Stratifications of derived categories from tilting modules over tame hereditary algebras

In this paper, we consider the endomorphism algebras of infinitely generated tilting modules of the form $R_{\mathcal{U}}\oplus R_{\mathcal{U}}\mathrm{/}R$ over tame hereditary $k$-algebras $R$ with $k$ an arbitrary field, where $R_{\mathcal{U}}$ is the universal localization of $R$ at an arbitrary set $\mathcal{U}$ of simple regular $R$-modules, and show that the derived module category of ${\text{End}}_R(R_{\mathcal{U}}\oplus R_{\mathcal{U}}\mathrm{/}R)$ is a recollement of the derived module category $\text{D}R$ of $R$ and the derived module category $\text{D}{\mathbb{A}}_{\mathcal{U}}$ of the ad`ele ring ${\mathbb{A}}_{\mathcal{U}}$ associated with $\mathcal{U}$. When $k$ is an algebraically closed field, the ring ${\mathbb{A}}_{\mathcal{U}}$ can be precisely described in terms of Laurent power series ring $k((x))$ over $k$. Moreover, if $\mathcal{U}$ is a union of finitely many cliques, we give two different stratifications of the derived category of ${\text{End}}_R(R_{\mathcal{U}}\oplus R_{\mathcal{U}}\mathrm{/}R)$ by derived categories of rings, such that the two stratifications are of different finite lengths.