Calculating the Minimal Free Resolution of the Stanley-Reisner Ideal of a Self-dual Simplicial Complex



Dang, Phuong T

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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.



Free resolution, Simplicial complex, Stanley-Reisner ideal, Gröbner basis