quasidegreesLocalCohomology -- returns the quasidegree sets of local cohomology modules

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 $A$.

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.

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

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).