Alexander Duality for Stanley–Reisner Rings and Squarefree Nn-Graded Modules

✍ Scribed by Kohji Yanagawa

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 146 KB
• Volume: 225
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.8130

✦ Synopsis

Let S = k x 1

x n be a polynomial ring, and let ω S be its canonical module. First, we will define squarefreeness for n -graded S-modules. A Stanley-Reisner ring k = S/I , its syzygy module Syz i k , and Ext i S k ω S are always squarefree. This notion will simplify some standard arguments in the Stanley-Reisner ring theory. Next, we will prove that the i-linear strand of the minimal free resolution of a Stanley-Reisner ideal I ⊂ S has the "same information" as the module structure of Ext i S k ∨ ω S , where ∨ is the Alexander dual of . In particular, if k has a linear resolution, we can describe its minimal free resolution using the module structure of the canonical module of k ∨ , which is Cohen-Macaulay in this case. We can also give a new interpretation of a result of Herzog and co-workers, which states that k is sequentially Cohen-Macaulay if and only if I ∨ is componentwise linear.