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
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