Semiorthogonal nilpotent-local decomposition

Remark 8.4.18. The previous two results show that the pair \((\LMod _R^{S\dnil }, \LMod _R^{\Loc (S)})\) satisfies the axioms of a t-structure on \(\LMod _R\) in the sense of Definition 6.3.1: closure under shifts is Observation 8.4.10, orthogonality is part (3) of Proposition 8.4.16, and the decomposition axiom is Proposition 8.4.17.

This t-structure is, however, of a completely different nature from the ones in Chapter 6: both halves are stable subcategories, closed under all shifts, so that \(\LMod _R^{S\dnil }[n] = \LMod _R^{S\dnil }\) for every \(n\). In particular the heart is zero, and there is no connectivity intuition to be had here. What the pair really encodes is a semiorthogonal decomposition of \(\LMod _R\): every module is functorially glued from an \(S\)-nilpotent part and an \(S\)-local part, and there are no maps from the nilpotent part to the local part.