For a monomial ideal $M$, minimally generated by a list of monomials $L$ in a polynomial ring $S$, the Scarf complex is the subcomplex of the Taylor resolution of $S/M$ that is induced on the multihomogeneous basis elements with unique multidegrees. If the Scarf Complex is a resolution, then it is the minimal free resolution of $S/M$. For more information on the Scarf complex and its construction, see Bayer, Dave; Peeva, Irena; Sturmfels, Bernd Monomial Resolutions. Math. Res. Lett. 5 (1998), no. 1-2, 31–46, or Jeff Mermin Three Simplicial Resolutions, (English summary) Progress in commutative algebra 1, 127–141, de Gruyter, Berlin, 2012.
i1 : S = QQ[x_0..x_3, Degrees => {{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}}];
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i2 : M = monomialIdeal(x_0*x_1,x_0*x_3,x_1*x_2,x_2*x_3);
o2 : MonomialIdeal of S
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i3 : T = taylorResolution M;
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i4 : C = scarfChainComplex M;
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i5 : T.dd
1 4
o5 = 0 : S <----------------------------------- S : 1
| x_2x_3 x_0x_3 x_1x_2 x_0x_1 |
4 6
1 : S <-------------------------------------------------------- S : 2
{0, 0, 1, 1} | -x_0 -x_1 -x_0x_1 0 0 0 |
{1, 0, 0, 1} | x_2 0 0 -x_1x_2 -x_1 0 |
{0, 1, 1, 0} | 0 x_3 0 x_0x_3 0 -x_0 |
{1, 1, 0, 0} | 0 0 x_2x_3 0 x_3 x_2 |
6 4
2 : S <-------------------------------------- S : 3
{1, 0, 1, 1} | x_1 x_1 0 0 |
{0, 1, 1, 1} | -x_0 0 x_0 0 |
{1, 1, 1, 1} | 0 -1 -1 0 |
{1, 1, 1, 1} | 1 0 0 1 |
{1, 1, 0, 1} | 0 x_2 0 -x_2 |
{1, 1, 1, 0} | 0 0 x_3 x_3 |
4 1
3 : S <----------------------- S : 4
{1, 1, 1, 1} | -1 |
{1, 1, 1, 1} | 1 |
{1, 1, 1, 1} | -1 |
{1, 1, 1, 1} | 1 |
o5 : ChainComplexMap
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i6 : C.dd
1 4
o6 = 0 : S <----------------------------------- S : 1
| x_2x_3 x_0x_3 x_1x_2 x_0x_1 |
4 4
1 : S <---------------------------------------- S : 2
{0, 0, 1, 1} | -x_0 -x_1 0 0 |
{1, 0, 0, 1} | x_2 0 -x_1 0 |
{0, 1, 1, 0} | 0 x_3 0 -x_0 |
{1, 1, 0, 0} | 0 0 x_3 x_2 |
o6 : ChainComplexMap
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i7 : flatten for i to length C list degrees C_i
o7 = {{0, 0, 0, 0}, {0, 0, 1, 1}, {1, 0, 0, 1}, {0, 1, 1, 0}, {1, 1, 0, 0},
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{1, 0, 1, 1}, {0, 1, 1, 1}, {1, 1, 0, 1}, {1, 1, 1, 0}}
o7 : List
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i8 : prune homology C
o8 = 0 : cokernel | x_2x_3 x_0x_3 x_1x_2 x_0x_1 |
1 : 0
1
2 : S
o8 : GradedModule
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i9 : T' = taylorResolution{x_0*x_1,x_0*x_2,x_0*x_3};
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i10 : C' = scarfChainComplex{x_0*x_1,x_0*x_2,x_0*x_3};
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i11 : T'.dd
1 3
o11 = 0 : S <---------------------------- S : 1
| x_0x_1 x_0x_2 x_0x_3 |
3 3
1 : S <----------------------------------- S : 2
{1, 1, 0, 0} | -x_2 -x_3 0 |
{1, 0, 1, 0} | x_1 0 -x_3 |
{1, 0, 0, 1} | 0 x_1 x_2 |
3 1
2 : S <------------------------- S : 3
{1, 1, 1, 0} | x_3 |
{1, 1, 0, 1} | -x_2 |
{1, 0, 1, 1} | x_1 |
o11 : ChainComplexMap
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i12 : C'.dd
1 3
o12 = 0 : S <---------------------------- S : 1
| x_0x_1 x_0x_2 x_0x_3 |
3 3
1 : S <----------------------------------- S : 2
{1, 1, 0, 0} | -x_2 -x_3 0 |
{1, 0, 1, 0} | x_1 0 -x_3 |
{1, 0, 0, 1} | 0 x_1 x_2 |
3 1
2 : S <------------------------- S : 3
{1, 1, 1, 0} | x_3 |
{1, 1, 0, 1} | -x_2 |
{1, 0, 1, 1} | x_1 |
o12 : ChainComplexMap
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i13 : prune homology C'
o13 = 0 : cokernel | x_0x_3 x_0x_2 x_0x_1 |
1 : 0
2 : 0
3 : 0
o13 : GradedModule
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i14 : flatten for i to length C list degrees C'_i
o14 = {{0, 0, 0, 0}, {1, 1, 0, 0}, {1, 0, 1, 0}, {1, 0, 0, 1}, {1, 1, 1, 0},
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{1, 1, 0, 1}, {1, 0, 1, 1}}
o14 : List
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