Interval endomorphism algebras of posets: Reedy structure, combinatorics, and homological theory

Abstract

Let $P$ be a finite connected poset and let $\Lambda_P$ 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 $\Lambda_P$ carries a Reedy algebra structure in the sense of Dalezios--\v{S}\v{t}ov\'{\i}\v{c}ek. 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 $\Lambda_P$, 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 $\operatorname{gldim}\Lambda_{G_{m,\ell}}=\min\{2\ell,m+\ell-2\}$. 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.