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CompleteIntersectionResolutions :: ExtModule

ExtModule -- Ext^*(M,k) over a complete intersection as module over CI operator ring

Synopsis

Description

Uses code of Avramov-Grayson described in Macaulay2 book

i1 : kk= ZZ/101

o1 = kk

o1 : QuotientRing
i2 : S = kk[x,y,z]

o2 = S

o2 : PolynomialRing
i3 : I1 = ideal "x3y"

            3
o3 = ideal(x y)

o3 : Ideal of S
i4 : R1 = S/I1

o4 = R1

o4 : QuotientRing
i5 : M1 = R1^1/ideal(x^2)

o5 = cokernel | x2 |

                              1
o5 : R1-module, quotient of R1
i6 : betti res (M1, LengthLimit =>5)

            0 1 2 3 4 5
o6 = total: 1 1 1 1 1 1
         0: 1 . . . . .
         1: . 1 . . . .
         2: . . 1 . . .
         3: . . . 1 . .
         4: . . . . 1 .
         5: . . . . . 1

o6 : BettiTally
i7 : E = ExtModule M1

             2
o7 = (kk[X ])
          0

o7 : kk[X ]-module, free, degrees {0..1}
         0
i8 : apply(toList(0..10), i->hilbertFunction(i, E))

o8 = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

o8 : List
i9 : Eeven = evenExtModule(M1)

             1
o9 = (kk[X ])
          0

o9 : kk[X ]-module, free
         0
i10 : apply(toList(0..5), i->hilbertFunction(i, Eeven))

o10 = {1, 1, 1, 1, 1, 1}

o10 : List
i11 : Eodd = oddExtModule(M1)

              1
o11 = (kk[X ])
           0

o11 : kk[X ]-module, free
          0
i12 : apply(toList(0..5), i->hilbertFunction(i, Eodd))

o12 = {1, 1, 1, 1, 1, 1}

o12 : List
i13 : use S

o13 = S

o13 : PolynomialRing
i14 : I2 = ideal"x3,yz"

              3
o14 = ideal (x , y*z)

o14 : Ideal of S
i15 : R2 = S/I2

o15 = R2

o15 : QuotientRing
i16 : M2 = R2^1/ideal"x2,y,z"

o16 = cokernel | x2 y z |

                               1
o16 : R2-module, quotient of R2
i17 : betti res (M2, LengthLimit =>10)

             0 1 2 3 4  5  6  7  8  9 10
o17 = total: 1 3 5 7 9 11 13 15 17 19 21
          0: 1 2 2 2 2  2  2  2  2  2  2
          1: . 1 3 4 4  4  4  4  4  4  4
          2: . . . 1 3  4  4  4  4  4  4
          3: . . . . .  1  3  4  4  4  4
          4: . . . . .  .  .  1  3  4  4
          5: . . . . .  .  .  .  .  1  3

o17 : BettiTally
i18 : E = ExtModule M2

                  8
o18 = (kk[X ..X ])
           0   1

o18 : kk[X ..X ]-module, free, degrees {0..1, 2:1, 3:2, 3}
          0   1
i19 : apply(toList(0..10), i->hilbertFunction(i, E))

o19 = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}

o19 : List
i20 : Eeven = evenExtModule M2

                  4
o20 = (kk[X ..X ])
           0   1

o20 : kk[X ..X ]-module, free, degrees {0..1, 2:1}
          0   1
i21 : apply(toList(0..5), i->hilbertFunction(i, Eeven))

o21 = {1, 5, 9, 13, 17, 21}

o21 : List
i22 : Eodd = oddExtModule M2

                  4
o22 = (kk[X ..X ])
           0   1

o22 : kk[X ..X ]-module, free, degrees {3:0, 1}
          0   1
i23 : apply(toList(0..5), i->hilbertFunction(i, Eodd))

o23 = {3, 7, 11, 15, 19, 23}

o23 : List

See also

Ways to use ExtModule :

For the programmer

The object ExtModule is a method function.