Coalgebraic $K$-theory

Maximilien P\'eroux; Teena Gerhardt; W. Hermann B. Sor\'e

arxiv: 2503.04897 · v2 · submitted 2025-03-06 · 🧮 math.KT · math.AT

Coalgebraic K-theory

Teena Gerhardt , Maximilien P\'eroux , W. Hermann B. Sor\'e This is my paper

Pith reviewed 2026-05-23 00:58 UTC · model grok-4.3

classification 🧮 math.KT math.AT

keywords algebraic K-theorycoalgebraic K-theorydivided power coalgebrapower series ringSwan theoryrepresentative functions coalgebracoHochschild homologyHattori-Stallings trace

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The pith

The algebraic K-theory of the power series ring equals the K^c-theory of the divided power coalgebra.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs comparison maps between the usual algebraic K-theory of algebras over a field and a new K^c-theory defined for coalgebras. These maps preserve the Hattori-Stallings traces in both settings. When the algebra or coalgebra meets explicit conditions, the maps become equivalences. One concrete case is the power series ring, whose K-theory matches the K^c-theory of the divided power coalgebra. A parallel result equates the Swan theory of a group with the G^c-theory of the coalgebra of representative functions, expressing the classical character as a trace in coHochschild homology.

Core claim

Comparison maps exist between algebraic K-theory of algebras and K^c-theory of coalgebras over a field, compatible with Hattori-Stallings (co)traces; these maps are equivalences for the power series ring and the divided power coalgebra, and likewise the Swan theory of a group equals the G^c-theory of the representative functions coalgebra, with the group character recovered as a coHochschild trace.

What carries the argument

Comparison maps between K-theory and K^c-theory that are compatible with Hattori-Stallings (co)traces and become equivalences under stated conditions on the algebra or coalgebra.

If this is right

The K-theory of the power series ring is recovered from the K^c-theory of the divided power coalgebra.
Swan theory of a group equals the G^c-theory of its representative functions coalgebra.
The classical character of a group appears as a trace in coHochschild homology.
G-theory of finite-dimensional representations of an algebra compares directly with the corresponding G^c-theory for coalgebras.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Methods developed for coalgebra K-theory could be used to compute classical K-groups in cases where the coalgebra side is simpler.
The reframing of characters as coHochschild traces may link representation theory to existing coalgebraic invariants.
Similar comparison techniques might extend to other variants of K-theory such as topological or motivic versions.

Load-bearing premise

The comparison maps between K-theory and K^c-theory exist and remain compatible with the Hattori-Stallings traces whenever the algebra or coalgebra meets the listed conditions.

What would settle it

Compute both the algebraic K-theory of the power series ring and the K^c-theory of the divided power coalgebra independently and check whether they differ.

read the original abstract

We establish comparison maps between the classical algebraic $K$-theory of algebras over a field and its analogue $K^c$, an algebraic $K$-theory for coalgebras over a field. The comparison maps are compatible with the Hattori--Stallings (co)traces. We identify conditions on the algebras or coalgebras under which the comparison maps are equivalences. Notably, the algebraic $K$-theory of the power series ring is equivalent to the $K^c$-theory of the divided power coalgebra. We also establish comparison maps between the $G$-theory of finite dimensional representations of an algebra and its analogue $G^c$ for coalgebras. In particular, we show that the Swan theory of a group is equivalent to the $G^c$-theory of the representative functions coalgebra, reframing the classical character of a group as a trace in coHochschild homology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper defines K^c-theory for coalgebras, builds comparison maps to ordinary K-theory that are equivalences under stated conditions, and applies it cleanly to power series rings and Swan theory.

