Important woman’s crime (6)
I believe the answer is:
bigamy
'crime' is the definition.
(bigamy is a kind of crime)
'important woman's' is the wordplay.
'important' becomes 'big' ('big' can be similar in meaning to 'important').
'woman' becomes 'amy'.
'big'+'amy'='BIGAMY'
(Other definitions for bigamy that I've seen before include "Crime involving marriage(s)" , "Having more than one spouse, illegally" , "Illegal marriage to more than one person" , "Having two concurrent spouses" , "overmatiness" .)