canonicalModule -- computes the canonical module of a module $M$ or an ideal $I$.

Synopsis

Usage:

canonicalModule M

canonicalModule I
Inputs:
- M, a module, module over the polynomial ring $S=K[x_1,\ldots,x_n]$, with $K$ a field or an ideal $I\subset S$
Outputs:
- K, a module, the canonical module

Description

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring. Given a $S$-module $M$ (or an ideal $I\subset S$), it returns the canonical module $\omega(M)$ (or $\omega(S/I)$), as $\mathrm{Ext}_S^{n-d}(M,S(-n))$, where $d=\dim M$ (or $d=\dim S/I$).

i1 : S=QQ[x_1..x_5];

i2 : I=(x_1^2*x_3,x_2*x_3^2*x_4,x_1*x_3^3*x_5);

i3 : M=S^1/I;

i4 : canonicalModule M

o4 = cokernel {4} | x_3 |

                            1
o4 : S-module, quotient of S

Ways to use canonicalModule:

canonicalModule(Ideal)
canonicalModule(Module)

For the programmer

The object canonicalModule is a method function.

canonicalModule -- computes the canonical module of a module $M$ or an ideal $I$.

Synopsis

Description

See also

Ways to use canonicalModule:

For the programmer