Alexander Duality for Stanley–Reisner Rings and Squarefree [formula omitted]n-Graded Modules

Abstract

Let S=k[x1,…,xn] be a polynomial ring, and let ωS be its canonical module. First, we will define squarefreeness for Nn-graded S-modules. A Stanley–Reisner ring k[Δ]=S/IΔ, its syzygy module Syzi(k[Δ]), and ExtiS(k[Δ],ωS) 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 IΔ⊂S has the “same information” as the module structure of ExtiS(k[Δ∨],ωS), where Δ∨ is the Alexander dual of Δ. In particular, if k[Δ] has a linear resolution, we can describe its minimal free resolution using the module structure of the canonical module of k[Δ∨], 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[Δ] is sequentially Cohen–Macaulay if and only if IΔ∨ is componentwise linear.