Perish, sadly, after personal possession (9)
I believe the answer is:
ownership
'possession' is the definition.
(I know that possession can be written as ownership)
'perish sadly after personal' is the wordplay.
'sadly' indicates anagramming the letters.
'after' says to put letters next to each other.
'personal' becomes 'own' (eg one's own possessions are personal possessions).
'perish' is an anagram of 'ership'.
'ership' put after 'own' is 'OWNERSHIP'.
(Other definitions for ownership that I've seen before include "Legal title" , "Proprietorship, possession" , "The right of possessing something" , "Proprietary right of possession" .)