On functors that detect $S_n$

TL;DR

This paper introduces specific functors on modules over Noetherian rings that can detect the Serre condition $S_n$ through the vanishing of their derived functors, providing a new tool for algebraic geometry and commutative algebra.

Contribution

The authors construct new left exact functors $D_k$ on module categories over Noetherian rings and establish their derived functors' vanishing as criteria for $S_n$ conditions.

Findings

Vanishing of certain derived functors characterizes $S_n$ conditions.

Construction of functors $D_k$ applicable to modules over Noetherian rings.

Provides a new method to detect $S_n$ properties via functor vanishing.

Abstract

Let $A$ be a Noetherian ring. For each $k$ where $0 \leq k \leq \dim A$ we construct left exact functors $D_k$ on $Mod(A)$. Let $D^i_k$ be the $i^{th}$-right derived functor of $D_k$. Let $M$ be a finitely generated $A$-module. Under mild conditions on $A$ and $M$ we prove that vanishing of some finitely many $D^i_k(M)$ is equivalent to $M$ satisfying $S_n$.