论文标题
在$ {\ mathrm {ext}}}^1 $的Drinfeld模块上
On ${\mathrm{Ext}}^1$ for Drinfeld modules
论文作者
论文摘要
令$ a = {\ mathbb f} _q [t] $是有限字段$ {\ mathbb f} _q $上的多项式环，让$ ϕ $和$ψ$ as $ a- $ a- $ drinfeld模块。在本文中，我们将组$ {\ mathrm {ext}}^1（ϕ，ψ）$与Baer添加一起考虑。我们表明，如果$ \ mathrm {rank} ϕ> \ mathrm {rank}ψ$，则$ \ mathrm {ext^1}（ϕ，ψ）$具有\ tm模块的结构。我们给出了描述这种结构的完整算法。我们将其概括为以下情况：$ \ MATHRM {ext^1}（φ，ψ）$，其中$φ$是a \ tm模块，$ψ$是Drinfeld模块，$ \ Mathrm {ext^1}（φ，c^{\ otimes e}） $ e $ - 卡利兹模块的张量产品。我们还建立了\ tm模块的$ \ ext $组与相应的伴随$ {\ mathbf t}^σ$ -Modules之间的二元性。最后，我们证明了\ tm模块和双\ tm动机的$“ \ hom- \ ext” $六项精确序列的存在。由于\ tm模块的类别仅是加性（不是abelian），因此该结果是不平凡的。
Let $A={\mathbb F}_q[t]$ be the polynomial ring over a finite field ${\mathbb F}_q$ and let $ϕ$ and $ψ$ be $A-$Drinfeld modules. In this paper we consider the group ${\mathrm{Ext}}^1(ϕ,ψ)$ with the Baer addition. We show that if $\mathrm{rank}ϕ>\mathrm{rank}ψ$ then $\mathrm{Ext^1}(ϕ,ψ)$ has the structure of a \tm module. We give complete algorithm describing this structure. We generalize this to the cases: $\mathrm{Ext^1}(Φ,ψ)$ where $Φ$ is a \tm module and $ψ$ is a Drinfeld module and $\mathrm{Ext^1}(Φ, C^{\otimes e})$ where $Φ$ is a \tm module and $C^{\otimes e}$ is the $e$-th tensor product of Carlitz module. We also establish duality between $\Ext$ groups for \tm modules and the corresponding adjoint ${\mathbf t}^σ$-modules. Finally, we prove the existence of $"\Hom-\Ext"$ six-term exact sequences for \tm modules and dual \tm motives. As the category of \tm modules is only additive (not abelian) this result is nontrivial.
