# On the definition and an example of silting/tilting subcategories in a triangulated categories according to a paper by Aihara and Iyama

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!

$$\operatorname{Hom}_{\mathcal{T}}(\mathcal{M}, \mathcal{M}[>0]) = 0$$ 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.
• @Jeremy_Rickard Thank you so much for your answer! But the condition $\operatorname{Hom}_{\mathcal{T}}\left(X, \Sigma^i(Y)\right) = 0$ isnt supposed to be for all objects $X,Y \in \mathcal{M}$ instead of all objects $X,Y$ of $\mathcal{T}$? Seems more intuitive.