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# taylorResolution -- create the Taylor resolution of a monomial ideal

## Synopsis

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

## Description

If $M$ is a monomial ideal, minimally generated by $L$, then the Taylor resolution of $M$ is the resolution of $M$ obtained by homogenizing the $(\#L - 1)$-simplex.

 i1 : S = QQ[vars(0..3)] o1 = S o1 : PolynomialRing i2 : M = monomialIdeal(a*b,c^3,c*d,b^2*c) 2 3 o2 = monomialIdeal (a*b, b c, c , c*d) o2 : MonomialIdeal of S i3 : T = taylorResolution M 1 4 6 4 1 o3 = S <-- S <-- S <-- S <-- S 0 1 2 3 4 o3 : ChainComplex i4 : T.dd 1 4 o4 = 0 : S <-------------------- S : 1 | cd ab c3 b2c | 4 6 1 : S <----------------------------------- S : 2 {2} | -ab -c2 -b2 0 0 0 | {2} | cd 0 0 -c3 -bc 0 | {3} | 0 d 0 ab 0 -b2 | {3} | 0 0 d 0 a c2 | 6 4 2 : S <-------------------------- S : 3 {4} | c2 b 0 0 | {4} | -ab 0 b2 0 | {4} | 0 -a -c2 0 | {5} | d 0 0 b | {4} | 0 d 0 -c2 | {5} | 0 0 d a | 4 1 3 : S <-------------- S : 4 {6} | -b | {5} | c2 | {6} | -a | {6} | d | o4 : ChainComplexMap

If $M$ is generated by a regular sequence $L$, then the Taylor resolution is the Koszul complex on $L$.

 i5 : L = gens S o5 = {a, b, c, d} o5 : List i6 : T = taylorResolution L; i7 : K = koszul matrix{L}; i8 : T.dd 1 4 o8 = 0 : S <--------------- S : 1 | a b c d | 4 6 1 : S <----------------------------- S : 2 {1} | -b -c -d 0 0 0 | {1} | a 0 0 -c -d 0 | {1} | 0 a 0 b 0 -d | {1} | 0 0 a 0 b c | 6 4 2 : S <----------------------- S : 3 {2} | c d 0 0 | {2} | -b 0 d 0 | {2} | 0 -b -c 0 | {2} | a 0 0 d | {2} | 0 a 0 -c | {2} | 0 0 a b | 4 1 3 : S <-------------- S : 4 {3} | -d | {3} | c | {3} | -b | {3} | a | o8 : ChainComplexMap i9 : K.dd 1 4 o9 = 0 : S <--------------- S : 1 | a b c d | 4 6 1 : S <----------------------------- S : 2 {1} | -b -c 0 -d 0 0 | {1} | a 0 -c 0 -d 0 | {1} | 0 a b 0 0 -d | {1} | 0 0 0 a b c | 6 4 2 : S <----------------------- S : 3 {2} | c d 0 0 | {2} | -b 0 d 0 | {2} | a 0 0 d | {2} | 0 -b -c 0 | {2} | 0 a 0 -c | {2} | 0 0 a b | 4 1 3 : S <-------------- S : 4 {3} | -d | {3} | c | {3} | -b | {3} | a | o9 : ChainComplexMap