N-fiber-full modules

TL;DR

This paper introduces the concept of N-fiber-full modules up to a certain level, characterizes them via Ext functors' flatness, and applies the theory to extend results on squarefree Gr"obner degenerations.

Contribution

It defines N-fiber-full modules up to h and establishes their characterization through Ext functor flatness, extending previous work on degenerations.

Findings

N-fiber-full modules are characterized by Ext flatness conditions.

The paper extends results on squarefree Gr"obner degenerations.

Provides new tools for studying module degenerations.

Abstract

Let $A$ be a Noetherian flat $K[t]$-algebra, $h$ an integer and let $N$ be a graded $K[t]$-module, we introduce and study "$N$-fiber-full up to $h$" $A$-modules. We prove that an $A$-module $M$ is $N$-fiber-full up to $h$ if and only if $\mathrm{Ext}^i_A(M, N)$ is flat over $K[t]$ for all $i\le h-1$. And we show some applications of this result extending the recent result on squarefree Gr\"obner degenerations by Conca and Varbaro.