## Abstract

We define the normal Hochschild cohomology of an admissible subcategory of the derived category
of coherent sheaves on a smooth projective variety *X*, a graded vector space
which controls the restriction morphism from the Hochschild cohomology of *X*
to the Hochschild cohomology of the orthogonal complement of this admissible subcategory.
When the subcategory is generated by an exceptional collection, we define a new invariant (the height)
and show that the orthogonal to an exceptional collection of height *h* in the derived category of a smooth
projective variety *X* has the same Hochschild cohomology as *X* in degrees up to *h* - 2.
We use this to describe the second Hochschild cohomology of quasiphantom categories
in the derived categories of some surfaces of general type. We also give necessary
and sufficient conditions for the fullness of an exceptional collection in terms of its
height and of its normal Hochschild cohomology.

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