Projective Modules and Classical Algebraic K-Theory of Non-Commutative Gamma Semirings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917140981874688 |
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| author | Gokavarapu, Chandrasekhar |
| author_facet | Gokavarapu, Chandrasekhar |
| contents | In this paper, we initiate the study of algebraic K-theory for non-commutative $Γ$-semirings, extending the classical constructions of Grothendieck and Bass to this setting. We first establish the categorical foundations by constructing the category of finitely generated projective bi-$Γ$-modules over a non-commutative $Γ$-semiring $T$. We prove that this category admits an exact structure, allowing for the definition of the Grothendieck group $K_0^Γ(T)$. Furthermore, we develop the theory of the Whitehead group $K_1^Γ(T)$ using elementary matrices and the Steinberg relations in the non-commutative $Γ$-semiring context. We establish the fundamental exact sequences linking $K_0$ and $K_1$ and provide explicit calculations for specific classes of non-commutative $Γ$-semirings. This work lays the algebraic groundwork for future studies on higher K-theory spectra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11097 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Projective Modules and Classical Algebraic K-Theory of Non-Commutative Gamma Semirings Gokavarapu, Chandrasekhar Rings and Algebras K-Theory and Homology 19A49, 19B14, 18G55, 16Y60, 16D40 19A49, 19B14, 18G55, 16Y60, 16D40 In this paper, we initiate the study of algebraic K-theory for non-commutative $Γ$-semirings, extending the classical constructions of Grothendieck and Bass to this setting. We first establish the categorical foundations by constructing the category of finitely generated projective bi-$Γ$-modules over a non-commutative $Γ$-semiring $T$. We prove that this category admits an exact structure, allowing for the definition of the Grothendieck group $K_0^Γ(T)$. Furthermore, we develop the theory of the Whitehead group $K_1^Γ(T)$ using elementary matrices and the Steinberg relations in the non-commutative $Γ$-semiring context. We establish the fundamental exact sequences linking $K_0$ and $K_1$ and provide explicit calculations for specific classes of non-commutative $Γ$-semirings. This work lays the algebraic groundwork for future studies on higher K-theory spectra. |
| title | Projective Modules and Classical Algebraic K-Theory of Non-Commutative Gamma Semirings |
| topic | Rings and Algebras K-Theory and Homology 19A49, 19B14, 18G55, 16Y60, 16D40 19A49, 19B14, 18G55, 16Y60, 16D40 |
| url | https://arxiv.org/abs/2512.11097 |