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

By Kohji Yanagawa. 2000

149 citations recorded by the source corpus.

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

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/98826e28dd07e2eb4917cc3efa535266930f0a10

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Citations in the corpus

Gröbner geometry of Schubert polynomials (2001, methodology): “which come from the Hilbert numerator of J rather than J⋆. However, these two are intimately related, as the next result demonstrates. It holds more generally for the “squarefree modules” of Yanagawa [Yan00], as shown in [Mil00b, Theorem 4.36]. A Z-graded version of Proposition 4.7.2 was proved by T”
Bass numbers of local cohomology modules with supports in monomial ideals (2001, methodology): “Note that we can describe the module structure of ExtiS(S/I∆, ωS) and H i I∆ (ωS) using the chain complexes of certain subcomplexes of ∆ (see [13, 18, 21]).”