Quillen’s Q-Construction in Noncommutative K-Theory

• Quillen’s Q-Construction is a categorical framework that organizes non-commutative n-ary Gamma-semirings using exact categories and span maps.
• It develops a spectrum-level K-theory by leveraging both Quillen and Waldhausen constructions with coherent spectral and long exact sequences.
• The approach ensures derived Morita invariance and functorial localization, thereby connecting homological invariants to geometric and algebraic structures.

A non-commutative $n$-ary $\Gamma$-semiring $(T, \Gamma)$ generalizes semiring theory by incorporating a parameter semiring $\Gamma$ and an $n$-ary, slot-sensitive multiplication law. Building on this algebraic foundation, the algebraic $K$-theory for such structures provides invariants sensitive to the intricate module and homological architecture of these non-commutative objects. Recent developments achieve a complete spectrum-level $K$-theory, clarify foundational exact categories, establish derived-geometric invariance, and provide computational tools via spectral and long exact sequences (Gokavarapu, 11 Dec 2025, Gokavarapu, 25 Nov 2025).

1. Foundations: Non-Commutative $n$-ary $\Gamma$-Semirings and Modules

A non-commutative $n$-ary $\Gamma$-semiring $(T,\Gamma)$ consists of:

• An additive commutative monoid $(T,+,0)$,
• A parameter semiring $(\Gamma,+,\cdot)$,
• An $n$-ary, slot-sensitive multiplication,

$$\mu : T^n \times \Gamma \to T, \qquad (t_1,\dots,t_n;\gamma) \mapsto \mu(t_1,\dots,t_n;\gamma)$$

which is distributive in each $T$-slot and in the $\Gamma$-parameter, 0-absorbing, and satisfies suitable associativity and unitality axioms (Gokavarapu, 11 Dec 2025, Gokavarapu, 25 Nov 2025).

Associated module notions are defined as follows:

• An $n$-ary left $(T,\Gamma)$-module $M$ is an additive commutative monoid with a compatible action,

$\mu_M \colon T^{n-1} \times M \times \Gamma \to M,$

subject to analogues of distributivity and associativity.

• The category $T\text{-Mod}$ of $n$-ary $(T, \Gamma)$-modules, and its subcategory $T\text{-Mod}^{\text{fgp}}$ of finitely generated projective modules, allow for projectivity notions: $M$ is projective iff $\operatorname{Hom}(M,-)$ preserves admissible epimorphisms.

Categories of bi-modules, whose morphisms respect positional actions, play a crucial role in formulating slot-sensitivity and ensuring the desired exact and additive properties (Gokavarapu, 25 Nov 2025).

2. Exact Categories and Quillen $Q$-Construction

Constructing algebraic $K$-theory relies on the precise exact category structure:

• The exact category $\mathcal{C} = T\text{-Mod}^{\text{bi}}$ of bi-finite, slot-sensitive modules consists of objects that are finite in both arguments and possess positional closure.
• Exact sequences are those that are exact as sequences of abelian monoids, with admissible monomorphisms and epimorphisms identified by positional closure under $T$-action.

The Quillen $Q$-construction forms the basis for defining higher $K$-groups:

• The category $Q(\mathcal{C})$ retains objects of $\mathcal{C}$; morphisms $X \to Y$ are isomorphism classes of spans $X \xleftarrow{p} Z \xrightarrow{i} Y$ with admissible maps.
• The $K$-theory spectrum is constructed as

$$K(\mathcal{C}) = \Omega |N_\bullet Q(\mathcal{C})|,$$

and homotopy groups define $K$-groups: $K_n(T, \Gamma) = \pi_n K(\mathcal{C})$ for $n \ge 0$.

Low-degree identifications recover:

• $K_0(T,\Gamma)$ as the Grothendieck group of finitely generated projectives,
• $K_1(T,\Gamma)$ as the Whitehead group approximated by the colimit of $$\operatorname{GL}_n(T, \Gamma)/[\operatorname{GL}_n, \operatorname{GL}_n]$$ (Gokavarapu, 11 Dec 2025).

3. Waldhausen Construction, dg-Enhancements, and Spectral Equivalences

The bounded chain complex category $Ch^b(\mathcal{C})$ inherits a Waldhausen category structure:

• Cofibrations: degreewise admissible monomorphisms,
• Weak equivalences: quasi-isomorphisms.

The $S_\bullet$-construction yields a simplicial category whose nerve, after group completion and looping, yields

$$K^{\mathrm{Wald}}(Ch^b(\mathcal{C})) = \Omega |wS_\bullet(Ch^b(\mathcal{C}))|$$

with $$K_n^{\mathrm{Wald}}(T,\Gamma) = \pi_n K^{\mathrm{Wald}}(Ch^b(\mathcal{C}))$$.

A central theorem is the Gillet–Waldhausen comparison, asserting a canonical weak equivalence of the Quillen and Waldhausen spectra,

$$K^Q(\mathcal{C}) \simeq K^{\mathrm{Wald}}(Ch^b(\mathcal{C})),$$

which implies isomorphism of all $K$-groups obtained from these models (Gokavarapu, 11 Dec 2025).

