Free resolutions and marked families
Cristina Bertone, Francesca Cioffi, Paolo Lella
TL;DR
The paper develops a comprehensive framework for marked bases over quasi-stable (Pommaret) modules and their syzygies, introducing the $U$-resolution and a rich functorial viewpoint. It proves key minimality criteria, relating minimal $U$-resolutions to componentwise linearity, and demonstrates how Betti numbers of marked-basis-generated modules are controlled by the underlying $U$-resolution. It then constructs and analyzes functors that parameterize marked bases, their syzygies, and whole resolutions, showing representability and natural isomorphisms with the marked scheme $\mathbf{Mf}_U$, and provides explicit schemes and projections (including the second-factor projection) that connect coefficients of syzygies to those of marked bases. The work culminates in detailed examples and an application to Hilbert schemes, illustrating how marked schemes encode loci of componentwise linear ideals and their syzygy structures, with explicit parameter-elimination results via depth conditions.
Abstract
Let $\mathbb{K}$ be a field and $A$ a Noetherian $\mathbb{K}$-algebra. In a paper of 2020, M. Albert, C. Bertone, M. Roggero and W. M. Seiler proved that, given a quasi-stable module $U \subset R^m$ with $R=\mathbb{K}[x_0,\dots,x_n]$, any submodule $M\subseteq (R\otimes A)^m$ generated by a marked basis over $U$ admits a special free resolution described in terms of marked bases as well, called the {\em $U$-resolution of $M$}. In this paper, we first investigate the minimality of the $U$-resolution and its structure. When $M$ is an ideal and $A=\mathbb{K}$, we show that $M$ is componentwise linear if and only if its $U$-resolution is minimal, up to a linear change of variables. Then, adopting a functorial approach to the construction of the $U$-resolution, we prove that certain functors naturally associated with the resolution are isomorphic. These isomorphisms arise from the fact that the marked basis of the $i$-th syzygy module in the $U$-resolution can be expressed in terms of the coefficients of the marked basis of $M$. Moreover, when $M$ is an ideal of depth at least 2, this correspondence can be reversed: in this case, the marked basis of $M$ itself can be written in terms of the coefficients of the marked basis of its first syzygy module.