Localized Persistent Commutative Algebra
We develop a localized persistent theory of commutative algebra for Stanley-Reisner rings, based on local cohomology supported at a coordinate prime rather than at the maximal ideal. The construction is modeled on the persistent Stanley-Reisner theory of Suwayyid and Wei (arXiv:2503.23482) and its functorial development for graphs and hypergraphs (arXiv:2512.17619), in which invariants of the face ring such as graded Betti numbers and f- and h-vectors are persisted across a filtration. That framework is built from the minimal free resolution and is thus Tor-theoretic; we work instead on the injective side, and the resulting modules record information localized at a single vertex, complementing the global picture given by maximal-support local cohomology. For a vertex prime $p_i = (x_j : j \neq i)$ we prove an exact $\mathbb{Z}^n$-graded decomposition of $H^q_{p_i}(k[\Delta])$ into the maximal-support local cohomology of the deletion and of the link of the vertex $i$, the first in $x_i$-degree zero and the second repeated in every positive $x_i$-degree; at the level of graded dimensions this recovers the vertex-prime case of Rahimi's bigraded formula. With Hochster's formula this yields a closed combinatorial description of every multigraded piece. Building on this structure we introduce per-vertex persistent local cohomology numbers, prove a persistent links-Hochster formula, obtain interval decompositions of the resulting reversed-arrow persistence modules and a bottleneck stability theorem, retain multiplication by the uninverted variable as a morphism of persistence modules that the two barcodes alone do not determine, and extend the theory to an arbitrary coordinate prime, where the multiplication maps of the uninverted variables assemble into a commuting Boolean diagram of persistence modules.