Algebraic K-Theory: Non-Commutative Γ-Semirings

• The paper establishes exact-categorical and spectral foundations to define higher algebraic K-theory for non-commutative n-ary Γ-semirings.
• It employs Quillen and Waldhausen constructions to connect classical K₀, K₁ invariants with non-commutative derived Γ-geometry.
• Explicit calculations for matrix, quiver, and upper-triangular Γ-semirings demonstrate the approach’s effectiveness and Morita invariance.

Algebraic K-theory for non-commutative $Γ$-semirings provides a systematic framework to paper and classify the structure and invariants of bi-modules over generalized semiring-like algebras where both the underlying multiplication and the $Γ$-action are non-commutative and $n$-ary. The field unifies and extends the Grothendieck–Bass–Quillen program from rings to a context of higher arities, positional asymmetry, and non-additive settings. Recent developments establish the exact-categorical, homological, and spectrum-level foundations necessary for defining and analyzing $K$-theory in this setting, and tie the resulting invariants to non-commutative derived $Γ$-geometry and categorical homotopy theory (Gokavarapu, 25 Nov 2025, Gokavarapu, 11 Dec 2025, Gokavarapu, 11 Dec 2025).

1. Foundations: Non-Commutative $\boldsymbol{Γ}$-Semirings and Bi-$Γ$-Modules

A non-commutative $n$-ary $Γ$-semiring is defined as a quadruple $(T, +, Γ, \cdot)$ where:

• $(T, +)$ is a commutative monoid with zero,
• $Γ$ is a commutative semigroup,
• $\cdot$ is an $n$-ary external multiplication $\cdot:T^n \times Γ^{n-1} \to T$,
• The multiplication is additive in each $T$-slot, zero-absorbed, $n$-ary associative, but typically non-symmetric in $T$-inputs.

A bi-$Γ$-module over $T$ is an abelian monoid $(M,+)$ equipped with left and right $n$-ary compatible actions, namely $T^{n-1}\times M \times Γ^{n-1}\to M$ and $M\times Γ^{n-1}\times T^{n-1}\to M$, each satisfying the same additivity and associativity with respect to the $Γ$-structure and $T$-actions.

Morphisms between bi-$Γ$-modules are additive and strictly compatible with all positional actions. The full subcategory of finitely generated projective bi-$Γ$-modules over $T$ is denoted $Proj^Γ(T)$ and is central for $K$-theory (Gokavarapu, 11 Dec 2025).

2. Exact Categories and Homological Tools

The categories of bi-$Γ$-modules (and, more generally, left or right $Γ$-modules) are shown to be additive and carry exact structures in the sense of Quillen. In the exact category $Exact_Γ(T)=T$-$\Gamma$Mod$_{\mathrm{bi}}$, conflations are those short exact sequences in Ab which are stable under the $T$–$Γ$-actions. Projective objects include, in particular, all free bi-$Γ$-modules constructed by generators and relations for arbitrary sets $X$; every projective is a direct summand of some $F(X)$. Finitely generated projective bi-$Γ$-modules can be realized as the image of idempotent endomorphisms of free bi-modules on finite sets (Gokavarapu, 25 Nov 2025).

Resolutions in these categories (both projective and injective) support the definition of derived functors $\mathrm{Ext}^Γ$ and $\mathrm{Tor}_Γ$, and their spectral sequences, facilitating deeper connections to derived and spectral algebraic geometry.

3. Classical Algebraic $\boldsymbol{K}$-Theory: $\boldsymbol{K_0}$ and $\boldsymbol{K_1}$

The Grothendieck group $K_0^Γ(T)$ is defined via generators $[P]$ for $P$ finitely generated projective, and relations $[P]=[P']+[P'']$ given by (split) exact sequences $0\to P'\to P\to P''\to0$. Additivity, idempotent splitting, and Morita invariance for $K_0^Γ$ extend naturally from the ring case. Explicit calculations of $K_0^Γ$ for matrix and quiver path $\Gamma$-semirings demonstrate the power of these constructions; for example, $K_0^Γ(\mathcal{T}_n(\mathbb{N}))\cong\mathbb{Z}^n$ for the upper-triangular matrix semiring over naturals (Gokavarapu, 11 Dec 2025).

The Whitehead group $K_1^Γ(T)$ is constructed using the stabilization $GL^{Γ}(T)=\mathrm{colim}_n GL_n^{Γ}(T)$ of the general linear group over $T$, modulo the subgroup $E^{Γ}(T)$ generated by elementary transvections with $Γ$-weights. The Steinberg group $St^{Γ}(T)$ provides a central extension: $1\to\ker(ϕ)\to St^{Γ}(T)\to E^{Γ}(T)\to1$, with $K_1^Γ(T)=GL^Γ(T)/E^Γ(T)$. Equivalently, $K_1^Γ(T)=\pi_1|\mathrm{Aut}(Proj^Γ(T))|$ via the automorphism category. The fundamental exact sequence relates $K_0$, $K_1$ for $T$, for $\Gamma$-ideals, and for quotients $T/I$ (Gokavarapu, 11 Dec 2025).

