eloisagrifo / levels Goto Github PK

add caveat that it only works over cis

• A method `coghost'
• A method `isCoghost'
  analogously as for ghost

isPerfect decides if a complex is perfect -- how would we do that?

As much as we can.

What is the proper way to overload an option?

Rewrite with the old code?

There are two ways to do this:

1. Find the canonical ghost map using `ghost', then check whether the given map factors through this one
2. Check it by hand: pre-compose with R, which is the same as taking homology of the map, and check whether this is zero

More generally, for ghost with respect to some G:

1. Works for regular rings or if G is perfect
2. Pre-compose with G, Problem: There might be infinitely many degrees to check

Make sure this is never infinity.

Document option length

Check that the support of F is contained in the support of G, if not the level is infinity.

Works when G is perfect (in particular if R is regular):

1. Change `ghost' to ghost wrt G, instead of finding a cover for ker(F.dd), find a generating set for Ext(G,F) in each degree
2. Don't need to change level', only ghost'

When R is artinian (or G perfect) it is possible to compute level via coghost maps

1. Implement `coghost'
2. In level, use coghost' instead of ghost', and do g*f' instead of f*g'