$\tau$-tilting theory via the morphism category of projective modules I: ICE-closed subcategories

This paper endeavors to explore certain distinguished modules and subcategories within mod$\mathrm{\Lambda }$. Let $\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}-\mathrm{\Lambda }$ denote the category of all finitely generated projective $\mathrm{\Lambda }$-modules and define $\mathcal{P}(\mathrm{\Lambda }):=\mathrm{M}\mathrm{o}\mathrm{r}(\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}-\mathrm{\Lambda })$. Due to the favorable homological properties of $\mathcal{P}(\mathrm{\Lambda })$, we initially examine several noteworthy objects and subcategories of $\mathcal{P}(\mathrm{\Lambda })$, subsequently relating these findings to $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }$. We demonstrate the existence of a bijection between tilting objects of $\mathcal{P}(\mathrm{\Lambda })$ and support $\tau$-tilting $\mathrm{\Lambda }$-modules. This bijection further suggests a correspondence between tilting objects of $\mathcal{P}(\mathrm{\Lambda })$ that possess a specific direct summand and $\tau$-tilting $\mathrm{\Lambda }$-modules. We establish a bijection between two-term silting complexes within ${\mathbb{K}}^b(\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}-\mathrm{\Lambda })$ and tilting objects of $\mathcal{P}(\mathrm{\Lambda })$. Following our examination of Image-Cokernel-Extension closed (hereafter referred to as ICE-closed) subcategories of $\mathcal{P}(\mathrm{\Lambda })$, we demonstrate a bijection between rigid objects in $\mathcal{P}(\mathrm{\Lambda })$ and ICE-closed subcategories of $\mathcal{P}(\mathrm{\Lambda })$ with enough Ext-projectives. Subsequently, we present bijections linking rigid objects in $\mathcal{P}(\mathrm{\Lambda })$ with a designated direct summand, $\tau$-rigid pairs within $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }$, and ICE-closed subcategories of $\mathcal{P}(\mathrm{\Lambda })$ that contain a special object and also have enough Ext-projectives. In order to translate the concept of ICE-closed subcategory from $\mathcal{P}(\mathrm{\Lambda })$ to $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }$, it is necessary to introduce the framework of ICE-closed subcategories of $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }$ relative to a projective module.