In this chapter we discuss free resolutions of monomial ideals; we call them monomial resolutions. The problem to describe the minimal free resolution of a monomial ideal (over a polynomial ring) was posed by Kaplansky in the early 1960’s. Despite the helpful combinatorial structure of monomial ideals, the problem turned out to be hard. The structure of a minimal free monomial resolution can be quite complex. There exists a minimal free monomial resolution which cannot be encoded in the structure of any CW-complex. In fact, even the minimal free resolutions of ideals generated by quadratic monomials are so complicated that it is beyond reach to obtain a description of them; we do not even know how to express the regularity of such ideals. In this situation, the guideline is to introduce new ideas and constructions which either have strong applications or/and are beautiful. Most proofs about monomial resolutions are easy. The key point is not to provide complicated proofs, but to introduce new beautiful ideas.
KeywordsSimplicial Complex Betti Number Monomial Ideal Free Resolution Order Complex
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