Relative delooping levels on intervals of torsion classes

Abstract

Let $U$ be a support $τ$-tilting module over a finite-dimensional algebra $A$, and let $\mathcal{V}\subseteq\operatorname{Fac}U$ be any torsion class. The exact heart $\mathcal{H}=\operatorname{Fac}U\cap\mathcal{V}^{\perp}$ has an Ext-progenerator $P$. We prove that $\operatorname{Hom}_A(P,-)$ realizes $\mathcal{H}$ as a submodule-closed exact subcategory of $\operatorname{mod}B$, where $B=\text{End}_A(P)$, and preserves minimal syzygies. The global relative $k$-delooping levels are independent of $k\geq1$. Writing $g=\operatorname{glo}\text{-}\operatorname{dell}\mathcal{H}$, we obtain the sharp comparison $g\leq\operatorname{glo}\text{-}\operatorname{dell}B\leq g+1$ and $\operatorname{findim}B^{\mathrm{op}}\leq g+1$. Thus the two global levels are finite simultaneously. For wide hearts the relative levels equal their classical counterparts. We also characterize syzygy finiteness of $B$ by a finite relative syzygy stratum and obtain explicit bounds for both finitistic dimensions. If the heart is Frobenius, it is exactly the category of finitely generated Gorenstein-projective $B$-modules under this realization, and both self-injective dimensions of $B$ are at most one. We also give examples to show the main results.