# quasidegrees -- compute the quasidegree set of a module

## Synopsis

• Usage:
quasidegrees I
quasidegrees M
• Inputs:
• I, an ideal,
• M, , a finitely generated module
• Outputs:
• a list, a list that indexes the quasidegrees of M

## Description

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)$\bullet t$.

## Ways to use quasidegrees :

• "quasidegrees(Ideal)"
• "quasidegrees(Module)"

## For the programmer

The object quasidegrees is .