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# CheckCM -- when to perform a Cohen-Macaulay check on the ideal

## Description

Whether to check that the ideal is geometrically vertex decomposable using the result of [KR, Corollary 4.5] which relates the geometrically vertex decomposable and Cohen-Macaulay properties. Set CheckCM=>"once" to perform this check once, that is, only for the ideal given in the input; CheckCM=>"always" (default) for all of the following $C_{y,I}$ and $N_{y,I}$ ideals as well; or CheckCM=>"never".

In particular, [KR, Corollary 4.5] states that if a homogeneous ideal $I$ is geometrically vertex decomposable then $I$ must be Cohen-Macaulay. Equivalently, if a homogeneous ideal $I$ is not Cohen-Macaulay, then it is not geometrically vertex decomposable. By the Auslander-Buchsbaum formula, if $I$ is homogeneous, then Cohen-Macualayness is equivalent to equality of projective dimension and codimension, which is in general a quicker check than using isCM from the Depth package.

Since the result of [KR, Corollary 4.5] holds only for homogeneous ideals, the Cohen-Macaulayness is checked only when the given ideal is homogeneous, no matter the value of CheckCM.

We set the default value for CheckCM to "always" for, if the ideal is homogeneous, it is quicker to check equality of projective dimension and codimension than it is to check unmixedness, and Cohen-Macaulay impilies unmixed. Hence if an ideal is homogeneous, we need only check unmixedness directly when it is not Cohen-Macaulay.

## References

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

## See also

• isGVD -- checks whether an ideal is geometrically vertex decomposable
• isLexCompatiblyGVD -- checks whether an ideal is <-compatibly geometrically vertex decomposable for a given order

## Functions with optional argument named CheckCM :

• isGVD(...,CheckCM=>...)
• isLexCompatiblyGVD(...,CheckCM=>...)

## For the programmer

The object CheckCM is .