Applying dg- and $\infty$-categorical enhancements, one identifies $K$-theory spectra with those of perfect complexes in the small stable $\infty$-category $\operatorname{Perf}(\mathcal{C})$. This enables the use of techniques from derived non-commutative geometry (Gokavarapu, 11 Dec 2025).

4. Fundamental Sequences, Spectral Sequences, and Homological Calculi

$K$-theory for non-commutative $n$-ary $\Gamma$-semirings admits:

• Localization Sequences: For an exact, extension-closed subcategory $\mathcal{A} \subset \mathcal{C}$, the quotient $\mathcal{C}/\mathcal{A}$ fits into a long exact sequence:

$$\cdots \to K_{n+1}(\mathcal{C}/\mathcal{A}) \to K_n(\mathcal{A}) \to K_n(\mathcal{C}) \to K_n(\mathcal{C}/\mathcal{A}) \to \cdots$$

• Excision: Pushouts in the category of $\Gamma$-semirings satisfying Tor-vanishing yield Mayer–Vietoris sequences for $K$-groups.

Homological tools are essential:

• Projective and injective resolutions: The presence of enough free and co-free bi-modules, especially under (bi-)Noetherian hypotheses, guarantees finite-length resolutions (Gokavarapu, 25 Nov 2025).
• Spectral sequences: Universal coefficient spectral sequences of the form

$$E^2_{p,q} = \operatorname{Tor}_p^{(j,k),\Gamma}(M, K_q(T)) \implies K_{p+q}(T; M)$$

and dually for $\operatorname{Ext}$, connect $K$-theory with classical derived functors.

• Long exact sequences: Short exact sequences in the module theory induce long exact Ext–Tor sequences for corresponding $K$-groups.

These calculi enable concrete computation of $K$-groups for specific $n$-ary $\Gamma$-semirings using geometric dévissage and homological techniques (Gokavarapu, 11 Dec 2025, Gokavarapu, 25 Nov 2025).

5. Derived Morita Invariance and Functoriality

Derived Morita invariance is a fundamental property:

• If a progenerator $M$ in $\operatorname{BiMod}_\Gamma(T)$ induces an equivalence via $\operatorname{Hom}_{T-\Gamma}(M,-)$ to $\operatorname{BiMod}_\Gamma(S)$, where $S = \operatorname{End}_{T-\Gamma}(M)$, then $K$-groups are isomorphic:

$K_i(T) \cong K_i(S) \qquad \forall i \geq 0.$

• More generally, any $\Gamma$-semiring map $(T, \Gamma) \to (T', \Gamma')$ induces exact functors between module categories and hence a canonical map on $K$-theory (Gokavarapu, 25 Nov 2025, Gokavarapu, 11 Dec 2025).

Functoriality, localization, and excision results show that $K_n(T,\Gamma)$ is a derived-geometric invariant of $\operatorname{Spec}_{\Gamma}^{\mathrm{nc}}(T)$, allowing reduction to geometric and homological invariants.

6. Low-Degree Formulas and Higher $K$-Groups

Explicit low-degree formulas complete the picture:

• $K_0(T, \Gamma)$ is the Grothendieck group of finitely generated projective $(T,\Gamma)$-modules,

$$K_0(T, \Gamma) = \langle [P]\,|\, [P]=[P']+[P''] \ \text{for exact} \ 0 \to P' \to P \to P'' \to 0 \rangle.$$

• $K_1(T, \Gamma)$ is the Whitehead group,

$$K_1(T,\Gamma) = \varinjlim_n \operatorname{GL}_n(T,\Gamma)/E(T,\Gamma),$$

where $E(T,\Gamma)$ is generated by elementary matrices.

• For $n \ge 2$,

$$K_n(T,\Gamma) = \pi_n |N_\bullet Q(\mathcal{C})| \cong \pi_n K^{\mathrm{Wald}}(Ch^b(\mathcal{C})) \cong \pi_n K^{\operatorname{Perf}}(\mathcal{C}).$$

The higher groups capture increasingly subtle information about the derived module category structure and its extensions.

7. Context, Significance, and Connections

The algebraic $K$-theory of non-commutative $n$-ary $\Gamma$-semirings unifies the general structural theory for non-commutative $n$-ary systems with derived $\Gamma$-geometry, paralleling Grothendieck's and Kontsevich's frameworks for classical and non-commutative algebraic geometry (Gokavarapu, 25 Nov 2025). The foundational exact categories and spectral sequences enable analyses of localization, Mayer–Vietoris, and Morita-type phenomena, yielding a robust toolkit for investigating the geometry and homological behavior of non-commutative spectra.

A plausible implication is that this framework provides the basis for further generalization to higher and $\infty$-categorical settings, as recent work already leverages stable $\infty$-categories and dg-enhancements. These invariants are expected to play a central role in derived non-commutative geometry, allowing for applications in both pure mathematics (e.g., non-commutative motives, categorifications) and possible connections to mathematical physics through non-commutative structural sheaf theory and index theorems.

References: (Gokavarapu, 11 Dec 2025, Gokavarapu, 25 Nov 2025)