# What's the meaning of “decreasing filtration”?

Page 152-153, "Algebraic Geometry" by Lei Fu.

The condition (e) of the definition of spectral sequence is listed as follows:

(e) A family of objects $H^n(n\in \Bbb Z)$ in $\mathcal C$ and each $H^n$ is provided with a decreasing filtration $$\cdots \supset F^pH^n \supset F^{p+1}H^n \supset\cdots.$$

What's the meaning of "decreasing filtration"?

## 1 Answer

It's explained in the line following the term. A sequence of indexed objects (I'd assume subsets of $H^n$ in this case) with the inclusion relation "$\supset$". The word decreasing refers to the direction of the inclusion. (If you'd replace "$\supset$" by "$\subset$" it would be increasing. )

(Sometimes additional requirements are imposed, e.g. that $H^n$ is the limit (in whatever sense) of the sequence in the increasing direction or $\{0\}$ the limit in the other direction, but I'd expect this to be stated).

If additional structure is relevant than usually one wants the sequence to respect that, e.g. in case of vector spaces you'd require that the inclusion relation is actually a subspace relation, in case of groups you'd want to have subgroups etc.