Alexander duality for Stanley-Reisner rings and squarefree Nn-graded modules

Let S = k[x(1),..., x(n)] be a polynomial ring, and let omega(s) be its canonical module. First, we will define squarefreeness for N-n-graded S-modules. A Stanley-Reisner ring k[Delta] = S/I-Delta, its syzygy module Syz(i)(k[Delta]), and Ex(s)(i)(k[Delta], omega(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-Delta subset of S has the "same information" as the module structure of Ext(s)(i)(k[Delta(v)], omega(s)), where Delta(v) is the Alexander dual of Delta. In particular, if k[Delta] has a linear resolution, we can describe its minimal free resolution using the module structure of the canonical module of k[Delta(v)], 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[Delta] is sequentially Cohen-Macaulay if and only if I(Delta)v is componentwise linear. (C) 2000 Academic Press.