4. Higher Algebraic $\boldsymbol{K}$-Theory: Quillen and Waldhausen Constructions

The development of higher $K$-theory utilizes the category $\mathcal{C}=T$-Mod$^{\rm bi}$ of bi-finite slot-sensitive $n$-ary $(T,\Gamma)$-modules with its exact structure. Two spectrum-level definitions are central:

• The Quillen $Q$-construction: where $Q\mathcal{C}$ has objects those of $\mathcal{C}$ and morphisms given by spans of admissible epimorphisms/monomorphisms. The realization $K(\mathcal{C})=\Omega |N_\bullet(Q\mathcal{C})|$ defines the $K$-theory space with homotopy groups $K_n(T,\Gamma)=\pi_n K(\mathcal{C})$.
• The Waldhausen $S_\bullet$-construction on $Ch^b(\mathcal{C})$: bounded chain complexes in $\mathcal{C}$ become a Waldhausen category (cofibrations = degreewise admissible monomorphisms, weak equivalences = quasi-isomorphisms), yielding a spectrum $\mathbf{K}_W(T,\Gamma)$ whose homotopy groups agree with the Quillen definition (Gokavarapu, 11 Dec 2025).

A central result establishes a canonical weak equivalence $$\mathbf{K}_Q(T,\Gamma)\simeq \mathbf{K}_W(T,\Gamma)$$; all higher $K$-theory groups are thus canonically identified as invariants of the stable $\infty$-category of perfect objects or the derived category of quasi-coherent sheaves on the non-commutative $\Gamma$-spectrum $\mathrm{Spec}_\Gamma^{nc}(T)$.

5. Localization, Dévissage, and Exact Sequences

Classical features of algebraic $K$-theory, including localization and dévissage, extend verbatim:

• For an exact subcategory $\mathcal{A}\subset \mathcal{C}$, inclusion yields a homotopy fiber sequence of $K$-theory spectra $$K(\mathcal{A})\to K(\mathcal{C})\to K(\mathcal{C}/\mathcal{A})$$, leading to the long exact sequence of $K$-theory groups.
• Standard dévissage hypotheses (where the filtration of an object lies in a simpler exact subcategory) yield that $K(\mathcal{A})\cong K(\mathcal{C})$ whenever the embedding induces $K$-theory equivalence.
• The presence of two-sided $\Gamma$-ideals and their quotient categories reproduces the classical machinery for localization and excision. Spectral sequences derived from projective/injective resolutions of bi-$Γ$-modules relate (co)homology groups and $K$-theoretic filtrations (Gokavarapu, 25 Nov 2025, Gokavarapu, 11 Dec 2025).

6. Non-Commutative Derived $\boldsymbol{Γ}$-Geometry and Morita Invariance

Algebraic $K$-theory for non-commutative $n$-ary $\Gamma$-semirings is constructed as a derived-geometric invariant of $\mathrm{Spec}_\Gamma^{nc}(T)$, the non-commutative $\Gamma$-spectrum. Morita-type invariance is established: every Morita equivalence of bi-$Γ$-module categories or derived equivalence of their stable/enhanced categories (e.g., $Perf(X)$ for $X=\operatorname{Spec}_{\Gamma}^{nc}(T)$) induces a weak equivalence of the corresponding $K$-theory spectra. The derived invariance is shown for all $n\ge0$: $K_n(T,\Gamma)\cong K_n(T',\Gamma')$ under derived Morita equivalence (Gokavarapu, 25 Nov 2025, Gokavarapu, 11 Dec 2025).

Furthermore, dg- and spectral enhancements allow comparison with topological invariants, in particular linking $K$-theory to topological cyclic homology ($TC$), and supporting trace methods and descent in non-additive and motivic contexts.

7. Explicit Examples, Calculations, and Research Directions

Algebraic $K$-theory for various explicit non-commutative $\Gamma$-semirings recovers and generalizes classical calculations:

• For matrix $\Gamma$-semirings, Morita invariance yields $K_i^\Gamma(\mathcal{M}_m(T))\cong K_i^\Gamma(T)$ for $i=0,1$.
• For upper-triangular matrix semirings, $$K_i^\Gamma(\mathcal{M}^+_n(T))\cong \bigoplus_{k=1}^n K_i^\Gamma(T)$$.
• For path semirings $TQ$ of quivers with $r$ vertices: $K_0^\Gamma(TQ)\cong\mathbb{Z}^r$ and $K_1^\Gamma(TQ)\cong (T^\times)^r$ (Gokavarapu, 11 Dec 2025).

Current directions include computation and structural analysis of higher $K_n^\Gamma(T)$ for quantum, graded, or filtered $Γ$-semirings; the development of explicit analogs of cyclotomic, differential-graded, and motivic $K$-theory in the non-commutative $\Gamma$-context; applications to descent problems and non-commutative motives; and deeper connections of $K$-theory with topological and derived invariants in the presence of $n$-ary and non-symmetric structure (Gokavarapu, 11 Dec 2025, Gokavarapu, 11 Dec 2025).