The Second Vanishing Theorem for Formal Local Cohomology Modules

TL;DR

This paper extends classical vanishing theorems to formal local cohomology modules over Noetherian local rings by introducing the formal dimension invariant and characterizing vanishing conditions.

Contribution

It introduces the formal dimension invariant and establishes a second vanishing theorem for formal local cohomology, extending classical results to the formal setting.

Findings

Vanishing of higher formal local cohomology characterized by quotient ring dimension

Introduction of the formal dimension invariant

Applications to derived category complexes

Abstract

This paper establishes a second vanishing theorem for formal local cohomology modules over Noetherian local rings. We introduce the \textit{formal dimension} invariant and characterize the vanishing of higher formal local cohomology in terms of the dimension of the quotient ring modulo minimal primes. Our main result extends classical vanishing theorems to the formal setting, with applications to the structure of complexes in derived categories. Necessary and sufficient conditions are given via spectral sequence analysis and duality arguments.