Wakamatsu倾斜模沿Frobenius扩张的转移
Transfer of Wakamatsu tilting modules along Frobenius extensions
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中文总结 AI 辅助
本文研究Frobenius扩张下Wakamatsu倾斜模的转移，给出诱导模仍为Wakamatsu倾斜模的充分条件，刻画倾斜情形，证明相关自同态环映射为Frobenius扩张并讨论其应用。
中文摘要 AI 辅助
设$\boldsymbol{\nu: R\to A}$为Frobenius扩张，$T$为左$R$-模Wakamatsu倾斜模。本文给出诱导模$A\bigotimes_R T$仍为Wakamatsu倾斜模的充分条件，建立分裂中心投射Frobenius扩张的上升-下降结果；还通过对所有$i>0$，$\boldsymbol{\text{Ext}}_R^i(T,A\bigotimes_R T)$消失刻画倾斜情形；此外，若$A\bigotimes_R T \text{属于}\boldsymbol{\text{add}}_R(T)$，则自同态环的自然映射$S=\text{End}_R(T)\to B=\text{End}_A(A\bigotimes_R T)$为Frobenius扩张，且$T\bigotimes_S B$作为$R$-$B$-双模同构于$A\bigotimes_R T$；文中还讨论了其在Brenner–Butler–Miyashita等价及若干特定类Frobenius扩张中的应用。
英文摘要
Let $ι: R\to A$ be a Frobenius extension and let $T$ be a Wakamatsu tilting left $R$-module. We give sufficient conditions for the induced module $A\otimes_R T$ to remain Wakamatsu tilting and establish an ascent--descent result for split centrally projective Frobenius extensions. We also characterize the tilting case by the vanishing of $\mathrm{Ext}_R^i(T,A\otimes_R T)$ for all $i>0$. Moreover, if $A\otimes_R T\in\mathrm{add}_R(T)$, then the natural map $S=\mathrm{End}_R(T)\to B=\mathrm{End}_A(A\otimes_R T)$ of endomorphism rings is a Frobenius extension and $T\otimes_S B\cong A\otimes_R T$ as $R$-$B$-bimodules. Applications to Brenner--Butler--Miyashita equivalences and some specific classes of Frobenius extensions are also discussed.
