Definition
A collection \(\{E_i\}\) of \(D^b(X)\) is
- full if \(\{E_i\}_i\) generate \(D^b(X)\).
- exceptional if \(\operatorname{Hom}_{D^b(X)}(E_i, E_i[n])\) is \(k\) when \(n = 0\) and is \(0\) otherwise. “It’s like Schur’s lemma” -David. Also we require \(\Hom_{D^b(X)}(E_i, E_j[n]) = 0\) for all \(n\) when \(i > j\) (slogan: Homs go up). Note that this requires an ordering on \(\{E_i\}_i\)
- ordered if the collection has an ordering
- strong if \(\Hom_{D(X)}(E_i, E_j[n]) = 0\) for all \(n \neq 0\) when \(i < j\). Notice we’ve swapped out Homs for \(D(X)\) rather than \(D^b(X)\).