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How to Compute Homology of a Graded Chain Complex https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=2615 |
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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: Code: LIB "homolog.lib";
ring R=2,(x,y),dp; //d2 will represent a homomorphism of free graded k[x,y]-modules module d2; d2[1]=x*gen(3)+x*gen(2)+x*gen(1); d2[2]=y*gen(3)+y*gen(2)+y*gen(1); intvec grades_C1=1,1,1; attrib(d2,"isHomog",grades_C1); //d1 will also represent a homomorphism of free graded k[x,y]-modules module d1; d1[1]=gen(3)+gen(2); d1[2]=gen(1)+gen(3); d1[3]=gen(1)+gen(2); intvec grades_C0=1,1,1; attrib(d1,"isHomog",grades_C0); module triv=0; //Compute a presentation for M:=ker(d1)/im(d2) def M=homology(d2,d1,triv,triv); //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. attrib(M,"isHomog"); //Compute a free resolution of M resolution resM=res(M,0); //Since the grading was lost for M, the Betti numbers are off by 1. print(betti(resM), "betti"); |
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