Bounds on the Projective Dimension of Ideals
Jason McCullough (Rider University)
MSRI: Simons Auditorium
Given a homogeneous ideal I in a polynomial ring R over a field, one can compute a minimal graded free resolution of I (or R/I). The length of this resolution, called the projective dimension, is finite and at most the number of variables of R by Hilbert's Syzygy Theorem. When I has few generators in low degree, one expects that projdim(R/I) cannot be arbitrarily large, even when R has many variables. Stillman asked precisely for a bound on projdim(R/I) in terms of the degrees of the minimal generators of I. In this talk I will survey the history of this problem and recent developments toward a solution, including the exponential bound of Ananyan-Hochster in the case of ideals generated by quadrics, specific bounds for three-generated ideals in low degree by Eisenbud-Huneke and Engheta, and families of ideals with large projective dimension.