论文标题
真诚的淤积模块和消失的条件
Sincere silting modules and vanishing conditions
论文作者
论文摘要
让$ r $成为一个完美的戒指，$ t $是$ r $ - 模块。我们研究了真诚模块，真诚的淤积模块和倾斜模块的特征。事实证明，$ t $是且仅当$ t $推荐满足消失条件$ \ mathrm {kerext}^{0 \ le i \ le 1} _r（t， - ）= 0 $，而$ t $仅在$ \ \ mathrm {kerrm {kerrm {kerrm} {0 i \ leqslant 1} _ {r}（t， - ）= 0 $和$ \ mathrm {gen} t \ subseteq \ subseteq \ mathrm {ker} \ mathrm {ext}^{1 \ leqslant i \ leqslant i \ leqslant 2}}作为一个应用程序，我们证明有限投影尺寸的真诚淤塞$ r $ -module $ t $在且仅当$ \ mathrm {ext}^{i}^{i} _ {r} _ {r} _ {r}（t，t，t，t^{（j）} = 0 = 0 $的所有设置$ j $ j $＆j $ j $＆j $ j $＆je $ i i \ ge 1 $ \ ge 1 $。这不仅扩展了张[14]的主要结果，从有限生成的模块上的Artin代数到无限生成的模块，在更通用的环上，而且在不使用函数$τ$和Auslander-Reiten公式的情况下提供了不同的证明。
Let $R$ be a perfect ring and $T$ be an $R$-module. We study characterizations of sincere modules, sincere silting modules and tilting modules in terms of various vanishing conditions. It is proved that $T$ is sincere silting if and only if $T$ is presilting satisfing the vanishing condition $\mathrm{KerExt}^{0\le i\le 1}_R(T,-)=0$, and that $T$ is tilting if and only if $\mathrm{Ker}\mathrm{Ext}^{0\leqslant i\leqslant 1}_{R}(T,-)=0$ and $\mathrm{Gen}T\subseteq \mathrm{Ker}\mathrm{Ext}^{1\leqslant i\leqslant 2}_{R}(T,-)$. As an application, we prove that a sincere silting $R$-module $T$ of finite projective dimension is tilting if and only if $\mathrm{Ext}^{i}_{R}(T,T^{(J)})=0$ for all sets $J$ and all integer $i\ge 1$. This not only extends a main result of Zhang [14]from finitely generated modules over Artin algebras to infinitely generated modules over more general rings, but also gives it a different proof without using the functor $τ$ and Auslander-Reiten formula.
