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We introduce the concept of edgewise domination in clutters and use it to provide an upper bound for the projective dimension of any squarefree monomial ideal. We then compare this bound to a bound given by Faltings. Finally, we study a family of clutters associated to graphs and compute dom-ination parameters for certain classes of these clutters. [ABSTRACT FROM AUTHOR]

We characterize the lcm lattices that support a monomial ideal with a pure resolution. Given such a lattice, we provide a construction that yields a monomial ideal with that lcm lattice and whose minimal free resolution is pure. [ABSTRACT FROM AUTHOR]

Let M and N be two monomials of the same degree, and let I be the smallest Borel ideal containing M and N. We show that the toric ring of I is Koszul by constructing a quadratic Gröbner basis for the associated toric ideal. Our proofs use the construction of graphs corresponding to fibers of the toric map. As a consequence, we conclude that the Rees algebra is also Koszul. [ABSTRACT FROM AUTHOR]