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.



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 $t\in R\_1$ 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.


