发表机构
Kobe University（神户大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限连通偏序集的区间端态射代数，证明其Reedy代数结构，给出标准模的显式投射分解，将Ext群计算化为单纯复形上同调，并推导全局维数公式，解决网格猜想。
AI 中文摘要
设$P$为有限连通偏序集，$\u039b_P$为$P$的所有区间表示在域上的直和的对偶端态射代数。通过投影化，该代数控制相对于区间可分解表示的分解，这在持续同调理论中自然出现。我们首先证明$\u039b_P$具有Dalezios–Šťovíček意义下的Reedy代数结构。其Reedy度由索引区间的基数给出，诱导的拟遗传序由区间基数的逆序给出。对于所得的拟遗传结构，我们给出标准模的具体组合描述，并构造这些模的显式投射分解。利用这些分解，我们将标准–单Ext群的计算归结为由区间组合决定的单纯复形的约化上同调。序反转给出相应的单–余标准公式。基于这些计算，我们确定所有单–单Ext群。当对应区间对饱和时，这些群在唯一度上为一维，否则为零。作为推论，我们得到$\u039b_P$的全局维数的精确组合公式，特别表明其与系数域无关。作为应用，对于$m$乘$\u2113$的网格$G_{m,\u2113}$（$m\geq\u2113\geq2$），我们给出显式公式$\u0067\u006c\u0064\u0069\u006d\u039b_{G_{m,\u2113}}=\min\{2\u2113,m+\u2113-2\}$。这也给出了这些网格的区间分解全局维数的显式公式，解决了Asashiba–Escolar–Nakashima–Yoshiwaki的相应网格猜想，并确定了稳定值和精确稳定阈值。
英文摘要
Let $P$ be a finite connected poset and let $Λ_P$ be the opposite endomorphism algebra of the direct sum of all interval representations of $P$ over a field. Via projectivization, this algebra governs resolutions relative to interval-decomposable representations, which arise naturally in persistence theory. We first show that $Λ_P$ carries a Reedy algebra structure in the sense of Dalezios--Šťov\'ıček. Its Reedy degree is given by the cardinality of the indexing interval, and the induced quasi-hereditary order is given by reverse interval cardinality. With respect to the resulting quasi-hereditary structure, we give a concrete combinatorial description of the standard modules and construct explicit projective resolutions of these modules. Using these resolutions, we reduce the calculation of standard--simple Ext groups to the reduced cohomology of simplicial complexes determined by the interval combinatorics. Order reversal gives the corresponding simple--costandard formula. Building on these calculations, we determine all simple--simple Ext groups. These groups are one-dimensional in a unique degree when the corresponding pair of intervals is saturated, and vanish otherwise. As a consequence, we obtain an exact combinatorial formula for the global dimension of $Λ_P$, which in particular shows that it is independent of the coefficient field. As an application, for the $m$ by $\ell$ grid $G_{m,\ell}$ with $m\geq\ell\geq2$, we give the explicit formula $\operatorname{gldim}Λ_{G_{m,\ell}}=\min\{2\ell,m+\ell-2\}$. This also gives an explicit formula for the interval-resolution global dimension of these grids, settling the corresponding grid conjectures of Asashiba--Escolar--Nakashima--Yoshiwaki and determining the stable value and the precise stabilization threshold.
Comments36 pages, 1 figure
