Free resolutions and marked families
Cristina Bertone, Francesca Cioffi, Paolo Lella

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 be a field and a Noetherian -algebra. In a paper of 2020, M. Albert, C. Bertone, M. Roggero and W. M. Seiler proved that, given a quasi-stable module with , any submodule generated by a marked basis over admits a special free resolution described in terms of marked bases as well, called the {\em -resolution of }. In this paper, we first investigate the minimality of the -resolution and its structure. When is an ideal and , we show that is componentwise linear if and only if its -resolution is minimal, up to a linear change of variables. Then, adopting a functorial approach to the construction of the -resolution, we prove that certain functors naturally associated with the resolution are isomorphic. These isomorphisms arise from the fact that…
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