论文标题
左根箭的Gorenstein平坦表示
Gorenstein flat representations of left rooted quivers
论文作者
论文摘要
我们研究类别中的gorenstein flat对象$ {\ sf rep}（q，r）$的表示左根颤抖$ q $的表示，其中$ {\ sf mod}（r）$，所有左$ r $ $ $ - 模型的类别，$ r $ $ r $的类别是一个任意的协会环。我们表明，$ {\ sf rep}（q，r）$ in $ x $ in gorenstein flat且仅当每个顶点$ i $ the pertex $ i $ the canonical同构$φ_i^x：\ oplus_ {a：j \ to i} x（j）x（j）$和x $ imective和左$ {\ rm coker}φ_i^x $是gorenstein flat。 As an application of this result, we show that there is a hereditary abelian model structure on ${\sf Rep}(Q,R)$ whose cofibrant objects are precisely the Gorenstein flat representations, fibrant objects are precisely the cotorsion representations, and trivial objects are precisely the representations with values in the right orthogonal category of all projectively coresolved Gorenstein flat left $ r $ - 模型。
We study Gorenstein flat objects in the category ${\sf Rep}(Q,R)$ of representations of a left rooted quiver $Q$ with values in ${\sf Mod}(R)$, the category of all left $R$-modules, where $R$ is an arbitrary associative ring. We show that a representation $X$ in ${\sf Rep}(Q,R)$ is Gorenstein flat if and only if for each vertex $i$ the canonical homomorphism $φ_i^X: \oplus_{a:j\to i}X(j)\to X(i)$ is injective, and the left $R$-modules $X(i)$ and ${\rm Coker}φ_i^X$ are Gorenstein flat. As an application of this result, we show that there is a hereditary abelian model structure on ${\sf Rep}(Q,R)$ whose cofibrant objects are precisely the Gorenstein flat representations, fibrant objects are precisely the cotorsion representations, and trivial objects are precisely the representations with values in the right orthogonal category of all projectively coresolved Gorenstein flat left $R$-modules.
