Algorithms for determination of t-module structures on some extension groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917957470257152 |
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| author | Głoch, Filip Kędzierski, Dawid E. Krasoń, Piotr |
| author_facet | Głoch, Filip Kędzierski, Dawid E. Krasoń, Piotr |
| contents | In \cite{kk04} the second and third author extended the methods of \cite{pr} and determined the \tm module structure on $\Ext^1(Φ,Ψ)$ where $Φ$ and $Ψ$ were Anderson \tm modules over $A={\mathbf F}_q[t]$ of some specific types. This approach involved the concept of biderivation and certain reduction algorithm. In this paper we generalize the results of \cite{pr} and \cite{kk04} and present complete algorithm for computation of \tm module structure on $\Ext^1(Φ,Ψ)$ for \tm modules $Φ$ and $Ψ$ such that $\rk Φ> \rk Ψ.$ The last condition is not sufficient for our algorithm to be executable. We show that it can be applied when the matrix at the biggest power of $τ$ in $Φ_t$ is invertible. We also introduce a notion of $τ$-composition series which we find suitable for the additive category of \tm modules and show that under certain assumptions on the composition series of $Φ$ and $Ψ$ our algorithm is also executable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08207 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algorithms for determination of t-module structures on some extension groups Głoch, Filip Kędzierski, Dawid E. Krasoń, Piotr Number Theory In \cite{kk04} the second and third author extended the methods of \cite{pr} and determined the \tm module structure on $\Ext^1(Φ,Ψ)$ where $Φ$ and $Ψ$ were Anderson \tm modules over $A={\mathbf F}_q[t]$ of some specific types. This approach involved the concept of biderivation and certain reduction algorithm. In this paper we generalize the results of \cite{pr} and \cite{kk04} and present complete algorithm for computation of \tm module structure on $\Ext^1(Φ,Ψ)$ for \tm modules $Φ$ and $Ψ$ such that $\rk Φ> \rk Ψ.$ The last condition is not sufficient for our algorithm to be executable. We show that it can be applied when the matrix at the biggest power of $τ$ in $Φ_t$ is invertible. We also introduce a notion of $τ$-composition series which we find suitable for the additive category of \tm modules and show that under certain assumptions on the composition series of $Φ$ and $Ψ$ our algorithm is also executable. |
| title | Algorithms for determination of t-module structures on some extension groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.08207 |