The Second Vanishing Theorem for Formal Local Cohomology Modules

This paper establishes a second vanishing theorem for formal local cohomology modules over Noetherian local rings. We introduce the $\mathit{\text{formal}}\mathit{\text{dimension}}$ 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.