forming a split exact sequence

$\mathbf C$6

and the map between the last two columns is an isomorphism of complexes induced by the relative algebraic Chern character. Termwise, for $\mathbf C$7,

$\mathbf C$8

The supported and Adams-decomposed forms

$\mathbf C$9

are also stated explicitly. This is the setting in which the phrase relative Chern character isomorphism is literally accurate (Dribus et al., 2015).

The same source clarifies the hypotheses under which the statement is valid: nilpotence is essential, the main applications are over characteristic zero, and the strongest local functorial formulation is for split nilpotent pairs. Without splitting, one should use homotopy fibers rather than kernels. In this precise sense, the relative Chern character isomorphism is a nilpotent theorem, not a general feature of all relative regulators.

4. Regulator compatibility and normalization phenomena

Even where no isomorphism theorem is available, relative Chern characters are often important because they compare different regulator formalisms. In the complex case $K$0, Tamme gives an explicit cocycle for the relative Chern character,

$K$1

and compares it with the Lie algebra cocycle for Borel’s regulator. Using surjectivity of

$K$2

the comparison theorem with Deligne–Beilinson Chern characters, and explicit cocycle computations, the dissertation reproves Burgos’ theorem

$K$3

The stated factor $K$4 depends on the normalization conventions adopted in the paper and compared carefully to Burgos’ sign conventions (Tamme, 2010).

In the $K$5-adic setting, for $K$6 with $K$7 a complete DVR, the relative Chern character is compared to the $K$8-adic Borel regulator of Huber–Kings through the commutative triangle

$K$9

so the relative Chern character agrees with the $$K_i(X)=\pi_i(BGL(A)^+), \qquad K^{-i}_{\mathrm{top}}(X)=\pi_i(BU^X), \qquad i>0.$$0-adic Borel regulator up to the explicit factor

$$K_i(X)=\pi_i(BGL(A)^+), \qquad K^{-i}_{\mathrm{top}}(X)=\pi_i(BU^X), \qquad i>0.$$1

Tamme emphasizes that this is a normalization issue: the relative Chern character uses Chern characters, whereas the Huber–Kings $$K_i(X)=\pi_i(BGL(A)^+), \qquad K^{-i}_{\mathrm{top}}(X)=\pi_i(BU^X), \qquad i>0.$$2-adic Borel regulator is normalized via Chern classes (Tamme, 2010).

The later $$K_i(X)=\pi_i(BGL(A)^+), \qquad K^{-i}_{\mathrm{top}}(X)=\pi_i(BU^X), \qquad i>0.$$3-adic paper then places the same relative map into the syntomic and étale regulator picture. For smooth projective $$K_i(X)=\pi_i(BGL(A)^+), \qquad K^{-i}_{\mathrm{top}}(X)=\pi_i(BU^X), \qquad i>0.$$4-schemes with finite residue field, one obtains the commutative diagram

$$K_i(X)=\pi_i(BGL(A)^+), \qquad K^{-i}_{\mathrm{top}}(X)=\pi_i(BU^X), \qquad i>0.$$5

where the lower horizontal map is the Bloch–Kato exponential. For $$K_i(X)=\pi_i(BGL(A)^+), \qquad K^{-i}_{\mathrm{top}}(X)=\pi_i(BU^X), \qquad i>0.$$6, this recovers the Huber–Kings theorem with the explicit factor $$K_i(X)=\pi_i(BGL(A)^+), \qquad K^{-i}_{\mathrm{top}}(X)=\pi_i(BU^X), \qquad i>0.$$7 coming from normalization differences between Chern classes and Chern characters (Tamme, 2011).

5. Universal, simplicial, and secondary models

A second major strand of the subject concerns uniqueness and cocycle-level realization rather than isomorphism. Tabuada’s universal characterization of the Chern character maps is not a theorem about relative groups $$K_i(X)=\pi_i(BGL(A)^+), \qquad K^{-i}_{\mathrm{top}}(X)=\pi_i(BU^X), \qquad i>0.$$8 or relative cyclic homology $$K_i(X)=\pi_i(BGL(A)^+), \qquad K^{-i}_{\mathrm{top}}(X)=\pi_i(BU^X), \qquad i>0.$$9, but it identifies the absolute Grothendieck-group-level Chern characters

$K$0

as the unique natural transformations determined by the unit class $K$1 in the universal additive framework. The key formulas are

$K$2

and

$K$3

The paper explicitly says that it does not define relative groups such as $K$4 or state an isomorphism theorem for them, but it strongly suggests that relative theories built as homotopy fibers or exact-sequence terms should inherit uniqueness and compatibility properties from the same additive formalism (Tabuada, 2010).

