On the finite generation of ideals in tensor triangular geometry

Key Points

• The research aims to explore the conditions under which ideals in tensor triangular geometry can be finitely generated.
• Examined properties of essentially small tensor triangulated categories.
• Analyzed the weakly Noetherian spectrum of these categories.
• Established conditions linking finite generation of prime ideals to the finiteness of the prime ideals set.
• Every prime ideal can be generated by finitely many objects under specific conditions.
• Proved that a finite set of prime ideals leads to finite generation in the examined categories.