Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings

TL;DR

This paper introduces a novel method using iterated mapping cones on the Koszul complex to explicitly construct minimal free resolutions over complete intersection rings, providing a new perspective and explicit matrix descriptions.

Contribution

It develops a new construction of minimal free resolutions using iterated mapping cones, independent of Tate's classical approach, with explicit differential maps.

Findings

Constructs exact sequences involving homology algebra components.

Expresses differentials explicitly as block matrices with combinatorial patterns.

Provides an alternative, explicit resolution method for the residue field over complete intersection rings.

Abstract

Let $(R,\mathfrak m, \mathsf k)$ be a complete intersection local ring, $K$ be the Koszul complex on a minimal set of generators of $\mathfrak m$, and $A=H(K)$ be its homology algebra. We establish exact sequences involving direct sums of the components of $A$ and express the images of the maps of these sequences as homologies of iterated mapping cones built on $K$. As an application of this iterated mapping cone construction, we recover a minimal free resolution of the residue field $\mathsf k$ over $R$, independent from the well-known resolution constructed by Tate by adjoining variables and killing cycles. Through our construction, the differential maps can be expressed explicitly as blocks of matrices, arranged in some combinatorial patterns.