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# isSCM -- determines if a (hyper)graph is sequentially Cohen-Macaulay

- Usage:
    `b = isSCM H`
- Inputs:
  - H, a hypergraph,
- Optional inputs:
  - Gins => ..., default value false, use gins inside isSCM
- Outputs:
  - b, a Boolean value, true if the edgeIdeal of H is sequentially Cohen-Macaulay

## Description

This uses the edge ideal notion of sequential Cohen-Macaulayness; a hypergraph is called SCM if and only if its edge ideal is SCM. The algorithm is based on work of Herzog and Hibi, using the Alexander dual. H is SCM if and only if the Alexander dual of the edge ideal of H is componentwise linear.

There is an optional argument called Gins for isSCM. The default value is false, meaning that isSCM takes the Alexander dual of the edge ideal of H and checks in all relevant degrees to see if the ideal in that degree has a linear resolution. In characteristic zero with the reverse-lex order, one can test for componentwise linearity using gins, which may be faster in some cases. This approach is based on work of Aramova-Herzog-Hibi and Conca. We make no attempt to check the characteristic of the field or the monomial order, so use caution when using this method.

    i1 : R = QQ[a..f];

    i2 : G = cycle(R,4)

    o2 = Graph{"edges" => {{a, b}, {b, c}, {c, d}, {a, d}}}
               "ring" => R
               "vertices" => {a, b, c, d, e, f}

    o2 : Graph

    i3 : isSCM G

    o3 = false

    i4 : H = graph(monomialIdeal(a*b,b*c,c*d,a*d,a*e)); --4-cycle with whisker

    i5 : isSCM H

    o5 = true

    i6 : isSCM(H,Gins=>true) --use Gins technique

    o6 = true

## See also

- isCM -- determines if a (hyper)graph is Cohen-Macaulay
- edgeIdeal -- creates the edge ideal of a (hyper)graph

## Ways to use isSCM:

- `isSCM(HyperGraph)`

## For the programmer

The object isSCM is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/EdgeIdeals.m2:3582:0.