Given a finitely generated module over a ℤd-graded polynomial ring R, quasidegreesAsVariables gives a representation of the quasidegree set of M using the variables of R. This method captures the plane arrangement of the quasidegree set of the module.
If the input is an ideal I, then quasidegreesAsVariables executes for the module R/I where R is the ring of I.
A synonym for this function is qav.
i1 : R = QQ[x,y,Degrees=>{{1,0},{0,1}}] o1 = R o1 : PolynomialRing |
i2 : I = ideal(x^2*y,x*y^2,y^3) 2 2 3 o2 = ideal (x y, x*y , y ) o2 : Ideal of R |
i3 : M = R^1/I o3 = cokernel | x2y xy2 y3 | 1 o3 : R-module, quotient of R |
i4 : quasidegreesAsVariables M 2 o4 = {{1, {x}}, {y, {}}, {x*y, {}}, {y , {}}} o4 : List |
In the above example, the first element in the list {1,{x}} corresponds to a line in the x direction with no shift. The element {y,{}} corresponds to a point shifted in the direction of the degree of y, the element {x*y,{}} corresponds to a point shifted in the direction of the degree xy, and the element {y2,{}}corresponds to a point shifted in the direction of the degree of y2.
The next example has a 2 dimensional quasidegree set.
i5 : R=QQ[x,y,z,Degrees=>{{1,0,0},{0,1,0},{0,0,1}}] o5 = R o5 : PolynomialRing |
i6 : I=ideal(y) o6 = ideal y o6 : Ideal of R |
i7 : M=R^1/I o7 = cokernel | y | 1 o7 : R-module, quotient of R |
i8 : quasidegreesAsVariables M o8 = {{1, {x, z}}} o8 : List |
The quasidegree set of ℚ[x,y,z]/<y> with the standard ℤ3-grading is the (unshifted) xz-plane.