论文标题
Posets模块的同源代数
Homological algebra of modules over posets
论文作者
论文摘要
在有限的演示和分辨率的方向上，开发了POSET上的模块的同源代数，与有限生成的模块在Noetherian的交换环上的有限生成的模块。在争议的中心是如何定义有限性来替换失败的赛车假设。为此目的引入的可驯服条件捕获了矢量空间家族的变化的有限性，该媒介空间由Posets索引，其特征在于不同的拓扑，代数，组合和同源表现。驯服性既有理论和计算目的：保证各种的有限演示和决议，为Syzygy Therorem都与算法操作相关。即使在$ \ mathbb {n}^n $ gradings的有限生成的离散设置中，驯服性及其同源理论也是新的，其中驯服比Noetherian更弱。在过滤的拓扑空间的持续同源性的背景下，尤其是在多个实际参数的背景下，驯服的代数理论在同源性类别的出生和死亡方面产生了拓扑解释的数据结构。
Homological algebra of modules over posets is developed, as closely parallel as possible to that of finitely generated modules over noetherian commutative rings, in the direction of finite presentations and resolutions. Centrally at issue is how to define finiteness to replace the noetherian hypothesis which fails. The tameness condition introduced for this purpose captures finiteness for variation in families of vector spaces indexed by posets in a way that is characterized equivalently by distinct topological, algebraic, combinatorial, and homological manifestations. Tameness serves both theoretical and computational purposes: it guarantees finite presentations and resolutions of various sorts, all related by a syzygy theorem, amenable to algorithmic manipulation. Tameness and its homological theory are new even in the finitely generated discrete setting of $\mathbb{N}^n$-gradings, where tame is materially weaker than noetherian. In the context of persistent homology of filtered topological spaces, especially with multiple real parameters, the algebraic theory of tameness yields topologically interpretable data structures in terms of birth and death of homology classes.
