In the paper "*Silting mutation in triangulated categories*" by *Aihara and Iyama*, I stumbled upon this nice definition( **Definition 2.1**) of a tilting/silting subcategory of a triangulated category $\mathcal{T}$. Let $\mathcal{M}$ be a subcategory of a triangulated category $\mathcal{T}$:


(1) We say that $\mathcal{M}$ is a silting subcategory of $\mathcal{T}$ if $Hom\_{\mathcal{T}}(\mathcal{M}, \mathcal{M}[>0]) = 0$ and **thick**$(\mathcal{M}) = \mathcal{T}$.


(2) We say that $\mathcal{M}$ is a tilting subcategory of $\mathcal{T}$ if it is silting and $Hom\_{\mathcal{T}}(\mathcal{M}, \mathcal{M}[< 0]) = 0$.


It is clear to me as it is stated on the paper that **thick**$(\mathcal{M})$ is the smallest thick subcategory of $\mathcal{T}$ containing $\mathcal{M}$. Where a thick subcategory is triangulated subcategory of $\mathcal{T}$ closed under direct summands. What is not clear to me is the notion of $Hom\_{\mathcal{T}}(\mathcal{M}, \mathcal{M}[>0]) = 0$ (respectively $Hom\_{\mathcal{T}}(\mathcal{M}, \mathcal{M}[\neq 0]) = 0$).


If I'm right $Hom\_{\mathcal{T}}(\mathcal{M}, \mathcal{M}[> 0]) = 0$ means that for every $Hom\_{T}(X,Y) =0$ for every $X \in \mathcal{M}$ and $Y \in \mathcal{M}[>0]$ where $\mathcal{M}[>0]$ stands for $\Sigma^{i}(\mathcal{M})$ for $i \in \mathbb{Z}^{+}, i \neq 0$ and $\Sigma$ is the translation functor associated to the triangulated category $T$. Is my understanding of this definition right? Also as a first example of tilting subcategory the stalk of complexes of a ring $A$ in the triangulated category $K^{b}(A)$ is considered. Can someone recall me the proper definition of a stalk complex of a ring in a bounded homotopy category of complexes? Thanks for the support!



On the definition and an example of silting/tilting subcategories in a triangulated categories according to a paper by Aihara and Iyama
$\operatorname{Hom}\_{\mathcal{T}}(\mathcal{M}, \mathcal{M}[>0]) = 0$ means that $\operatorname{Hom}\_{\mathcal{T}}\left(X, \Sigma^i(Y)\right) = 0$ for all objects $X,Y$ of $\mathcal{M}$ and all integers $i>0$.


A stalk complex is a complex with only one nonzero term. So "$A$ considered as a stalk complex" means the complex with $A$ in degree zero and with $0$ in all other degrees.


