Minimal resolutions and algebraic properties of some classes of graded ideals

This thesis investigates the interaction between Combinatorics and Commutative Algebra, in order to study minimal resolutions and homological properties of graded ideals. In the exterior algebra, we construct minimal graded resolutions of monomial ideals with linear quotients via iterated mapping cones, describing their Betti numbers and differentials. In the polynomial ring, we develop the theory of vector-spread strongly stable ideals, generalizing both Macaulay and Kruskal-Katona theorems and describe an upper bound for their Betti numbers. For the class of squarefree principal vector-spread Borel ideals, we describe the primary decomposition and characterize the sequentially Cohen-Macaulay property. Moreover, we completely classify the ideals in this class having the property that their ordinary and symbolic powers coincide. For edge ideals of forests, we apply Betti splitting techniques to their squarefree powers, proving the non-increasingness of the normalized depth function and computing regularity in terms of matching numbers, thus confirming a conjecture of Erey and Hibi. We further study binomial edge ideals, establishing the sequentially Cohen-Macaulayness of cycles, wheels, block graphs, and cones. Finally, we present the Macaulay2 package SCMAlgebras, which provides computational tools for testing the sequentially Cohen-Macaulayness.