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# regularitySequence -- regularity of Ext modules for a sequence of MCM approximations

## Synopsis

• Usage:
L = regularitySequence (R,M)
• Inputs:
• R, a list, list of rings R_i = S/(f_0..f_{(i-1)}), complete intersections
• M, , module over R_c where c = length R - 1.
• Outputs:
• L, a list, List of pairs {regularity evenExtModule M_i, regularity oddExtModule M_i)

## Description

Computes the non-free parts M_i of the MCM approximation to M over R_i, stopping when M_i becomes free, and returns the list whose elements are the pairs of regularities, starting with M_{(c-1)} Note that the first pair is for the

 i1 : c = 3;d=2 o2 = 2 i3 : R = setupRings(c,d); i4 : Rc = R_c o4 = Rc o4 : QuotientRing i5 : M = coker matrix{{Rc_0,Rc_1,Rc_2},{Rc_1,Rc_2,Rc_0}} o5 = cokernel | x_0 x_1 x_2 | | x_1 x_2 x_0 | 2 o5 : Rc-module, quotient of Rc i6 : regularitySequence(R,M) reg even ext, soc degs even ext, reg odd ext, soc degs odd ext {3, {1, 1, 1}, 2, {1, 1}} {2, {0, 0, 0, 1}, 2, {0, 0, 0}} {0, {}, 0, {}}