Rigid, maximal rigid and cluster-tilting objects

Question

Let $\mathcal{D}$ be a $k$-linear, Hom-finite triangulated category with a Serre functor $\mathbb{S}$. An important class of objects in $\mathcal{D}$ are the cluster-tilting objects, which have many nice properties.

Definition 2.10. (1) An object $T$ in $\mathcal{D}$ is called rigid if $\operatorname{Ext}_{\mathcal{D}}^{1}(T, T)=0$.

(2) An object $T$ in $\mathcal{D}$ is called maximal rigid if it is rigid and maximal with respect to the property: $$ \operatorname{add} T=\left\{X \in \mathcal{D} \mid \operatorname{Ext}_{\mathcal{D}}^{1}(T \oplus X, T \oplus X)=0\right\}. $$

(3) An object $T$ in $\mathcal{D}$ is called cluster-tilting if $$ \operatorname{add} T=\left\{X \in \mathcal{D} \mid \operatorname{Ext}_{\mathcal{D}}^{1}(T, X)=0\right\}=\left\{X \in \mathcal{D} \mid \operatorname{Ext}_{\mathcal{D}}^{1}(X, T)=0\right\}. $$

Could you please tell me the reason for these three names and motivation for these definitions? Thank you!

Answer 1

For rigid and maximal rigid, you can think instead in the category of quiver representations.

The term "rigid" comes from the fact that the vanishing of Ext^1 can be understood as the absence of any true deformation of the module. It is a general fact in the study of quiver representations that the tangent space of the orbit of one representation, considered inside the space of all representations with the same dimension vector, has for codimension the dim of Ext^1.

Then "maximal rigid" means that there is no way to add another indecomposable summand. In other words, every indecomposable summand that has no Ext^1 with itself and with the module already appears as a summand in the module.