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Topic review - Possible bug in Betti diagram
Author Message
  Post subject:  Re: Possible bug in Betti diagram  Reply with quote
Your ideal is nor homogeneous,
see
https://www.singular.uni-kl.de:8002/trac/ticket/810#comment:1
Post Posted: Sat Sep 23, 2017 1:09 pm
  Post subject:  Possible bug in Betti diagram  Reply with quote
We know that if $I$ is a graded ideal in a polynomial ring, then the graded Betti numbers of $I$ are at most the corresponding graded Betti numbers of the initial ideal of $I$ with respect to any monomial order (See Corollary 3.3.3 of the book by Herzog and Hibi).

The following code in Singular says differently. Where is my mistake?
Code:

>  ring r = 0, (x, y,z, u,v, w, a, b, c, d), Dp;
> ideal P = zw+u2+uv, ya+zu+uv, xb+y2+yz+zu, uvcd+wac+wb2, zvcd-uac-ub2-vac-vb2;

> P;
P[1]=zw+u2+uv
P[2]=ya+zu+uv
P[3]=xb+y2+yz+zu
P[4]=uvcd+wac+wb2
P[5]=zvcd-uac-ub2-vac-vb2
> P = std(P);
> ideal P' = lead(P);
> P';
P'[1]=zw
P'[2]=ya
P'[3]=xb
P'[4]=uvcd
P'[5]=zvcd
> resolution R = mres(P, 0);
> resolution R' = mres(P', 0);
> print(betti(R), "betti");
           0     1     2     3     4
------------------------------------
    0:     1     -     -     -     -
    1:     -     3     -     -     -
    2:     -     -     4     -     -
    3:     -     2     1      3     -
    4:     -     -     4      2     1
    5:     -     -     -       2     1
------------------------------------
total:     1     5     9     7     2

> print(betti(R'), "betti");
           0     1     2     3     4
------------------------------------
    0:     1     -     -     -     -
    1:     -     3     -     -     -
    2:     -     -     3     -     -
    3:     -     2     2     1     -
    4:     -     -     4     4     -
    5:     -     -     -      2     2
------------------------------------
total:     1     5     9     7     2

Post Posted: Fri Sep 22, 2017 3:26 pm


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