Nothing fits with information detectives last gave on killing (8)
I believe the answer is:
genocide
'killing' is the definition.
(genocide is a kind of killing)
'nothing fits with information detectives last gave' is the wordplay.
'nothing' becomes 'o' (looks like zero - 0).
'fits with' means one lot of letters go next to another.
'information' becomes 'gen' (gen can informally mean information).
'detectives' becomes 'cid'.
'last' indicates one should take the final letters.
The final letter of 'gave' is 'e'.
'o' after 'gen' is 'geno'.
'geno'+'cid'+'e'='GENOCIDE'
'on' is the link.
(Other definitions for genocide that I've seen before include "Racial extermination" , "Large-scale killing" , "Racial murder" , "Extermination of ethnic group or nationality" , "Deliberate extermination of a race of people" .)