Algebraic properties of ideals of poset homomorphisms

Given finite posets $P$ and $Q$, we consider a specific ideal $L(P,Q)$, whose minimal monomial generators correspond to order-preserving maps $\varphi :P\to Q$. We study algebraic invariants of those ideals. In particular, sharp lower and upper bounds for the Castelnuovo-Mumford regularity and the projective dimension are provided. Precise formulas are obtained for a large subclass of these ideals. Moreover, we provide complete characterizations for several algebraic properties of $L(P,Q)$, including being Buchsbaum, Cohen-Macaulay, Gorenstein and having a linear resolution. We also give a partial characterization for Golod property of $L(P,Q)$. Using those results, we derive applications for other important classes of monomial ideals, such as initial ideals of determinantal ideals and multichain ideals.