Additive Ternary $Γ$-Modules and Homological Algebra
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911393097187328 |
|---|---|
| author | Gokavarapu, Chandrasekhar Dasari, Madhusudhana Rao |
| author_facet | Gokavarapu, Chandrasekhar Dasari, Madhusudhana Rao |
| contents | Fix a commutative monoid $(T,+,0)$, a commutative monoid $(Γ,+,0_Γ)$, and a map \[ (a,α,b,β,c)\longmapsto a\,α\,b\,β\,c\in T \] which is additive in each variable and associative in the ternary sense. A left additive ternary $Γ$-module is an abelian group $M$ equipped with an action $(a,α,b,β,m)\mapsto a\,α\,b\,β\,m$ satisfying the same associativity constraints. Two scalars are intrinsic, so the action is genuinely polyadic; in particular, we do not assume a canonical unary action $T\times M\to M$. The first part constructs the \emph{operator ring} $\mathcal O_{T,Γ}$ generated by the left translations $(a,α,b,β)$. It is shown that $T\!\!\;\!-\!Γ\mathrm{Mod}$ is equivalent to the ordinary module category $\mathcal O_{T,Γ}\mathrm{Mod}$. The category is therefore abelian. Under the unital operator hypothesis \emph{(U)} (i.e. when $\mathcal O_{T,Γ}$ is unital), it has enough projectives and enough injectives. The second part defines a tensor product over $T$ by a \emph{bi-balanced} universal property forced by the ternary action, proves right exactness of $-\otimes_T-$, and establishes a Tensor--Hom adjunction for bimodules. Assuming \emph{(U)}, derived functors $\mathrm{Ext}$ and $\mathrm{Tor}$ are developed inside the additive track. A finite example with $T=\ZZ/4\ZZ$ gives explicit computations \[\mathrm{Ext}^1_T(\ZZ/2\ZZ,\ZZ/2\ZZ)\cong \ZZ/2\ZZ,\qquad
\mathrm{Tor}^T_1(\ZZ/2\ZZ,\ZZ/2\ZZ)\cong \ZZ/2\ZZ,\] with a concrete nonsplit extension and an explicit failure of tensor exactness. Counterexamples isolate the precise points where naive binary-module arguments break. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02544 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Additive Ternary $Γ$-Modules and Homological Algebra Gokavarapu, Chandrasekhar Dasari, Madhusudhana Rao Rings and Algebras Primary 16Y60, Secondary 18G15, 16D90, 18E10 Fix a commutative monoid $(T,+,0)$, a commutative monoid $(Γ,+,0_Γ)$, and a map \[ (a,α,b,β,c)\longmapsto a\,α\,b\,β\,c\in T \] which is additive in each variable and associative in the ternary sense. A left additive ternary $Γ$-module is an abelian group $M$ equipped with an action $(a,α,b,β,m)\mapsto a\,α\,b\,β\,m$ satisfying the same associativity constraints. Two scalars are intrinsic, so the action is genuinely polyadic; in particular, we do not assume a canonical unary action $T\times M\to M$. The first part constructs the \emph{operator ring} $\mathcal O_{T,Γ}$ generated by the left translations $(a,α,b,β)$. It is shown that $T\!\!\;\!-\!Γ\mathrm{Mod}$ is equivalent to the ordinary module category $\mathcal O_{T,Γ}\mathrm{Mod}$. The category is therefore abelian. Under the unital operator hypothesis \emph{(U)} (i.e. when $\mathcal O_{T,Γ}$ is unital), it has enough projectives and enough injectives. The second part defines a tensor product over $T$ by a \emph{bi-balanced} universal property forced by the ternary action, proves right exactness of $-\otimes_T-$, and establishes a Tensor--Hom adjunction for bimodules. Assuming \emph{(U)}, derived functors $\mathrm{Ext}$ and $\mathrm{Tor}$ are developed inside the additive track. A finite example with $T=\ZZ/4\ZZ$ gives explicit computations \[\mathrm{Ext}^1_T(\ZZ/2\ZZ,\ZZ/2\ZZ)\cong \ZZ/2\ZZ,\qquad \mathrm{Tor}^T_1(\ZZ/2\ZZ,\ZZ/2\ZZ)\cong \ZZ/2\ZZ,\] with a concrete nonsplit extension and an explicit failure of tensor exactness. Counterexamples isolate the precise points where naive binary-module arguments break. |
| title | Additive Ternary $Γ$-Modules and Homological Algebra |
| topic | Rings and Algebras Primary 16Y60, Secondary 18G15, 16D90, 18E10 |
| url | https://arxiv.org/abs/2511.02544 |