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

- Usage:
    `L = regularitySequence (R,M)`
- Inputs:
  - R, a list, list of rings R_i = S/(f_0..f_{(i-1)}), complete intersections
  - M, a module, 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, {}}

## See also

- approximation -- returns pair of components of the map from the MCM approximation
- auslanderInvariant -- measures failure of surjectivity of the essential MCM approximation

## Ways to use regularitySequence:

- `regularitySequence(List,Module)`

## For the programmer

The object regularitySequence is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/CompleteIntersectionResolutions.m2:2600:0.