DBMS recommandation for C# applications with provides triggers, procedures and if possible free
I think you have two that I would recommend:

The first is PostgreSQL, which runs on your local computer as a database server.  It has an impressive feature set and allows stored procedures to be written in a number of languages (including PL/PGSQL, PL/Perl, and PL/Python).  It is free and it is enterprise-grade.  PostgreSQL ideas have made their way into Informix and elsewhere.

The second is Firebird which is the open sourced version of Interbase.  It contains triggers, stored procedures, and more, and is free.  The feature set is not as large as it is with PostgreSQL in terms of programmability, but it has the additional feature of being able to be run as an embedded library in the process of your application.  PostgreSQL also has a larger community.