CoCoA » CoCoALib

Mario suggested that the internal representation should be vector< map<degree, long> >.

The resolution is a list of modules (with homomorphisms between them) traditionally indexed from 0 on the right, and with higher integers as ones goes to the left (in the traditional way of writing them).

Since the list of modules is "dense" (i.e. there are no "holes") is seems most sensible to use a vector with one position for each module. In the entry for each module we must find the shifts and their multiplicities: each degree(-shift) appears at most once, and has a positive integer multiplicity. Using a std::map will ensure that each degree appears only once. Degrees must be ordered to use them as keys in a map, but in fact they are already (lex) ordered.

Note that this design also allows an empty resolution (e.g. the resolution of the zero ideal).

Note: normally the shifts are written prefixed with a minus sign. In CoCoALib we have gradings over some cartesian power of ZZ, so we can negate degrees. What do we want to store in the Betti structure? A shift which is to be subtracted from the plain degree, or a shift which is to be added to a plain degree?