Knowlpedia

Filtered ring

A ring equipped with an increasing multiplicative filtration.

A filtered ring is a RR together with an increasing family of additive subgroups {FnR}n∈Z\{F_nR\}_{n\in\mathbb Z} such that:

  • FnR⊆Fn+1RF_nR\subseteq F_{n+1}R for all nn,
  • 1∈F0R1\in F_0R,
  • FnR⋅FmR⊆Fn+mRF_nR\cdot F_mR \subseteq F_{n+m}R for all m,nm,n,
  • often ⋃nFnR=R\bigcup_n F_nR = R, in which case the filtration is exhaustive.

Filtrations measure “order” or “size” of elements and produce graded approximations via the ; many structural arguments pass from RR to its graded shadow.

Examples
  • The degree filtration on k[x1,…,xn]k[x_1,\dots,x_n] given by Fd={polynomials of degree≤d}F_d=\{\text{polynomials of degree}\le d\} is multiplicative.
  • The order filtration on a ring of differential operators is increasing and multiplicative.
Remarks

Some authors instead use decreasing filtrations. For example, the powers R⊇I⊇I2⊇⋯R\supseteq I\supseteq I^2\supseteq\cdots of an ideal form the decreasing II-adic filtration; its indexing convention should not be confused with the increasing convention above.