Extremal behavior of ideals of minors

Abstract

Ideals of minors arising from minimal free resolutions—equivalently, the Fitting ideals of syzygy modules—are natural invariants of interest in commutative algebra and algebraic geometry. A surprising observation in recent years by Brown–Dao–Sridhar is that for many nice classes of rings, such as complete intersection rings and Golod rings, ideals of minors tend to become eventually periodic. In this article, we establish similar periodicity phenomena for further classes of rings, namely fiber products and artinian stretched Gorenstein rings. In fact, we show that when the embedding dimension is at least 3, under a mild characteristic assumption, the ideals of minors stabilize to powers of the maximal ideal, exhibiting extremal behavior. We also study the transfer of periodicity between rings. Specifically, we prove that for any local ring and a super‐regular element , if the ideals of minors of an ‐module have extremal behavior, then so do the ideals of minors of over .