Positivity and nonstandard graded Betti numbers

• Published in: The Bulletin of the London Mathematical Society 2024-01, Vol.56 (1), p.111-123
• Main Authors: Brown, Michael K., Erman, Daniel
• Format: Article
• Language: English
• ISSN: 0024-6093; 1469-2120
• DOI: 10.1112/blms.12917

A foundational principle in the study of modules over standard graded polynomial rings is that geometric positivity conditions imply vanishing of Betti numbers. The main goal of this paper is to determine the extent to which this principle extends to the nonstandard Z$\mathbb {Z}$‐graded case. In this setting, the classical arguments break down, and the results become much more nuanced. We introduce a new notion of Castelnuovo–Mumford regularity and employ exterior algebra techniques to control the shapes of nonstandard Z$\mathbb {Z}$‐graded minimal free resolutions. Our main result reveals a unique feature in the nonstandard Z$\mathbb {Z}$‐graded case: the possible degrees of the syzygies of a graded module in this setting are controlled not only by its regularity, but also by its depth. As an application of our main result, we show that given a simplicial projective toric variety and a module M$M$ over its coordinate ring, the multigraded Betti numbers of M$M$ are contained in a particular polytope when M$M$ satisfies an appropriate positivity condition.