# quasidegreesLocalCohomology -- returns the quasidegree sets of local cohomology modules

## Synopsis

• Usage:
quasidegreesLocalCohomology(I)
quasidegreesLocalCohomology(M)
quasidegreesLocalCohomology(i,I)
quasidegreesLocalCohomology(i,M)
• Inputs:
• i, an integer, the cohomological degree to be computed
• I, an ideal, an ideal in a multigraded polynomial ring
• M, , a module over a multigraded polynomial ring
• Outputs:
• a list, that represents the quasidegree set of local cohomology modules

## 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 :

• "quasidegreesLocalCohomology(Ideal)"
• "quasidegreesLocalCohomology(Module)"
• "quasidegreesLocalCohomology(ZZ,Ideal)"
• "quasidegreesLocalCohomology(ZZ,Module)"

## For the programmer

The object quasidegreesLocalCohomology is .