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

Let S = k[x1,…,xn] be a polynomial ring, and let ωS be its canonical module. First, we will define squarefreeness for n-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.