Free resolutions and marked families

TL;DR

This paper explores the structure and minimality of $U$-resolutions for modules generated by marked bases over quasi-stable modules, linking these resolutions to properties like componentwise linearity and establishing functorial isomorphisms.

Contribution

It investigates the minimality and structure of $U$-resolutions, relates them to componentwise linearity, and introduces functorial isomorphisms based on marked basis coefficients.

Findings

$U$-resolution minimality characterizes componentwise linearity for ideals.

Functorial isomorphisms relate syzygy modules and marked basis coefficients.

Reversal of the basis-coefficient relationship for ideals of depth ≥ 2.

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.