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Associated graded ring

The graded ring gr_F(R)=⊕ F_nR/F_{n-1}R attached to a filtered ring.

Given a (R,F∙)(R,F_\bullet) with Fn−1R⊆FnRF_{n-1}R\subseteq F_nR and FnR FmR⊆Fn+mRF_nR\,F_mR\subseteq F_{n+m}R, the associated graded ring is

grF(R)  :=  ⨁nFnR/Fn−1R,\mathrm{gr}_F(R) \;:=\; \bigoplus_{n} F_nR/F_{n-1}R,

with multiplication induced from RR: if x∈FnRx\in F_nR and y∈FmRy\in F_mR, then the product of their classes in the quotients defines an element of Fn+mR/Fn+m−1RF_{n+m}R/F_{n+m-1}R. Each summand is a of additive groups, and the direct sum is an internal .

The ring grF(R)\mathrm{gr}_F(R) is naturally a ; it captures the leading-order behavior of elements of RR and often has simpler algebraic structure.

Examples
  • For the decreasing II-adic filtration, the analogous convention gives grI(R)=⨁n≥0In/In+1\mathrm{gr}_I(R)=\bigoplus_{n\ge 0} I^n/I^{n+1}.
  • If the filtration is already split by degree (as in a graded ring viewed with Fn=⨁i≤nRiF_n=\bigoplus_{i\le n}R_i), then grF(R)≅R\mathrm{gr}_F(R)\cong R as graded rings.