Add Foreign Key relationship between two Databases

If you need rock solid integrity, have both tables in one database, and use an FK constraint. If your parent table is in another database, nothing prevents anyone from restoring that parent database from an old backup, and then you have orphans.

This is why FK between databases is not supported.

You would need to manage the referential constraint across databases using a Trigger.

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Basically you create an insert, update trigger to verify the existence of the Key in the Primary key table. If the key does not exist then revert the insert or update and then handle the exception.

Example:

Create Trigger dbo.MyTableTrigger ON dbo.MyTable, After Insert, Update
As
Begin

   If NOT Exists(select PK from OtherDB.dbo.TableName where PK in (Select FK from inserted) BEGIN
      -- Handle the Referential Error Here
   END

END

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Edited: Just to clarify. This is not the best approach with enforcing referential integrity. Ideally you would want both tables in the same db but if that is not possible. Then the above is a potential work around for you.

In my experience, the best way to handle this when the primary authoritative source of information for two tables which are related has to be in two separate databases is to sync a copy of the table from the primary location to the secondary location (using T-SQL or SSIS with appropriate error checking - you cannot truncate and repopulate a table while it has a foreign key reference, so there are a few ways to skin the cat on the table updating).

Then add a traditional FK relationship in the second location to the table which is effectively a read-only copy.

You can use a trigger or scheduled job in the primary location to keep the copy updated.

You could use check constraint with a user defined function to make the check. It is more reliable than a trigger. It can be disabled and reenabled when necessary same as foreign keys and rechecked after a database2 restore.

CREATE FUNCTION dbo.fn_db2_schema2_tb_A
(@column1 INT) 
RETURNS BIT
AS
BEGIN
    DECLARE @exists bit = 0
    IF EXISTS (
      SELECT TOP 1 1 FROM DB2.SCHEMA2.tb_A 
      WHERE COLUMN_KEY_1 =  @COLUMN1
    ) BEGIN 
         SET @exists = 1 
      END;
      RETURN @exists
END
GO

ALTER TABLE db1.schema1.tb_S
  ADD CONSTRAINT CHK_S_key_col1_in_db2_schema2_tb_A
    CHECK(dbo.fn_db2_schema2_tb_A(key_col1) = 1)