The method quasidegrees takes a finitely generated module $M$ over the polynomial ring that is presented by a monomial matrix and computes the quasidegree set of $M$. The quasidegrees of $M$ are indexed by a list of pairs $(v,F)$ where $v$ is a vector and $F$ is a list of vectors f1,...,fl. The pair $(v,F)$ indexes the plane $v+span_\CCF$. The quasidegree set of M is the union of all such planes that the pairs (v,F) index.
If the input is an ideal $I$, then quasidegrees executes for the module $R/I$ where $R$ is the ring of $I$.
The following example computes the quasidegree set of $\QQ[x,y]/<x^2,y^2>$ under the standard $\ZZ^2$-grading.
i1 : A = matrix{{1,0},{0,1}}
o1 = | 1 0 |
| 0 1 |
2 2
o1 : Matrix ZZ <--- ZZ
|
i2 : R = QQ[x,y, Degrees => entries transpose A] o2 = R o2 : PolynomialRing |
i3 : I = ideal(x^2,y^2)
2 2
o3 = ideal (x , y )
o3 : Ideal of R
|
i4 : M = R^1/I
o4 = cokernel | x2 y2 |
1
o4 : R-module, quotient of R
|
i5 : quasidegrees M
o5 = {{| 0 |, {}}, {| 1 |, {}}, {| 1 |, {}}, {0, {}}}
| 1 | | 0 | | 1 |
o5 : List
|
The quasidegree set is given to be the points (0,1), (1,0), (1,1), and (0,0).
The next example takes $R$ computes the quasidegrees of the above module after twisting $R$ by multidegree (3,2).
i6 : R = R^{{-3,-2}}
1
o6 = (QQ[x..y])
o6 : QQ[x..y]-module, free, degrees {{3, 2}}
|
i7 : M = R^1/I
o7 = cokernel {3, 2} | x2 y2 |
1
o7 : QQ[x..y]-module, quotient of (QQ[x..y])
|
i8 : quasidegrees M
o8 = {{| 3 |, {}}, {| 3 |, {}}, {| 4 |, {}}, {| 4 |, {}}}
| 3 | | 2 | | 2 | | 3 |
o8 : List
|
The following demonstrates a quasidegree set that is not a finite number of points.
i9 : A = matrix{{1,0},{0,1}}
o9 = | 1 0 |
| 0 1 |
2 2
o9 : Matrix ZZ <--- ZZ
|
i10 : R = QQ[x,y] o10 = R o10 : PolynomialRing |
i11 : R = toGradedRing(A,R) o11 = R o11 : PolynomialRing |
i12 : I = ideal(x^2*y,y^2)
2 2
o12 = ideal (x y, y )
o12 : Ideal of R
|
i13 : M=R^1/I
o13 = cokernel | x2y y2 |
1
o13 : R-module, quotient of R
|
i14 : quasidegrees M
o14 = {{| 1 |, {}}, {| 0 |, {}}, {0, {| 1 |}}}
| 1 | | 1 | | 0 |
o14 : List
|
In the above example, the quasidegree set of the module M consists of the points (1,1) and (0,1) along with the parameterized line (1,0)$\bullet t$.
The object quasidegrees is a method function.