Free resolutions over commutative Koszul algebras

TL;DR

This paper investigates the properties of minimal resolutions over Koszul algebras, establishing new bounds on the slopes and vanishing of certain Tor modules, advancing understanding of algebraic resolutions.

Contribution

It introduces new relationships between slopes of resolutions and proves vanishing results for Tor modules over Koszul algebras with specific conditions.

Findings

Proves Tor^Q_i(R,k)_j=0 for j>2i when Q and R are Koszul and J_1=0.

Establishes vanishing of Tor modules for j=2i under certain dimensional conditions.

Provides bounds on the slopes of minimal resolutions over Koszul algebras.

Abstract

For R=Q/J with Q a commutative graded algebra over a field and J non-zero, we relate the slopes of the minimal resolutions of R over Q and of k=R/R_{+} over R. When Q and R are Koszul and J_1=0 we prove Tor^Q_i(R,k)_j=0 for j>2i, for each non-negative integer i, and also for j=2i when i>dim Q-dim R and pd_QR is finite.