Geometric interpretation of filtered rings and modules Let $A$ be a commutative algebra, say over $\mathbb{C}$. 
Giving a grading on $A$ corresponds at least morally to giving a $\mathbb{C}^*$ action on spec(A): $A_i$ can be thought of as  those functions on which $t$ acts by multiplication with $t^i$. Similary a graded $A$ module is just a $\mathbb{C}^*$ equivariant sheaf.
Now I want to know, if there is also a geometric interpretation of filtered rings/modules.
 A: 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 $t\in R_1$ corresponding to the element $1\in F_1A$.  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.
