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Governor Limits, Summaries, and Triggers

I am wondering if DML statements and SOQL statements in a trigger in the following sequence count against governor limits:


Master object A has detail object B.  Object A has summary fields summarizing fields from object b.  There is an after update trigger on object A.


As I understand it, the updating of the summary fields on A following and update to the B object will cause the A object's after update trigger to fire.  Will the DML associated with that firing (and any SOQL embedded in the trigger) count against my governor limits?


If object B updates a large number of rows at the same time, will the trigger for object A be somehow bulkified as well?  If not, and the triggers DML and SOQL statements could against the limits, is there any technique I might use to get around this?


Thanks for your help,