|How to Compute Homology of a Graded Chain Complex
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|Author:||mpl [ Mon Jun 05, 2017 4:27 am ]|
|Post subject:||How to Compute Homology of a Graded Chain Complex|
I'm new to singular. I'd like to compute the Betti numbers of a graded k[x,y]-module M arising as a homology module of a chain complex of free graded k[x,y]-modules, and I'm running into difficulties: I'm using the "homology" command to obtain a presentation of M, but this seems to discard information about the graded structure on M. I wonder if my trouble is related to bug #67 in the Singular issue tracker.
In any case, I'd appreciate your advice on how to circumvent this problem. Below is some Singular code which demonstrates the issue in detail:
//d2 will represent a homomorphism of free graded k[x,y]-modules
//d1 will also represent a homomorphism of free graded k[x,y]-modules
//Compute a presentation for M:=ker(d1)/im(d2)
//Next command outputs the grade information for the generators of M.
//Note the null output; homology apparently is not returning a graded object, whereas
//the generator for M should live at grade 1.
//Compute a free resolution of M
//Since the grading was lost for M, the Betti numbers are off by 1.
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