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  "body": "\n## Summary\n\nAn exceptional collection in a linear [[triangulated category|T2V7GJJ2EN]] is a finite ordered list of exceptional objects.\nEach of them has only scalar endomorphisms and no self-extensions.\nThere are no morphisms, in any degree, from later objects to earlier ones.\nA full exceptional collection splits the category into a semiorthogonal decomposition whose pieces are copies of the derived category of vector spaces.\nThe basic example is Beilinson's collection $\\mathcal{O}, \\mathcal{O}(1), \\dots, \\mathcal{O}(n)$ on projective space.\n\n## Explanations\n\n### L1 human\n\nAn exceptional object behaves like a single point: its only endomorphisms are scalars and it has no deformations or extensions with itself.\nAn exceptional collection is a list of such objects in which maps only go forward along the list.\nWhen the collection generates everything, the whole category is built from these simple pieces in a triangular way, much like a matrix that is upper triangular.\nFor projective space, the line bundles $\\mathcal{O}, \\mathcal{O}(1), \\dots, \\mathcal{O}(n)$ form such a list, which makes its derived category very concrete.\n",
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