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gorenstein_nonci
Interests
- Gorenstein, but not a complete intersection
Description
Binomial ideal $I$ of $S=\mathbb{k}[x_1,x_2, x_3]$ generated by
$$
x_1^2,
x_2^2,
x_1x_3,
x_2x_3,
x_3^2 -x_1x_2
$$
Numerical properties
| - | no. of variables | : | 3 |
| - | dimension $S/I$ | : | 0 |
| - | height of $I$ | : | 3 |
| - | projdim $S/I$ | : | 3 |
| - | depth $S/I$ | : | 0 |
| - | degree of generators | : | 2 |
| - | h-vector | : | (1,3,1) |
| - | Betti table | |
| | 0 | 1 | 2 |
|---|
| total: | 5 | 5 | 1 |
| 2: | 5 | 5 | . |
| 3: | . | . | 1 |
|
Further properties
| - | radical | no |
| - | prime | no |
| - | complete intersection | no |
| - | $S/I$ Gorenstein | yes |
| - | $S/I$ level | yes |
| - | $S/I$ Cohen-Macaulay | yes |
| - | $S/I$ Buchsbaum | yes |
| - | $I$ connected in codim one | yes |
Primary decomposition
It is $(x_1,x_2,x_3)$-primary.
Computer algebra code
For simplicity, we use $\mathbb{Z}_n$ with $n=101$ as field of coefficients.
Macaulay 2:
S=ZZ/101[x_1..x_3]
I=ideal(x_1^2, x_2^2, x_1*x_3, x_2*x_3, x_3^2 -x_1*x_2)
CoCoA:
Use S::=ZZ/(101)[x[1..3]];
I:=Ideal(x[1]^2, x[2]^2, x[1]x[3], x[2]x[3], x[3]^2 -x[1]x[2]));
Singular:
ring S = 101, x(1..3), dp;
ideal I=x(1)^2, x(2)^2, x(1)x(3), x(2)x(3), x(3)^2 -x(1)x(2);
References
D. Eisenbud, Commutative Algebra with a View towards Algebraic Geometry,
Example 21.7, p. 532