The applications of derived categories and topological methods to combinatorial commutative algebra

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Developing a result of Nagel and Reiner, the author and R. Okazaki studied the non-standard polarization of a Borel fixed ideal, and gave its minimal free resolution. This resolution is supported by a regular CW complex, which can be “interpreted” by the discrete Morse theory in the framework constructed by Welker et al.The author also studied the dualizing complexes of seminormal affine semigroup rings and toric face rings. In this work, the preceding results of Bruns et al and Nguyes are re-formulated and improved using derived categories.