A study of maximal shifts in the minimal graded free resolution of the residue field

We study two invariants, rate and slant, that describe the extremal behavior of the maximal shifts in the minimal free resolution of the residue field over a standard graded algebra. We provide bounds and computations of these invariants for a variety of rings, including compressed level algebras, rings defined by general forms and rings defined by monomials. We also prove an asymptotic additivity property of the maximal shifts for certain classes of rings, including Golod rings, which allows to interpret slant as a limit.