nLab filtered complex

Redirected from "concrete objects".
Note: codiscrete object and concrete object both redirect for "concrete objects".

This needs to be merged with filtered chain 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.

Last revised on June 5, 2014 at 00:13:51. See the history of this page for a list of all contributions to it.