ag.algebraic geometry - Geometric interpretation of filtered rings and modules

To a filtered algebra $(A,F)$ one can assign its Rees algebra $R=bigoplus_i F_iA$. It is a graded algebra containing the algebra of polynomials in one variable $mathbb{C}[t]$ naturally embedded as the subalgebra generated by the element $tin R_1$ corresponding to the element $1in F_1A$. So the algebra $R$ defines a $mathbb{C}^*$-equivariant quasi-coherent sheaf of algebras $mathcal{R}$ over the affine line $operatorname{Spec}mathbb{C}[t]$. The algebra $A$ can be recovered as the fiber of $mathcal{R}$ at the point $t=1$, and the associated graded algebra $operatorname{gr}_FA$ is the fiber of $mathcal{R}$ at $t=0$. Filtered $A$-modules correspond to $mathbb{C}^*$-equivariant quasi-coherent sheaves of modules over $mathcal{R}$.

The algebra $R$ is a torsion-free $mathbb{C}[t]$-module, so the quasi-coherent sheaf $mathcal{R}$ over $operatorname{Spec}mathbb{C}[t]$ has to be torsion-free. This description does not take into accout the issue of completeness of the filtration $F$ (in case it extends also in the decreasing direction), which requires a separate consideration.