# 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.

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.
• No, one does not need $F_m=A$ for $m$ nonnegative at all. My answer simply presumes the notation in which $F_{i-1} A$ is contained in $F_i A$, and of course 1 is in $F_0 A$, so 1 is in $F_1 A$, to. E.g., the filtration can be increasing, with $F_iA=0$ for $i<0$, or it can be decreasing, with $F_iA=A$ for $i\geq0$, or it can extend nontrivially in both directions. All these cases are covered. – Leonid Positselski May 27 '10 at 9:54
• This question and answer is an old favorite of mine. Maybe it's worth adding that if $\mathcal R$ is the Rees algebra of a filtered ring $R$. then a filtered module over $R$ is a $\mathbb G_m$-equivariant quasicoherent sheaf over $Spec \mathcal R$, perhaps with some torsionfreeness condition. – Tim Campion May 16 '19 at 21:29