An arrangement of finitely many tropical hyperplanes in the tropical torus leads to a notion of `type' data for points, with the underlying unlabeled arrangement giving rise to `coarse type'. It is shown that the decomposition of the tropical torus induced by types gives rise to minimal cocellular resolutions of certain associated monomial ideals. Via the Cayley trick from geometric combinatorics this also yields cellular resolutions supported on mixed subdivisions of dilated simplices, extending previously known constructions. Moreover, the methods developed lead to an algebraic algorithm for computing the facial structure of arbitrary tropical complexes from point data.
Cellular resolutions of second powers of square-free monomial ideals with divisibility relations
University of Manitoba（曼尼托巴大学）；King Abdullah University Of Science and Technology（阿卜杜拉国王科技大学）；Dalhousie University（达尔豪斯大学）；Texas State University（德克萨斯州立大学）；University of Missouri-Kansas City（密苏里大学堪萨斯城分校）；University of Mississippi（密西西比大学）
Minimal free resolutions of monomial ideals and of toric rings are supported on posets
我们引入了\emph{resolution supported on a poset}的概念。当偏序集是CW-偏序集时，即正则CW-复形的面偏序集，我们恢复了Bayer和Sturmfels引入的细胞分辨的概念。Reiner和Welker以及Velasco的工作表明，存在单项式理想，其极小自由分解不是胞腔的，因此不能在任何CW-偏序集上得到支持。我们表明，对于任何单项理想有一个\emph{homology CW-poset}，支持最小的自由分解的理想。这使得人们可以扩展到每一个最小的分辨率，基本上逐字，技术最初开发的研究细胞的决议。作为这一过程的两个演示，我们证明了环面环的最小决议支持我们所谓的环面hcw-偏序集，我们给出了一个新的组合证明米勒的一个基本结果之间的关系Artininizations和亚历山大对偶的单项理想。
