Bounding Projective Dimension
نویسندگان
چکیده
The use of algorithms in algebra as well as the study of their complexity was initiated before the advent of modern computers. Hermann [25] studied the ideal membership problem, i.e determining whether a given polynomial is in a fixed homogeneous ideal, and found a doubly exponential bound on its computational complexity. Later Mayr and Meyer [31] found examples which show that her bound was nearly optimal. Their examples were further studied by Bayer and Stillman [3] and Koh [28] who showed that these ideals also had syzygies whose degrees are doubly exponential in the number of variables of the ambient ring. This survey addresses a different measure of the complexity of an ideal, approaching the problem from the perspective of computing the minimal free resolution of the ideal. Among invariants of free resolutions, we focus on the projective dimension, which counts the number of steps one needs to undertake in finding a minimal resolution; the precise definition of projective dimension is given in Section 2. In this paper we discuss estimates on the projective dimension of cyclic graded modules over a polynomial ring in terms of the degrees of the minimal generators of the defining ideal. We also establish connections to another well-known invariant, namely regularity. The investigation of this problem was initiated by Stillman who posed the following question:
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