I am studying Licci squarefree monomial ideals, where the definition is typically given in the polynomial ring localized at the maximal ideal (Licci ideals can be defined more generally in a regular local ring).
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Let R be a regular local ring, and let I and J be proper ideals of R. Then I and J are said to be directly linked, denoted by I ~ J, if there exists a regular sequence z = z_1, ..., z_g in I ∩ J such that: J = (z) : I and I = (z) : J.
Moreover, I is said to be linked to J if there exists a sequence of direct links:
I = I_0 ~ I_1 ~ I_2 ~ ... ~ I_m = J.
In this case, I is in the linkage class of J.
In particular, if one can choose J to be a complete intersection ideal (i.e., an ideal generated by a regular sequence), then I is called Licci.
In Macaulay2, there is a function isLicci to check if an ideal is Licci, but the computation is performed in the polynomial ring, which is not local. What does this function actually check?
For example, consider the ideal:
I = (x_1 * x_2, x_3 * x_4, x_5 * x_6, x_7 * x_8) in R = K[x_1, ..., x_8], and let m = (x_1, ..., x_8).
According to Theorem 3.7 of Licci squarefree monomial ideals generated in degree two or with deviation two, the localization of I at m is a Licci ideal. However, when I ask Macaulay2 if I is Licci using isLicci(I), it returns false.
It seems that Macaulay2 checks whether I is Licci only in the polynomial ring without considering the localization at the maximal ideal, which makes the result inconsistent with the theoretical definition in a local setting.
Is this understanding correct? Or is there something about the behavior of the isLicci function in Macaulay2 that I am missing?
Many thanks, my warm regards!