On $d$-term silting objects, torsion classes, and cotorsion classes

Abstract

For a finite-dimensional algebra $\hat{I}›$ over an algebraically closed field $K$, it is known that the poset of $2$-term silting objects in ${\mathrm{K}}^b(\mathrm{proj}\hat{I}›)$ is isomorphic to the poset of functorially finite torsion classes in $\mathrm{mod}\hat{I}›$, and to that of complete cotorsion classes in ${\mathrm{K}}^{[-1,0]}(\mathrm{proj}\hat{I}›)$. In this work, we generalise this result to the case of $d$-term silting objects for arbitrary $d\ge 2$ by introducing the notion of torsion classes for extriangulated categories. In particular, we show that the poset of $d$-term silting objects in ${\mathrm{K}}^b(\mathrm{proj}\hat{I}›)$ is isomorphic to the poset of complete and hereditary cotorsion classes in ${\mathrm{K}}^{[-d+1,0]}(\mathrm{proj}\hat{I}›)$, and to that of positive and functorially finite torsion classes in $D^{[-d+2,0]}(\mathrm{mod}\hat{I}›)$, an extension-closed subcategory of $D^b(\mathrm{mod}\hat{I}›)$. We further show that the posets $\mathrm{cotors}{\mathrm{K}}^{[-d+1,0]}(\mathrm{proj}\hat{I}›)$ and $\mathrm{tors}{D}^{[-d+2,0]}(\mathrm{mod}\hat{I}›)$ are lattices, and that the truncation functor $\ddot{I}{„}_{\ge -d+2}$ gives an isomorphism between the two.

30 pages, added Section 6 to prove connection with arXiv:2602.03659