The Alexander Duality Functors and Local Duality with Monomial Support
The Alexander Duality Functors and Local Duality with Monomial Support
复制标题
DOI：
10.1006/jabr.2000.8359
复制
发表时间：
2000-09
影响因子：
0.9
通讯作者：
Ezra Miller
中科院分区：
文献类型：
--
作者：
Ezra Miller
抽象的亚历山大对偶被制成一个函子，它的概念扩展为单项式理想的任何n阶生成的N n-分次模。与亚历山大对偶相关的函子提供了自由和内射分解水平上的对偶性，并且许多巴斯和贝蒂数关系作为推论。模M的极小内射分解等价于它的亚历山大对偶的内射分解，并且包含M在每个Z n-分次局部化上的极小自由分解中的所有映射.结果得到的对偶决议与细胞的决议和lcm格的相互作用。利用内射分解，将Eagon，Reiner和Terai的定理推广到所有N个n-分次模：M的投射维数等于其亚历山大对偶的支集正则性，M是Cohen-Macaulay当且仅当其亚历山大对偶有支集线性自由分解.亚历山大对偶被应用于多项式环S中无平方项理想I的Z n-分次局部上同调函子HiI（−）的上下文中，证明了一个对偶直接推广了局部对偶，这是当I = m是极大的情况.在这个过程中，引入了一个新的平坦复形来计算单项式理想上的局部上同调，结果表明寺井对HiI（S）的希尔伯特级数的公式等价于Hn − i m（S/I）的Hochster公式。
Abstract Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated N n-graded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as corollaries. A minimal injective resolution of a module M is equivalent to the injective resolution of its Alexander dual and contains all of the maps in the minimal free resolution of M over every Z n-graded localization. Results are obtained on the interaction of duality for resolutions with cellular resolutions and lcm-lattices. Using injective resolutions, theorems of Eagon, Reiner, and Terai are generalized to all N n-graded modules: the projective dimension of M equals the support-regularity of its Alexander dual, and M is Cohen–Macaulay if and only if its Alexander dual has a support-linear free resolution. Alexander duality is applied in the context of the Z n-graded local cohomology functors HiI(−) for squarefree monomial ideals I in the polynomial ring S, proving a duality directly generalizing local duality, which is the case when I = m is maximal. In the process, a new flat complex for calculating local cohomology at monomial ideals is introduced, showing, as a consequence, that Terai's formula for the Hilbert series of HiI(S) is equivalent to Hochster's for Hn − i m (S/I).