nLab
filtered complex

Idea

It’s easy to say what chain complex and homology mean (that is, these notions are definable); where things get tricky is, when calculating them, to figure out what the modules and differentials, kernels and images actually are. Sometimes there’s extra structure, e.g. a further hierarchy beyond the usual grading, that lets us figure these things out one layer at a time. Then we have to glue the layers back together, and that’s one place a spectral sequence is handy

Definition

For a poset II and an abelian category AA, an II-filtered complex is a functor F:IMon(Ch(A))F:I\to Mon(Ch(A)), from II to monomorphisms of chain complexes in AA. Roughly, this boils down to

  • A Complex d:CCd:C\to C, d 2=0d^2=0
  • with a submodules F i<CF_i \lt C for ii an object of II
  • such that F i<F jF_i \lt F_{j} for i<ji\lt j in II
  • such that dF i<F i d F_i \lt F_i .

(You may have noticed this isn’t the usual notation for functors. It’s traditional.)

The most frequent examples have I={}I=\mathbb{N}\cup\{\infty\}, I=I=-\mathbb{N}, or I={}I=\mathbb{Z}\cup\{\infty\}, with their usual total orderings; in this connection see spectral sequence of a filtered complex

Usually CC is a graded complex, with d j:C jC j1d_j:C_j\to C_{j-1}, and in this case we ask

d j:F iC jF iC j1.d_j:F_i\cap C_j \to F_i\cap C_{j-1}.

(If you prefer cohomology differentials, read ++ for -.)

Associated Graded complex

In the special case of a discrete totally-ordered filtration, one defines the associated graded complex G i(F)=F i+1/F iG_i(F) = F_{i+1}/F_i with differential induced by d[x]=[dx]d[x] = [d x]; again, if (C,d)(C,d) is graded, we have a bigraded complex with components G iC jG_i\cap C_j and differential of bidegree (±1,0)(\pm 1, 0).

References

Any book introducing spectral sequences.

Revised on January 14, 2011 03:57:04 by JCMc Keown (142.151.171.122)