Initially Cohen-Macaulay Modules

In this paper, we introduce initially Cohen-Macaulay modules over a commutative Noetherian local ring $R$, a new class of $R$-modules that generalizes both Cohen-Macaulay and sequentially Cohen-Macaulay modules. A finitely generated $R$-module $N$ is initially Cohen-Macaulay if its depth is equal to its initial dimension, an invariant defined as the infimum of the coheights of the associated primes of $N$. We develop the theory of these modules, providing homological, combinatorial, and topological characterizations and confirming their compatibility with regular sequences, localization, and dimension filtrations. When this theory is applied to simplicial complexes, we establish analogues of Reisner's criterion, the Eagon-Reiner theorem, and Duval's characterization of sequentially Cohen-Macaulay complexes. Finally, we classify certain classes of initially Cohen-Macaulay graphs of interest and those whose projective dimension coincides with their maximum vertex degree.