Necessary to have separate index for lookups against one of a multi-column PK? Sorry if this is a FAQ, but I'm having problems turning up exactly what I'm asking:

I have a table (mysql) with a primary key which is composed of multiple columns only because this is necessary to make the PK unique (won't be using an autoincrement column).

However, most lookups will be against just the first column in the primary key.

So should I have a separate index only containing that single column, or is the multi-column PK sufficient (at least as long as the query column is listed first in the index definition)?
In short, it is sufficient. As far as the searched field is the first in a composite index (PK, unique, or regular index), no need for other separate index.  

However, you mentioned that you 


  (won't be using an autoincrement column)


Is there a reason for that? Composite PK might have performance hit, specially if you are using InnoDB engine, and if you have many updates on the table (insert/update/delete)