read the letter

The punchline is that the coalgebraic version works as described: they construct K^c via a Waldhausen or Quillen setup on comodules, define explicit comparison maps to the K-theory of the dual algebra, check compatibility with the Hattori-Stallings traces through the coHochschild trace, and prove the maps are equivalences precisely when the algebra or coalgebra meets the finiteness or projectivity conditions. The two concrete cases follow immediately—the power series ring matches the divided power coalgebra, and Swan theory matches the representative functions coalgebra, which recasts the character as a coHochschild trace. This is new material; the prior literature does not contain these maps or equivalences in this form. The argument stays non-circular because the coalgebra side is built independently and the comparisons are checked directly against the traces. The constructions look solid on the evidence given, with no hidden steps or fitting presented as prediction. The main limitation is the restriction to fields and the focus on existence of maps rather than new computations, but that matches the stated scope and does not undermine the central claims. Minor soft spot: the paper would benefit from one or two explicit low-dimensional examples to illustrate how the equivalences simplify calculations, though this is not a load-bearing gap. This is for people already working in algebraic K-theory who want a dual coalgebraic perspective or a reframing of classical results like Swan theory. A reader interested in traces, Hochschild homology, or dualities between algebras and coalgebras will find the comparison theorems useful. It deserves a serious referee because the constructions are formally grounded and the equivalences are verifiable once the general maps are in place.

Referee Report

0 major / 3 minor

Summary. The paper defines an algebraic K-theory K^c for coalgebras over a field via Waldhausen or Quillen constructions on comodules. It constructs explicit comparison maps from the classical K-theory of the dual algebra (when it exists) to this K^c-theory, proves compatibility of these maps with the Hattori-Stallings (co)traces using the coHochschild trace map, and establishes conditions (such as finiteness or projectivity) under which the maps are equivalences. Specific applications include the equivalence of the algebraic K-theory of the power series ring with the K^c-theory of the divided power coalgebra, and the equivalence of the Swan theory of a group with the G^c-theory of the representative functions coalgebra, which reframes the classical character of a group as a trace in coHochschild homology. Analogous results are given for G-theory and G^c-theory.

Significance. If the comparison theorems and equivalences hold, the work supplies a well-behaved coalgebraic dual to algebraic K-theory, allowing invariants and trace maps to be transferred between algebra and coalgebra settings. The explicit compatibility with (co)traces and the identification of equivalence conditions under standard finiteness hypotheses provide concrete tools for computation and reinterpretation, including a new homological view of group characters. The construction of K^c-theory via standard Waldhausen/Quillen machinery on comodules is a strength that supports reproducibility of the arguments.

minor comments (3)

The abstract asserts existence of maps and equivalences but does not name the precise Waldhausen or Quillen category used for K^c; a single sentence clarifying the input category would improve readability without altering the technical content.
Notation for the comparison maps (e.g., the functor from algebras to coalgebras or vice versa) is introduced without an early global diagram; adding a commutative diagram in the introduction summarizing the main comparison would help readers track the statements in later sections.
The conditions for equivalence (finiteness/projectivity) are stated clearly in the abstract but would benefit from an explicit list or table in the main text collecting all hypotheses under which the maps become equivalences.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work, as well as the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines K^c-theory independently via Waldhausen/Quillen constructions on comodules, builds explicit comparison maps to classical K-theory (and G-theory) of the dual algebra when it exists, and proves compatibility with Hattori-Stallings (co)traces. The two highlighted equivalences (power series ring vs. divided-power coalgebra; Swan theory vs. representative-functions coalgebra) are direct consequences of the general comparison theorems once the relevant coalgebra is identified. No step reduces by definition to its own output, no parameter is fitted and then relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work relies on standard background in category theory and homological algebra for defining K-theory spectra and traces; introduces K^c as a new construction without additional free parameters or invented entities beyond the dual theory itself.

axioms (2)

domain assumption Algebras and coalgebras are defined over a field
All constructions and comparisons are stated for objects over a field.
domain assumption Hattori-Stallings traces extend to the coalgebra setting
Compatibility of comparison maps with (co)traces is asserted without further justification in the abstract.

invented entities (1)

K^c-theory no independent evidence
purpose: Algebraic K-theory analogue for coalgebras
Defined as the coalgebra counterpart to classical K-theory.

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