Each has left a motive for crime (5)
I believe the answer is:
arson
'crime' is the definition.
(crime of illegally starting a fire)
'each has left a motive' is the wordplay.
'each' becomes 'ea' (abbreviation).
'has left' is a deletion indicator.
'motive' becomes 'reason' (reason is a kind of motive).
'a'+'reason'='areason'
'areason' with 'ea' taken away is 'ARSON'.
'for' is the link.
(Other definitions for arson that I've seen before include "The act of firing" , "Incendiary crime" , "major crime" , "Pyromaniac crime" , "Crime of fire setting" .)