At the cocycle level, several papers provide explicit transgression data that are structurally close to relative characteristic class constructions. Suzuki constructs explicit cocycles in the simplicial de Rham complex $K$5 representing $K$6 and proves Brylinski’s conjecture by producing a cocycle $K$7 in a local truncated complex whose connecting image is the global Bott–Shulman–Stasheff Chern character cocycle. The short exact sequence

$K$8

induces a connecting morphism

$K$9

and Suzuki proves

$p$00

The paper does not formulate this as a relative $p$01-theoretic isomorphism, but it gives a boundary or transgression realization that is closely analogous to relative characteristic class constructions (Suzuki, 2013).

Takhtajan’s work on explicit computation of Chern character forms belongs to the same secondary tradition. For a holomorphic Hermitian bundle $p$02, with $p$03 and $p$04, the paper decomposes the Chern–Simons form

$p$05

and then solves ascent equations yielding a Bott–Chern representative

$p$06

For $p$07 and $p$08, explicit formulas are obtained in Cholesky coordinates. This does not prove an isomorphism theorem, but it supplies concrete secondary representatives of the type used in Deligne, Bott–Chern, and differential $p$09-theoretic settings (Takhtajan, 2014).

A further simplicial and homotopy-coherent version appears in the Hodge Chern character paper. It defines a simplicial-presheaf map

$p$10

that assigns to a sequence of composable holomorphic bundle isomorphisms with holomorphic connections explicit holomorphic forms. In simplicial degree $p$11, it gives the comparison form

$p$12

for a bundle isomorphism $p$13, and after passage to a Čech nerve and totalization the higher simplices govern compatibilities among lower-degree data. The paper explicitly states that it does not prove a relative Chern character isomorphism, but it does provide rich relative cocycle data (Glass et al., 2019).

6. Generalized forms: twists, matrix factorizations, and scope

The phrase Chern character isomorphism does become literally correct again in certain generalized settings, although not always under the same hypotheses as the nilpotent algebraic theorem. In higher twisted $p$14-theory, for any finite CW complex $p$15 equipped with a higher twist arising from a cohomotopy class $p$16, the paper on higher twisted $p$17-theory constructs twisted Chern character maps

$p$18

where the target is the twisted cohomology of

$p$19

Its main theorem states that the twisted Chern character induces an isomorphism of the realized Atiyah–Hirzebruch spectral sequence onto the spectral sequence computing higher twisted cohomology and consequently yields a real isomorphism

$p$20

The same paper develops relative twisted $p$21-theory and relative twisted cohomology for pairs and proves naturality with respect to six-term exact sequences, but the final isomorphism theorem is written only for absolute groups, not as a separate theorem

$p$22

Thus it provides an explicit relative framework with an absolute real isomorphism theorem (MacDonald et al., 2020).

An analogous curved comparison appears in the matrix-factorization setting. For a smooth separated scheme $p$23 of finite type over a characteristic-zero field and a function $p$24, the paper constructs a chain-level HKR-type quasi-isomorphism

$p$25

and proves that it is a quasi-isomorphism. The target computes the hypercohomology of the twisted de Rham complex

$p$26

and under this identification the negative cyclic Chern character of a matrix factorization $p$27 is represented by

$p$28

The paper is explicit that this is not a classical relative algebraic $p$29-theory theorem, but rather a comparison isomorphism in a curved Landau–Ginzburg setting, together with a support-localized and finite-group equivariant extension (Chung et al., 2021).

Taken together, these examples show that the phrase relative Chern character isomorphism has a stratified meaning. In the narrowest and most literal sense, it refers to the nilpotent equivalence

$p$30

and its consequences for infinitesimal deformation theory. In a broader regulator-theoretic sense, it refers to comparison diagrams identifying relative Chern characters with Deligne–Beilinson, syntomic, Borel, or $p$31-adic regulators up to explicit signs and factorials. In an even broader homotopical and cocycle-theoretic sense, it designates the transgression, uniqueness, and explicit form-level structures that make such comparison theorems possible.