### Abstract:

Combinatorial Commutative Algebra is a popular subject for investigation and for application.
It has various types of benefits widely acknowledged in some other branches of
Mathematics and other Sciences. One of those is calculating the minimal free resolution
of a Stanley-Reisner ring, the main purpose for this project. To be convenient for readers,
this thesis is divided into two parts, the theory part and the example part. In the theory,
we concentrate building the process in two ways, the Gröb nerbasis way and the Hochster's
theorem way. However, to understand both of them, we need to cover the basic knowledge
about abstract simplicial complex, minimal non-faces, Stanley-Reisner ring and etc. After
defining these, the two ways can be accessed. For the Gröbner basis way, we assume
that the readers know the concepts of an ideal over a polynomial ring, especially in this
case the Noetherian domain (because, the polynomial ring is over a field), and have some
intuition for divisibility from Number theory. Then we have a foundation to comprehend
the definitions of syzygy, Gröbner basis, S-polynomial etc. From those, the algorithm is
displayed effectively. Schreyer's theorem and the Macaulay matrix help a lot for doing the
computation.
However, the second way is more direct. Observing instead directly the bases of the
modules, we can go through the computation of Betti numbers by using the reduced homology.
This is acomplished through several concepts of upper Koszul complex and link to
determine the Betti number of the grade that we are looking at. Then, gradually, both are
enough to detect the resolution. Gaining these theories is sufficient for the second part. In
this, we process some problems for self-dual simplicial complexes to identify their minimal
free resolutions, using the two ways above. With the solution I for the Gröbner basis way
and solution II for Hochster's theorem way, we can distinguish and analyze the interaction
between them. This is a good thing to better understand and obtain a good orientation
for the new application in computer science and maybe in the real life. In conclusion, this
topic is very useful and realistic for several new approaches. Hence, this thesis is written to
clarify the computation and serve as material for advanced investigations. I hope through
this work, we can obtain enough ingredients to continue on the research and to develop
the skills for interesting areas like this, especially Combinatorial Commutative Algebra and
Computational Algebra.