Suppose there is a tableA and trigger defined for Insert, Update and delete to populate audit table say tableB. Now If i want to give a user U1 write permissions on table tableA, will the trigger work on any kind of operation to populate audit table ? The case here is that user has access only to main table and not audit table.
If tableA and tableB have the same owner (or are in the same schema, e.g. dbo) then you do not need to grant any explicit permissions to U1 on tableB.
Read more about ownership chaining here
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