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

Let $P$ be a finite connected poset and let ${\mathrm{\Lambda }}_P$ 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 ${\mathrm{\Lambda }}_P$ carries a Reedy algebra structure in the sense of Dalezios–Šťovíček. 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 ${\mathrm{\Lambda }}_P$, which in particular shows that it is independent of the coefficient field. As an application, for the $m$ by $\mathrm{\ell }$ grid $G_{m,\mathrm{\ell }}$ with $m\ge \mathrm{\ell }\ge 2$, we give the explicit formula $\mathrm{gldim}{\mathrm{\Lambda }}_{G_{m,\mathrm{\ell }}}=\min \{2\mathrm{\ell },m+\mathrm{\ell }-2\}$. 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.