Possible bug in the Betti diagram

[agmath@…]

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?

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

[steenpass]

Your input is not homogeneous with respect to total degree, so asking for the graded Betti numbers does not make sense.