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In this note we show that for a ${*}^n$-module, in particular, an almost $n$-tilting module, $P$ over a ring $R$ with $A={\mathrm{E}nd}_{R}P$ such that $P_A$ has finite flat dimension, the upper bound of the global dimension of $A$ can be estimated by the global dimension of $R$ and hence generalize the corresponding results in tilting theory and the ones in the theory of $*$-modules. As an application, we show that for a finitely generated projective module over a VN regular ring $R$, the global dimension of its endomorphism ring is not more...

Let $U$ be a dg-$A$-module, $B$ the endomorphism dg-algebra of $U$. We know that if $U$ is a good silting object, then there exist a dg-algebra $C$ and a recollement among the derived categories $𝐃(C,d)$ of $C$, $𝐃(B,d)$ of $B$ and $𝐃(A,d)$ of $A$. We investigate the condition under which the induced dg-algebra $C$ is weak nonpositive. In order to deal with both silting and cosilting dg-modules consistently, the notion of weak silting dg-modules is introduced. Thus, similar results for good cosilting dg-modules are obtained. Finally, some...

Let $𝒳$ be a semibrick in an extriangulated category. If $𝒳$ is a $\tau$-semibrick, then the Auslander-Reiten quiver $\Gamma \left(ℱ\right(𝒳\left)\right)$ of the filtration subcategory $ℱ\left(𝒳\right)$ generated by $𝒳$ is $\mathbb{Z}{𝔸}_{\infty }$. If $𝒳={\left\{X_i\right\}}_{i=1}^t$ is a $\tau$-cycle semibrick, then $\Gamma \left(ℱ\right(𝒳\left)\right)$ is $\mathbb{Z}{𝔸}_{\infty }/{\tau }_{𝔸}^t$.