Whether the input ideal is homogeneous, if known. The value of this input is only used if CheckCM=>true.

If an ideal $I$ is homogeneous and has a geometric vertex decomposition with respect to an indeterminate $y$, which is to say that ${\rm in}_y(I) = C_{y, I} \cap (N_{y, I} + \langle y \rangle)$, then both $C_{y, I}$ and $N_{y, I}$ are also homogeneous. Also, an ideal that is both homogeneous and geometrically vertex decomposable is Cohen-Macaulay [KR, Corollary 4.5].

[KR] Patricia Klein and Jenna Rajchgot. Geometric vertex decomposition and liaison. Forum Math. Sigma, 9 (2021) e70:1-23.

- CheckCM -- when to perform a Cohen-Macaulay check on the ideal
- isGVD -- checks whether an ideal is geometrically vertex decomposable
- isLexCompatiblyGVD -- checks whether an ideal is <-compatibly geometrically vertex decomposable for a given order

- isGVD(...,IsIdealHomogeneous=>...)
- isLexCompatiblyGVD(...,IsIdealHomogeneous=>...)

The object IsIdealHomogeneous is a symbol.