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# scarfChainComplex -- create the Scarf chain complex for a list of monomials.

## Synopsis

• Usage:
scarfChainComplex L
scarfChainComplex M
• Inputs:
• L, a list, of monomials in a polynomial ring, that minimally generate a monomial ideal.
• M, ,
• Outputs:
• C, ,

## Description

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}}]; i2 : M = monomialIdeal(x_0*x_1,x_0*x_3,x_1*x_2,x_2*x_3); o2 : MonomialIdeal of S i3 : T = taylorResolution M; i4 : C = scarfChainComplex M; 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 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 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}, ------------------------------------------------------------------------ {1, 0, 1, 1}, {0, 1, 1, 1}, {1, 1, 0, 1}, {1, 1, 1, 0}} o7 : List 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 i9 : T' = taylorResolution{x_0*x_1,x_0*x_2,x_0*x_3}; i10 : C' = scarfChainComplex{x_0*x_1,x_0*x_2,x_0*x_3}; 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 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 i13 : prune homology C' o13 = 0 : cokernel | x_0x_3 x_0x_2 x_0x_1 | 1 : 0 2 : 0 3 : 0 o13 : GradedModule 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}, ----------------------------------------------------------------------- {1, 1, 0, 1}, {1, 0, 1, 1}} o14 : List