Reconstruction from Koszul homology and applications to module and derived categories

TL;DR

This paper develops a method to reconstruct finitely generated modules over a noetherian ring from their Koszul homology, and applies this to classify subcategories in module and derived categories.

Contribution

It introduces a reconstruction technique from Koszul homology and explores its implications for classifying subcategories in module, derived, and singularity categories.

Findings

Modules can be reconstructed from Koszul homology via specific operations.

The method aids in classifying subcategories of module and derived categories.

Applications include understanding the structure of singularity categories.

Abstract

Let R be a commutative noetherian ring. Let M be a finitely generated R-module. In this paper, we reconstruct M from its Koszul homology with respect to a suitable sequence of elements of R by taking direct summands, syzygies and extensions, and count the number of those operations. Using this result, we consider generation and classification of certain subcategories of the category of finitely generated R-modules, its bounded derived category and the singularity category of R.