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ChainComplexExtras :: scarfComplex

scarfComplex -- constructs the algebraic Scarf complex of a monomial ideal

Synopsis

Description

The algebraic Scarf complex of a monomial ideal is the sub-chain complex of the taylorResolution supported on subsets of generators with unique LCMs.

i1 : R = QQ[a,b,c,d,e];
i2 : I = monomialIdeal(b^4*c^3, a*b^3*c*d^2*e, a*b^2*c^2*d*e^2, a^2*d^3*e^5, b*c^2*d^5*e^4);

o2 : MonomialIdeal of R
i3 : s = scarfComplex I

      1      5      8      3
o3 = R  <-- R  <-- R  <-- R  <-- 0
                                  
     0      1      2      3      4

o3 : ChainComplex
i4 : s.dd

          1                                                5
o4 = 0 : R  <-------------------------------------------- R  : 1
               | b4c3 ab3cd2e ab2c2de2 bc2d5e4 a2d3e5 |

          5                                                               8
     1 : R  <----------------------------------------------------------- R  : 2
               {7}  | -ad2e -ade2 0   -d5e4 0     0     0      0     |
               {8}  | bc2   0     -ce 0     0     -ade4 0      0     |
               {8}  | 0     b2c   bd  0     -d4e2 0     -ad2e3 0     |
               {12} | 0     0     0   b3c   ab    0     0      -a2e  |
               {10} | 0     0     0   0     0     b3c   b2c2   bc2d2 |

          8                            3
     2 : R  <------------------------ R  : 3
               {11} | e  0    0   |
               {11} | -d 0    0   |
               {10} | bc ade3 0   |
               {16} | 0  0    0   |
               {14} | 0  0    ae  |
               {14} | 0  -c   0   |
               {14} | 0  b    -d2 |
               {15} | 0  0    b   |

          3
     3 : R  <----- 0 : 4
               0

o4 : ChainComplexMap

The Scarf complex of I is always a subcomplex of the minimal free resolution of I, computed in M2 with the command res I. In this first example the Scarf complex is strictly smaller.

i5 : (betti s, betti res I)

             0 1 2 3         0 1 2 3
o5 = (total: 1 5 8 3, total: 1 5 8 4)
          0: 1 . . .      0: 1 . . .
          1: . . . .      1: . . . .
          2: . . . .      2: . . . .
          3: . . . .      3: . . . .
          4: . . . .      4: . . . .
          5: . . . .      5: . . . .
          6: . 1 . .      6: . 1 . .
          7: . 2 . .      7: . 2 . .
          8: . . 1 .      8: . . 1 .
          9: . 1 2 1      9: . 1 2 1
         10: . . . .     10: . . . .
         11: . 1 . .     11: . 1 . .
         12: . . 3 1     12: . . 3 1
         13: . . 1 1     13: . . 1 1
         14: . . 1 .     14: . . 1 1

o5 : Sequence

In some cases, such as when I is a generic monomial ideal, the Scarf complex of I is a minimal free resolution of I. In this case scarfComplex I and res I will be isomorphic but not necessarily equal.

i6 : I = monomialIdeal(a^2*b^11*c^7*d*e, a^5*b^10*c^2*d^3*e^2, a^6*b^8*c^11*d^2*e^3, a^3*b^5*c^3*d^5*e^4, a^8*b^2*c*d^4*e^7);

o6 : MonomialIdeal of R
i7 : isExact(prependZeroMap scarfComplex I)

o7 = true
i8 : isMinimalChainComplex scarfComplex I

o8 = true
i9 : betti scarfComplex I == betti res I

o9 = true
i10 : scarfComplex I == res I

o10 = false

See chain complexes (missing documentation) for an overview of chain complexes in Macaulay2.

See also

Ways to use scarfComplex :

For the programmer

The object scarfComplex is a method function.