Some properties of the gluing of schemes - Calixto, Rejiane Aparecida (2024)
The study of the properties of collage constructions spans various categories, being particularly relevant in the schemes category, where it allows the formation of new schemes. This work investigates the properties of gluing two schemes along a third, specifically along a closed sub-scheme. Based on properties of the fiber product of rings found in the literature, the analysis reveals that this gluing results in a locally Noetherian scheme. Additionally, conditions are established for the resulting scheme from the gluing to be Cohen-Macaulay, and partially Gorenstein and singular. The study also addresses the multiplicity of this scheme gluing, presenting corresponding formulas. As an application, formulas are derived for the gluing of schemes related to affine algebraic varieties. In the context of the formal schemes category, it is demonstrated that, under specific conditions, the gluing of formal schemes constitutes a formal scheme. This work provides a comprehensive understanding of the properties and characteristics resulting from scheme gluing, contributing to the advancement of knowledge in this domain.