About the isRegularSequence method.

[Kumar Sannidhya Shukla]

Hello Macaulay2 users,

I had a question about the isRegularSequence method from the Depth package. How does this method work for homogeneous sequences?

I have looked at the source code, and it seems that for homogeneous sequences, the method proceeds as follows: suppose we want to check if the sequence (x_1, ..., x_n)  is M-regular, let g be the Hilbert series of M and and f be the Hilbert series of M/(x_1, ..., x_n), then it checks whether f == g * product (1 - t^d_i) where d_i is the degree of x_i.

A comment following this method refers to an Exercise in Eisenbud's book. I believe exercise being referred to is 21.17.b :

b. Suppose that A is Cohen-Macaulay of dimension d, and that f_1,..., f_d
is a regular sequence of homogeneous elements in A with degrees
d_1, ... ,d_d. Set B = A/(f_1, ... , f_d). Show that

h_A(t) = h_B(t) Π( 1 - t^(d_i) ).

However, this exercise only applies to CM rings/modules and the converse is not stated. But the isRegularSequence method is really using the converse of the exercise and for any arbitrary (not necessarily CM) rings/modules.

So my question is: what properties of Hilbert series and regular sequences are being used here?

I have tried to check regularity by vanishing of some Tor's but the answer doesn't seem to agree with the isRegularSequence method. So either my Tor computation was incorrect or the converse of the exercise above does not hold for non CM-modules. If needed I can provide an explicit example where Tor's don't vanish but isRegularSequence returns true.

Best regards,

Kumar.