The counterexamples are geometric, we fix the triangulated category \(\mathbf{D}^\text{b}\text{Coh}(X)\) associated to a smooth variety \(X\). Why might a generator in \(\mathbf{D}^\text{b} \text{Coh} (X)\) not be a classical generator? In other words, what can allow a classical non-generator to have trivial orthogonal-complement?

Support is one good reason; the set of all skyscrapers on \(X\) spans but certainly doesn’t classically generate the derived category. Concretely we may take a regular local Noetherian ring \(R\) with the maximal ideal \(\mathfrak{m}\), and set \(X=\text{Spec}(R)\): the simple module \(R/\mathfrak{m}\) is a generator of \(\mathscr{T}\) by Nakayama’s lemma, but the subcategory \(\langle R/\mathfrak{m}\rangle\subset\mathscr{T}\) consists only of objects supported at \(\mathfrak{m} \in\text{Spec}(R)\) and does not include, for example, \(R\) itself.

The Reimann–Roch theorem is another good reason. Let \(X\) be a smooth projective curve of genus \(g\geq 2\), with two rationally inequivalent points \(p,q\in X\), and consider the line bundle \(L=\mathscr{O}_X(2p-q)\) which has degree \(1\) and satisfies \(\text{Hom}(\mathscr{O}_X,L)=\text{Hom}(L,\mathscr{O}_X)=0\). Now \(V=\mathscr{O}_X\oplus L\) certainly doesn’t classically generate \(\mathbf{D}^\text{b}\text{Coh} (X)\) (since otherwise the heart \(\text{Coh} X\), which would contain a generating semibrick, would be Artinian). But \(V\) does span the category: otherwise there is a complex \(W\neq 0\) such that \(\text{Hom}^\bullet(V,W)=0\). Since \(W\) splits into its cohomologies, we may assume \(W\) is a coherent sheaf. Further \(W\) splits into its torsion and torsion-free parts, and \(V\) clearly has a morphism to every torsion sheaf so we may assume \(W\) is locally free. In other words, we have bundles \(W\) and \(W'=W(q-2p)\) satisfying \(\text{h}^0(W)=\text{h}^1(W)=0\), \(\text{h}^0(W')=\text{h}^1(W')=0\).

But by the Hirzebruch–Riemann–Roch theorem, we see that \(0=\chi(W)=(1-g)\text{rk}W- \text{deg} W\) and \(0=\chi(W')=(1-g)\text{rk}W'- \text{deg} W'\), which cannot both simultaneously hold since \(\text{rk}W'=\text{rk}W\) but \(\text{deg} W'= \text{deg} W-1\).