ABSTRACT
In this paper we study monomial ideals using the operation “polarization” to first turn them into square-free monomial ideals. Various forms of polarization appear throughout the literature and have been used for different purposes in algebra and algebraic combinatorics (for example, Weyman [17], Fro¨berg [8], Schwartau [13] or Rota and Stein [11]). One of the most useful features of polarization is that the chain of substitutions that turn a given monomial ideal into a square-free one can be described in terms of a regular sequence (Fro¨berg [8]). This fact allows many properties of a monomial ideal to transfer to its polarization. Conversely, to study a given monomial ideal, one could examine its polarization. The advantage of this latter approach is that there are many combinatorial tools dealing with square-free
monomial ideals. One of these tools is Stanley-Reisner theory: Schwartau’s thesis [13] and the book by Stu¨ckrad and Vogel [15] discuss how the Stanley-Reisner theory of square-free monomial ideals produces results about general monomial ideals using polarization. Another tool for studying square-free monomial ideals, which will be our focus here, is facet ideal theory, developed by the author in [5], [6] and [7].
