Uniform bounds on projective dimension and Castelnuovo–Mumford regularity

In this article, we obtain uniform effective upper bounds for the projective dimension and the Castelnuovo–Mumford regularity of homogeneous ideals inside a standard graded polynomial ring S over a field. Such bounds are independent of the number of variables of S, in the spirit of Stillman’s conjecture and of the Ananyan–Hochster’s theorem, and depend on partial data extracted from the beginning or the end of the resolution. The main result is an extension of a theorem due to McCullough from 2012. Namely, we bound the projective dimension and the regularity of an ideal in terms of the regularity of a fraction of the syzygies.