Algebraic Approach to Formal Microstructure Sheaves and Formal Quantum Sheaves Over Projective Schemes

Abstract

The I-adic non-commutative filtrations have crucial applications in the study of Lie algebras and the integral group rings. The I-completion R of a commutative ring R is considered a useful tool in algebraic geometry. Here in this paper we study and prove certain algebraic properties of the filtered (Rees-graded) formal micro-sheaves over Y = specs(G(R)) or p(Y) = proj(G(R)). We observed that the microlocalizations of I-adic completions appear as solutions for the formal sheaves over Y or p(Y). We prove that the formal sheaves over Y or p(Y) are strongly filtered flat sheaves.