Alexander Duality for Stanley–Reisner Rings and Squarefree Nn-Graded Modules
Alexander Duality for Stanley–Reisner Rings and Squarefree Nn-Graded Modules
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DOI：
10.1006/jabr.1999.8130
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发表时间：
2000-03
影响因子：
0.9
通讯作者：
Kohji Yanagawa
中科院分区：
文献类型：
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作者：
Kohji Yanagawa
摘要 设 S = k[x1,…,xn] 为多项式环，并设 ωS 为其规范模。首先，我们将为 N 个 n 级 S 模块定义无平方性。 Stanley–Reisner 环 k[Δ] = S/IΔ，其合子模 Syzi(k[Δ]) 和 ExtiS(k[Δ], ωS) 始终是无平方的。这个概念将简化斯坦利-赖斯纳环理论中的一些标准论证。接下来，我们将证明 Stanley–Reisner 理想 IΔ ⊂ S 的最小自由分辨率的 i 线性链与 ExtiS(k[Δ∨ ], ωS) 的模结构具有“相同的信息”，其中 Δ∨ 是 Δ 的亚历山大对偶。特别是，如果 k[Δ] 具有线性分辨率，我们可以使用 k[Δ∨ ] 的规范模的模结构来描述其最小自由分辨率，在本例中为 Cohen-Macaulay。我们还可以对 Herzog 及其同事的结果给出新的解释，该结果指出，当且仅当 IΔ∨ 是分量线性时，k[Δ] 是顺序 Cohen-Macaulay。
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.