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

For a finite-dimensional algebra Λ over an algebraically closed field K, it is known that the poset of $2$-term silting objects in Kᵇ(projΛ) is isomorphic to the poset of functorially finite torsion classes in modΛ, and to that of complete cotorsion classes in K[-1,0](projΛ). In this work, we generalise this result to the case of d-term silting objects for arbitrary d≥ 2 by introducing the notion of torsion classes for extriangulated categories. In particular, we show that the poset of d-term silting objects in Kᵇ(projΛ) is isomorphic to the poset of complete and hereditary cotorsion classes in K[-d+1,0](projΛ), and to that of positive and functorially finite torsion classes in D[-d+2,0](modΛ), an extension-closed subcategory of Dᵇ(modΛ). We further show that the posets cotorsK[-d+1,0](projΛ) and tors D[-d+2,0](modΛ) are lattices, and that the truncation functor τ≥ -d+2 gives an isomorphism between the two.