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Quasidegrees :: quasidegreesLocalCohomology

quasidegreesLocalCohomology -- returns the quasidegree sets of local cohomology modules

Synopsis

Description

The input for this method is a module $M$ over a multigraded polynomial ring whose local cohomology modules can be presented by monomial matrices. If an integer $i$ is also included in the input, quasidegreesLocalCohomology(i,M) computes the quasidegree set of the $i-th$ local cohomology module, supported at the maximal irrelevant ideal, of $M$. If an integer is excluded from the input, then quasidegreesLocalCohomology(M) computes the quasidegree set of $H_{\mathbf m}^0(M)\oplus\cdots\oplus H_{\mathbf m}^{d-1}(M)$. The quasidegrees of local cohomology are indexed by a list of pairs $(v,F)$ where $v$ is a vector and $F$ is a list of vectors. The pair $(v,F)$ indexes the plane $v+span_\CCF$. The quasidegree set of the local cohomology modules is the union of all such planes that the pairs $(v,F)$ index.

If the input is an ideal $I$ in a multigraded polynomial ring $R$, then the method executes for the module $R/I$ where $R$ is the ring of $I$.

A synonym for this function is qlc.

The first example computes the quasidegree set of $H_{\mathbf m}^0(R/I)\oplus H_{\mathbf m}^1(R/I)$ where $I$ is the toric ideal associated to the matrix $A$.

i1 : A = matrix{{1,1,1,1},{0,1,5,11}}

o1 = | 1 1 1 1  |
     | 0 1 5 11 |

              2        4
o1 : Matrix ZZ  <--- ZZ
i2 : R = QQ[a..d]

o2 = R

o2 : PolynomialRing
i3 : R = toGradedRing(A,R)

o3 = R

o3 : PolynomialRing
i4 : I = toricIdeal(A,R)

               2    2    5    3 2   2 3    4    5    4
o4 = ideal (b*c  - a d, c  - b d , a c  - b d, b  - a c)

o4 : Ideal of R
i5 : M = R^1/I

o5 = cokernel | bc2-a2d c5-b3d2 a2c3-b4d b5-a4c |

                            1
o5 : R-module, quotient of R
i6 : quasidegreesLocalCohomology M

o6 = {{| 2 |, {}}, {| 3 |, {}}, {| 3 |, {}}, {| 4 |, {}}}
       | 4 |        | 4 |        | 9 |        | 9 |

o6 : List

The above example gives that the quasidegrees of the non-top local cohomology of $M$ are (4,9), (3,9), (2,4), and (3,4). We can see that these all come from the first local cohomology module.

i7 : quasidegreesLocalCohomology(1,M)

o7 = {{| 2 |, {}}, {| 3 |, {}}, {| 3 |, {}}, {| 4 |, {}}}
       | 4 |        | 4 |        | 9 |        | 9 |

o7 : List

The next example shows a module whose quasidegree set of its second local cohomology module at the irrelevant ideal, is a line.

i8 : A = matrix{{1,1,1,1,1},{0,0,1,1,0},{0,1,1,0,-2}}

o8 = | 1 1 1 1 1  |
     | 0 0 1 1 0  |
     | 0 1 1 0 -2 |

              3        5
o8 : Matrix ZZ  <--- ZZ
i9 : R = QQ[a..e]

o9 = R

o9 : PolynomialRing
i10 : R = toGradedRing(A,R)

o10 = R

o10 : PolynomialRing
i11 : I = toricIdeal(A,R)

                           2    2    2            3    2
o11 = ideal (a*c - b*d, a*d  - c e, a d - b*c*e, a  - b e)

o11 : Ideal of R
i12 : M = R^1/I

o12 = cokernel | ac-bd ad2-c2e a2d-bce a3-b2e |

                             1
o12 : R-module, quotient of R
i13 : quasidegreesLocalCohomology(2,M)

o13 = {{| 0 |, {|  1 |}}}
        | 0 |   |  0 |
        | 1 |   | -2 |

o13 : List

The above example gives that the quasidegrees of the second local cohomology module of $M$ at the irrelevant ideal is the complex parameterized line (0,0,1)+$t\bullet$(1,0,-2).

Ways to use quasidegreesLocalCohomology :

For the programmer

The object quasidegreesLocalCohomology is a method function.